Theorem itg1addlem3 22656

Description: Lemma for itg1add 22659. (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 2455	. . . . 5 |- ( i = A -> ( i = 0 <-> A = 0 ) )
2		eqeq1 2455	. . . . 5 |- ( j = B -> ( j = 0 <-> B = 0 ) )
3	1, 2	bi2anan9 884	. . . 4 |- ( ( i = A /\ j = B ) -> ( ( i = 0 /\ j = 0 ) <-> ( A = 0 /\ B = 0 ) ) )
4		sneq 3978	. . . . . . 7 |- ( i = A -> { i } = { A } )
5	4	imaeq2d 5168	. . . . . 6 |- ( i = A -> ( `' F " { i } ) = ( `' F " { A } ) )
6		sneq 3978	. . . . . . 7 |- ( j = B -> { j } = { B } )
7	6	imaeq2d 5168	. . . . . 6 |- ( j = B -> ( `' G " { j } ) = ( `' G " { B } ) )
8	5, 7	ineqan12d 3636	. . . . 5 |- ( ( i = A /\ j = B ) -> ( ( `' F " { i } ) i^i ( `' G " { j } ) ) = ( ( `' F " { A } ) i^i ( `' G " { B } ) ) )
9	8	fveq2d 5869	. . . 4 |- ( ( i = A /\ j = B ) -> ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) = ( vol ` ( ( `' F " { A } ) i^i ( `' G " { B } ) ) ) )
10	3, 9	ifbieq2d 3906	. . 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 9637	. . . 4 |- 0 e. _V
13		fvex 5875	. . . 4 |- ( vol ` ( ( `' F " { A } ) i^i ( `' G " { B } ) ) ) e. _V
14	12, 13	ifex 3949	. . 3 |- if ( ( A = 0 /\ B = 0 ) , 0 , ( vol ` ( ( `' F " { A } ) i^i ( `' G " { B } ) ) ) ) e. _V
15	10, 11, 14	ovmpt2a 6427	. 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 3890	. 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 2505	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 371   = wceq 1444   e. wcel 1887   i^i cin 3403   ifcif 3881   {csn 3968   `'ccnv 4833   dom cdm 4834   "cima 4837   ` cfv 5582  (class class class)co 6290   |-> cmpt2 6292   RRcr 9538   0cc0 9539   volcvol 22415   S.1citg1 22573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-mulcl 9601  ax-i2m1 9607

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295

This theorem is referenced by:  itg1addlem4  22657  itg1addlem5  22658