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 -> _l11 = _l12
_G -> nth i _l11 = nth i _l12
_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		hyp _l1h	_G -> _l11 = _l12
2	1	ntheq2d	_G -> nth i _l11 = nth i _l12
3	2	subeq1d	_G -> nth i _l11 - 1 = nth i _l12 - 1
4		hyp _l2h	_G -> _l21 = _l22
5	4	ntheq2d	_G -> nth i _l21 = nth i _l22
6	5	subeq1d	_G -> nth i _l21 - 1 = nth i _l22 - 1
7	3, 6	preqd	_G -> nth i _l11 - 1, nth i _l21 - 1 = nth i _l12 - 1, nth i _l22 - 1
8	7	lameqd	_G -> \ i, nth i _l11 - 1, nth i _l21 - 1 == \ i, nth i _l12 - 1, nth i _l22 - 1
9	1	leneqd	_G -> len _l11 = len _l12
10	4	leneqd	_G -> len _l21 = len _l22
11	9, 10	mineqd	_G -> min (len _l11) (len _l21) = min (len _l12) (len _l22)
12	8, 11	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))
13	12	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)