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Theorem itg1addlem3 19580
Description: Lemma for itg1add 19583. (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
StepHypRef Expression
1 eqeq1 2441 . . . . 5  |-  ( i  =  A  ->  (
i  =  0  <->  A  =  0 ) )
2 eqeq1 2441 . . . . 5  |-  ( j  =  B  ->  (
j  =  0  <->  B  =  0 ) )
31, 2bi2anan9 844 . . . 4  |-  ( ( i  =  A  /\  j  =  B )  ->  ( ( i  =  0  /\  j  =  0 )  <->  ( A  =  0  /\  B  =  0 ) ) )
4 sneq 3817 . . . . . . 7  |-  ( i  =  A  ->  { i }  =  { A } )
54imaeq2d 5195 . . . . . 6  |-  ( i  =  A  ->  ( `' F " { i } )  =  ( `' F " { A } ) )
6 sneq 3817 . . . . . . 7  |-  ( j  =  B  ->  { j }  =  { B } )
76imaeq2d 5195 . . . . . 6  |-  ( j  =  B  ->  ( `' G " { j } )  =  ( `' G " { B } ) )
85, 7ineqan12d 3536 . . . . 5  |-  ( ( i  =  A  /\  j  =  B )  ->  ( ( `' F " { i } )  i^i  ( `' G " { j } ) )  =  ( ( `' F " { A } )  i^i  ( `' G " { B } ) ) )
98fveq2d 5724 . . . 4  |-  ( ( i  =  A  /\  j  =  B )  ->  ( vol `  (
( `' F " { i } )  i^i  ( `' G " { j } ) ) )  =  ( vol `  ( ( `' F " { A } )  i^i  ( `' G " { B } ) ) ) )
103, 9ifbieq2d 3751 . . 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 9075 . . . 4  |-  0  e.  _V
13 fvex 5734 . . . 4  |-  ( vol `  ( ( `' F " { A } )  i^i  ( `' G " { B } ) ) )  e.  _V
1412, 13ifex 3789 . . 3  |-  if ( ( A  =  0  /\  B  =  0 ) ,  0 ,  ( vol `  (
( `' F " { A } )  i^i  ( `' G " { B } ) ) ) )  e.  _V
1510, 11, 14ovmpt2a 6196 . 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 3738 . 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 } ) ) ) )
1715, 16sylan9eq 2487 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A I B )  =  ( vol `  ( ( `' F " { A } )  i^i  ( `' G " { B } ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3311   ifcif 3731   {csn 3806   `'ccnv 4869   dom cdm 4870   "cima 4873   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   RRcr 8979   0cc0 8980   volcvol 19350   S.1citg1 19497
This theorem is referenced by:  itg1addlem4  19581  itg1addlem5  19582
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-mulcl 9042  ax-i2m1 9048
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078
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