Theorem zipeqd ≪ | index | src | ≫

theorem zipeqd (_G: wff) (_l11 _l12 _l21 _l22: nat):
  $ _G -> _l11 = _l12 $ >
  $ _G -> _l21 = _l22 $ >
  $ _G -> zip _l11 _l21 = zip _l12 _l22 $;


_G -> i = i
_G -> _l11 = _l12
_G -> nth i _l11 = nth i _l12
_G -> 1 = 1
_G -> nth i _l11 - 1 = nth i _l12 - 1
_G -> _l21 = _l22
_G -> nth i _l21 = nth i _l22
_G -> nth i _l21 - 1 = nth i _l22 - 1
_G -> nth i _l11 - 1, nth i _l21 - 1 = nth i _l12 - 1, nth i _l22 - 1
_G -> \ i, nth i _l11 - 1, nth i _l21 - 1 == \ i, nth i _l12 - 1, nth i _l22 - 1
_G -> len _l11 = len _l12
_G -> len _l21 = len _l22
_G -> min (len _l11) (len _l21) = min (len _l12) (len _l22)
_G -> lfn (\ i, nth i _l11 - 1, nth i _l21 - 1) (min (len _l11) (len _l21)) = lfn (\ i, nth i _l12 - 1, nth i _l22 - 1) (min (len _l12) (len _l22))
_G -> zip _l11 _l21 = zip _l12 _l22

Step	Hyp	Ref	Expression
1		eqidd	_G -> i = i
2		hyp _l1h	_G -> _l11 = _l12
3	1, 2	ntheqd	_G -> nth i _l11 = nth i _l12
4		eqidd	_G -> 1 = 1
5	3, 4	subeqd	_G -> nth i _l11 - 1 = nth i _l12 - 1
6		hyp _l2h	_G -> _l21 = _l22
7	1, 6	ntheqd	_G -> nth i _l21 = nth i _l22
8	7, 4	subeqd	_G -> nth i _l21 - 1 = nth i _l22 - 1
9	5, 8	preqd	_G -> nth i _l11 - 1, nth i _l21 - 1 = nth i _l12 - 1, nth i _l22 - 1
10	9	lameqd	_G -> \ i, nth i _l11 - 1, nth i _l21 - 1 == \ i, nth i _l12 - 1, nth i _l22 - 1
11	2	leneqd	_G -> len _l11 = len _l12
12	6	leneqd	_G -> len _l21 = len _l22
13	11, 12	mineqd	_G -> min (len _l11) (len _l21) = min (len _l12) (len _l22)
14	10, 13	lfneqd	_G -> lfn (\ i, nth i _l11 - 1, nth i _l21 - 1) (min (len _l11) (len _l21)) = lfn (\ i, nth i _l12 - 1, nth i _l22 - 1) (min (len _l12) (len _l22))
15	14	conv zip	_G -> zip _l11 _l21 = zip _l12 _l22

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12), axs_set (elab, ax_8), axs_the (theid, the0), axs_peano (peano2, addeq, muleq)