Theorem srecpauxeqd ≪ | index | src | ≫

theorem srecpauxeqd (_G: wff) (_A1 _A2: set) (_n1 _n2: nat):
  $ _G -> _A1 == _A2 $ >
  $ _G -> _n1 = _n2 $ >
  $ _G -> srecpaux _A1 _n1 = srecpaux _A2 _n2 $;


_G -> size (Dom f), lower {x | true (f @ x)} = size (Dom f), lower {x | true (f @ x)}
_G -> _A1 == _A2
_G -> (size (Dom f), lower {x | true (f @ x)} e. _A1 <-> size (Dom f), lower {x | true (f @ x)} e. _A2)
_G -> nat (size (Dom f), lower {x | true (f @ x)} e. _A1) = nat (size (Dom f), lower {x | true (f @ x)} e. _A2)
_G -> \ f, nat (size (Dom f), lower {x | true (f @ x)} e. _A1) == \ f, nat (size (Dom f), lower {x | true (f @ x)} e. _A2)
_G -> _n1 = _n2
_G -> srec (\ f, nat (size (Dom f), lower {x | true (f @ x)} e. _A1)) _n1 = srec (\ f, nat (size (Dom f), lower {x | true (f @ x)} e. _A2)) _n2
_G -> srecpaux _A1 _n1 = srecpaux _A2 _n2

Step	Hyp	Ref	Expression
1		eqidd	_G -> size (Dom f), lower {x \| true (f @ x)} = size (Dom f), lower {x \| true (f @ x)}
2		hyp _Ah	_G -> _A1 == _A2
3	1, 2	eleqd	_G -> (size (Dom f), lower {x \| true (f @ x)} e. _A1 <-> size (Dom f), lower {x \| true (f @ x)} e. _A2)
4	3	nateqd	_G -> nat (size (Dom f), lower {x \| true (f @ x)} e. _A1) = nat (size (Dom f), lower {x \| true (f @ x)} e. _A2)
5	4	lameqd	_G -> \ f, nat (size (Dom f), lower {x \| true (f @ x)} e. _A1) == \ f, nat (size (Dom f), lower {x \| true (f @ x)} e. _A2)
6		hyp _nh	_G -> _n1 = _n2
7	5, 6	sreceqd	_G -> srec (\ f, nat (size (Dom f), lower {x \| true (f @ x)} e. _A1)) _n1 = srec (\ f, nat (size (Dom f), lower {x \| true (f @ x)} e. _A2)) _n2
8	7	conv srecpaux	_G -> srecpaux _A1 _n1 = srecpaux _A2 _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)