Theorem itg1addlem3 23465

Description: Lemma for itg1add 23468. (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 2626	. . . . 5 |- ( i = A -> ( i = 0 <-> A = 0 ) )
2		eqeq1 2626	. . . . 5 |- ( j = B -> ( j = 0 <-> B = 0 ) )
3	1, 2	bi2anan9 917	. . . 4 |- ( ( i = A /\ j = B ) -> ( ( i = 0 /\ j = 0 ) <-> ( A = 0 /\ B = 0 ) ) )
4		sneq 4187	. . . . . . 7 |- ( i = A -> { i } = { A } )
5	4	imaeq2d 5466	. . . . . 6 |- ( i = A -> ( `' F " { i } ) = ( `' F " { A } ) )
6		sneq 4187	. . . . . . 7 |- ( j = B -> { j } = { B } )
7	6	imaeq2d 5466	. . . . . 6 |- ( j = B -> ( `' G " { j } ) = ( `' G " { B } ) )
8	5, 7	ineqan12d 3816	. . . . 5 |- ( ( i = A /\ j = B ) -> ( ( `' F " { i } ) i^i ( `' G " { j } ) ) = ( ( `' F " { A } ) i^i ( `' G " { B } ) ) )
9	8	fveq2d 6195	. . . 4 |- ( ( i = A /\ j = B ) -> ( vol ` ( ( `' F " { i } ) i^i ( `' G " { j } ) ) ) = ( vol ` ( ( `' F " { A } ) i^i ( `' G " { B } ) ) ) )
10	3, 9	ifbieq2d 4111	. . 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 10034	. . . 4 |- 0 e. _V
13		fvex 6201	. . . 4 |- ( vol ` ( ( `' F " { A } ) i^i ( `' G " { B } ) ) ) e. _V
14	12, 13	ifex 4156	. . 3 |- if ( ( A = 0 /\ B = 0 ) , 0 , ( vol ` ( ( `' F " { A } ) i^i ( `' G " { B } ) ) ) ) e. _V
15	10, 11, 14	ovmpt2a 6791	. 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 4095	. 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 2676	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 384   = wceq 1483   e. wcel 1990   i^i cin 3573   ifcif 4086   {csn 4177   `'ccnv 5113   dom cdm 5114   "cima 5117   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   RRcr 9935   0cc0 9936   volcvol 23232   S.1citg1 23384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-mulcl 9998  ax-i2m1 10004

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  itg1addlem4  23466  itg1addlem5  23467