Theorem takeeqd ≪ | index | src | ≫

theorem takeeqd (_G: wff) (_l1 _l2 _n1 _n2: nat):
  $ _G -> _l1 = _l2 $ >
  $ _G -> _n1 = _n2 $ >
  $ _G -> take _l1 _n1 = take _l2 _n2 $;


_G -> i = i
_G -> _l1 = _l2
_G -> nth i _l1 = nth i _l2
_G -> 1 = 1
_G -> nth i _l1 - 1 = nth i _l2 - 1
_G -> \ i, nth i _l1 - 1 == \ i, nth i _l2 - 1
_G -> len _l1 = len _l2
_G -> _n1 = _n2
_G -> min (len _l1) _n1 = min (len _l2) _n2
_G -> lfn (\ i, nth i _l1 - 1) (min (len _l1) _n1) = lfn (\ i, nth i _l2 - 1) (min (len _l2) _n2)
_G -> take _l1 _n1 = take _l2 _n2

Step	Hyp	Ref	Expression
1		eqidd	_G -> i = i
2		hyp _lh	_G -> _l1 = _l2
3	1, 2	ntheqd	_G -> nth i _l1 = nth i _l2
4		eqidd	_G -> 1 = 1
5	3, 4	subeqd	_G -> nth i _l1 - 1 = nth i _l2 - 1
6	5	lameqd	_G -> \ i, nth i _l1 - 1 == \ i, nth i _l2 - 1
7	2	leneqd	_G -> len _l1 = len _l2
8		hyp _nh	_G -> _n1 = _n2
9	7, 8	mineqd	_G -> min (len _l1) _n1 = min (len _l2) _n2
10	6, 9	lfneqd	_G -> lfn (\ i, nth i _l1 - 1) (min (len _l1) _n1) = lfn (\ i, nth i _l2 - 1) (min (len _l2) _n2)
11	10	conv take	_G -> take _l1 _n1 = take _l2 _n2

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)