Theorem grecaux1eqd ≪ | index | src | ≫

theorem grecaux1eqd (_G: wff) (_K1 _K2: set) (_x1 _x2 _z1 _z2 _n1 _n2: nat):
  $ _G -> _K1 == _K2 $ >
  $ _G -> _x1 = _x2 $ >
  $ _G -> _z1 = _z2 $ >
  $ _G -> _n1 = _n2 $ >
  $ _G -> grecaux1 _K1 _x1 _z1 _n1 = grecaux1 _K2 _x2 _z2 _n2 $;


_G -> _z1 = _z2
_G -> _K1 == _K2
_G -> _x1 = _x2
_G -> suc u = suc u
_G -> _x1 - suc u = _x2 - suc u
_G -> i = i
_G -> _x1 - suc u, i = _x2 - suc u, i
_G -> _K1 @ (_x1 - suc u, i) = _K2 @ (_x2 - suc u, i)
_G -> \ i, _K1 @ (_x1 - suc u, i) == \ i, _K2 @ (_x2 - suc u, i)
_G -> (\\ u, \ i, _K1 @ (_x1 - suc u, i)) == (\\ u, \ i, _K2 @ (_x2 - suc u, i))
_G -> _n1 = _n2
_G -> recn _z1 (\\ u, \ i, _K1 @ (_x1 - suc u, i)) _n1 = recn _z2 (\\ u, \ i, _K2 @ (_x2 - suc u, i)) _n2
_G -> grecaux1 _K1 _x1 _z1 _n1 = grecaux1 _K2 _x2 _z2 _n2

Step	Hyp	Ref	Expression
1		hyp _zh	_G -> _z1 = _z2
2		hyp _Kh	_G -> _K1 == _K2
3		hyp _xh	_G -> _x1 = _x2
4		eqidd	_G -> suc u = suc u
5	3, 4	subeqd	_G -> _x1 - suc u = _x2 - suc u
6		eqidd	_G -> i = i
7	5, 6	preqd	_G -> _x1 - suc u, i = _x2 - suc u, i
8	2, 7	appeqd	_G -> _K1 @ (_x1 - suc u, i) = _K2 @ (_x2 - suc u, i)
9	8	lameqd	_G -> \ i, _K1 @ (_x1 - suc u, i) == \ i, _K2 @ (_x2 - suc u, i)
10	9	slameqd	_G -> (\\ u, \ i, _K1 @ (_x1 - suc u, i)) == (\\ u, \ i, _K2 @ (_x2 - suc u, i))
11		hyp _nh	_G -> _n1 = _n2
12	1, 10, 11	recneqd	_G -> recn _z1 (\\ u, \ i, _K1 @ (_x1 - suc u, i)) _n1 = recn _z2 (\\ u, \ i, _K2 @ (_x2 - suc u, i)) _n2
13	12	conv grecaux1	_G -> grecaux1 _K1 _x1 _z1 _n1 = grecaux1 _K2 _x2 _z2 _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)