Theorem itg1addlem3 22231

Description: Lemma for itg1add 22234. (Contributed by Mario Carneiro, 26-Jun-2014.)

Hypotheses

Ref	Expression
i1fadd.1	|- ( ph -> F e. dom S.1 )
i1fadd.2	|- ( ph -> G e. dom S.1 )
itg1add.3	|- I = ( i e. RR , j e. RR |-> if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) )

Assertion

Ref	Expression
itg1addlem3	|- ( ( ( A e. RR /\ B e. RR ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A I B ) = ( vol ` ( ( `' F " { A } ) i^i ( `' G " { B } ) ) ) )

Distinct variable groups:   i, j, A   B, i, j   i, F, j   i, G, j   ph, i, j

Allowed substitution hints:   I( i, j)

Proof of Theorem itg1addlem3

Step	Hyp	Ref	Expression
1		eqeq1 2461	. . . . 5 |- ( i = A -> ( i = 0 <-> A = 0 ) )
2		eqeq1 2461	. . . . 5 |- ( j = B -> ( j = 0 <-> B = 0 ) )
3	1, 2	bi2anan9 873	. . . 4 |- ( ( i = A /\ j = B ) -> ( ( i = 0 /\ j = 0 ) <-> ( A = 0 /\ B = 0 ) ) )
4		sneq 4042	. . . . . . 7 |- ( i = A -> { i } = { A } )
5	4	imaeq2d 5347	. . . . . 6 |- ( i = A -> ( `' F " { i } ) = ( `' F " { A } ) )
6		sneq 4042	. . . . . . 7 |- ( j = B -> { j } = { B } )
7	6	imaeq2d 5347	. . . . . 6 |- ( j = B -> ( `' G " { j } ) = ( `' G " { B } ) )
8	5, 7	ineqan12d 3698	. . . . 5 |- ( ( i = A /\ j = B ) -> ( ( `' F " { i } ) i^i ( `' G " { j } ) ) = ( ( `' F " { A } ) i^i ( `' G " { B } ) ) )
9	8	fveq2d 5876	. . . 4 |- ( ( i = A /\ j = B ) -> ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) = ( vol ` ( ( `' F " { A } ) i^i ( `' G " { B } ) ) ) )
10	3, 9	ifbieq2d 3969	. . 3 |- ( ( i = A /\ j = B ) -> if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) = if ( ( A = 0 /\ B = 0 ) , 0 , ( vol ` ( ( `' F " { A } ) i^i ( `' G " { B } ) ) ) ) )
11		itg1add.3	. . 3 |- I = ( i e. RR , j e. RR |-> if ( ( i = 0 /\ j = 0 ) , 0 , ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) ) )
12		c0ex 9607	. . . 4 |- 0 e. _V
13		fvex 5882	. . . 4 |- ( vol ` ( ( `' F " { A } ) i^i ( `' G " { B } ) ) ) e. _V
14	12, 13	ifex 4013	. . 3 |- if ( ( A = 0 /\ B = 0 ) , 0 , ( vol ` ( ( `' F " { A } ) i^i ( `' G " { B } ) ) ) ) e. _V
15	10, 11, 14	ovmpt2a 6432	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A I B ) = if ( ( A = 0 /\ B = 0 ) , 0 , ( vol ` ( ( `' F " { A } ) i^i ( `' G " { B } ) ) ) ) )
16		iffalse 3953	. 2 |- ( -. ( A = 0 /\ B = 0 ) -> if ( ( A = 0 /\ B = 0 ) , 0 , ( vol ` ( ( `' F " { A } ) i^i ( `' G " { B } ) ) ) ) = ( vol ` ( ( `' F " { A } ) i^i ( `' G " { B } ) ) ) )
17	15, 16	sylan9eq 2518	1 |- ( ( ( A e. RR /\ B e. RR ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A I B ) = ( vol ` ( ( `' F " { A } ) i^i ( `' G " { B } ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1395   e. wcel 1819   i^i cin 3470   ifcif 3944   {csn 4032   `'ccnv 5007   dom cdm 5008   "cima 5011   ` cfv 5594  (class class class)co 6296   |-> cmpt2 6298   RRcr 9508   0cc0 9509   volcvol 22001   S.1citg1 22150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-mulcl 9571  ax-i2m1 9577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301

This theorem is referenced by:  itg1addlem4  22232  itg1addlem5  22233