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Theorem i1fmullem 21131
Description: Decompose the preimage of a product. (Contributed by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
i1fadd.1  |-  ( ph  ->  F  e.  dom  S.1 )
i1fadd.2  |-  ( ph  ->  G  e.  dom  S.1 )
Assertion
Ref Expression
i1fmullem  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( `' ( F  oF  x.  G
) " { A } )  =  U_ y  e.  ( ran  G 
\  { 0 } ) ( ( `' F " { ( A  /  y ) } )  i^i  ( `' G " { y } ) ) )
Distinct variable groups:    y, A    y, F    y, G    ph, y

Proof of Theorem i1fmullem
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 i1fadd.1 . . . . . . . . 9  |-  ( ph  ->  F  e.  dom  S.1 )
2 i1ff 21113 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
31, 2syl 16 . . . . . . . 8  |-  ( ph  ->  F : RR --> RR )
4 ffn 5556 . . . . . . . 8  |-  ( F : RR --> RR  ->  F  Fn  RR )
53, 4syl 16 . . . . . . 7  |-  ( ph  ->  F  Fn  RR )
6 i1fadd.2 . . . . . . . . 9  |-  ( ph  ->  G  e.  dom  S.1 )
7 i1ff 21113 . . . . . . . . 9  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
86, 7syl 16 . . . . . . . 8  |-  ( ph  ->  G : RR --> RR )
9 ffn 5556 . . . . . . . 8  |-  ( G : RR --> RR  ->  G  Fn  RR )
108, 9syl 16 . . . . . . 7  |-  ( ph  ->  G  Fn  RR )
11 reex 9369 . . . . . . . 8  |-  RR  e.  _V
1211a1i 11 . . . . . . 7  |-  ( ph  ->  RR  e.  _V )
13 inidm 3556 . . . . . . 7  |-  ( RR 
i^i  RR )  =  RR
145, 10, 12, 12, 13offn 6330 . . . . . 6  |-  ( ph  ->  ( F  oF  x.  G )  Fn  RR )
1514adantr 462 . . . . 5  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( F  oF  x.  G )  Fn  RR )
16 fniniseg 5821 . . . . 5  |-  ( ( F  oF  x.  G )  Fn  RR  ->  ( z  e.  ( `' ( F  oF  x.  G ) " { A } )  <-> 
( z  e.  RR  /\  ( ( F  oF  x.  G ) `  z )  =  A ) ) )
1715, 16syl 16 . . . 4  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( z  e.  ( `' ( F  oF  x.  G ) " { A } )  <-> 
( z  e.  RR  /\  ( ( F  oF  x.  G ) `  z )  =  A ) ) )
185adantr 462 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  ->  F  Fn  RR )
1910adantr 462 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  ->  G  Fn  RR )
2011a1i 11 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  ->  RR  e.  _V )
21 eqidd 2442 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  z  e.  RR )  ->  ( F `  z )  =  ( F `  z ) )
22 eqidd 2442 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  z  e.  RR )  ->  ( G `  z )  =  ( G `  z ) )
2318, 19, 20, 20, 13, 21, 22ofval 6328 . . . . . 6  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  z  e.  RR )  ->  ( ( F  oF  x.  G
) `  z )  =  ( ( F `
 z )  x.  ( G `  z
) ) )
2423eqeq1d 2449 . . . . 5  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  z  e.  RR )  ->  ( ( ( F  oF  x.  G ) `  z
)  =  A  <->  ( ( F `  z )  x.  ( G `  z
) )  =  A ) )
2524pm5.32da 636 . . . 4  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( ( z  e.  RR  /\  ( ( F  oF  x.  G ) `  z
)  =  A )  <-> 
( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z )
)  =  A ) ) )
2610ad2antrr 720 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  ->  G  Fn  RR )
27 simprl 750 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
z  e.  RR )
28 fnfvelrn 5837 . . . . . . . . 9  |-  ( ( G  Fn  RR  /\  z  e.  RR )  ->  ( G `  z
)  e.  ran  G
)
2926, 27, 28syl2anc 656 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( G `  z
)  e.  ran  G
)
30 eldifsni 3998 . . . . . . . . . . 11  |-  ( A  e.  ( CC  \  { 0 } )  ->  A  =/=  0
)
3130ad2antlr 721 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  ->  A  =/=  0 )
32 simprr 751 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( ( F `  z )  x.  ( G `  z )
)  =  A )
333ad2antrr 720 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  ->  F : RR --> RR )
3433, 27ffvelrnd 5841 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( F `  z
)  e.  RR )
3534recnd 9408 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( F `  z
)  e.  CC )
3635mul01d 9564 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( ( F `  z )  x.  0 )  =  0 )
3731, 32, 363netr4d 2633 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( ( F `  z )  x.  ( G `  z )
)  =/=  ( ( F `  z )  x.  0 ) )
38 oveq2 6098 . . . . . . . . . 10  |-  ( ( G `  z )  =  0  ->  (
( F `  z
)  x.  ( G `
 z ) )  =  ( ( F `
 z )  x.  0 ) )
3938necon3i 2648 . . . . . . . . 9  |-  ( ( ( F `  z
)  x.  ( G `
 z ) )  =/=  ( ( F `
 z )  x.  0 )  ->  ( G `  z )  =/=  0 )
4037, 39syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( G `  z
)  =/=  0 )
41 eldifsn 3997 . . . . . . . 8  |-  ( ( G `  z )  e.  ( ran  G  \  { 0 } )  <-> 
( ( G `  z )  e.  ran  G  /\  ( G `  z )  =/=  0
) )
4229, 40, 41sylanbrc 659 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( G `  z
)  e.  ( ran 
G  \  { 0 } ) )
438ad2antrr 720 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  ->  G : RR --> RR )
4443, 27ffvelrnd 5841 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( G `  z
)  e.  RR )
4544recnd 9408 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( G `  z
)  e.  CC )
4635, 45, 40divcan4d 10109 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( ( ( F `
 z )  x.  ( G `  z
) )  /  ( G `  z )
)  =  ( F `
 z ) )
4732oveq1d 6105 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( ( ( F `
 z )  x.  ( G `  z
) )  /  ( G `  z )
)  =  ( A  /  ( G `  z ) ) )
4846, 47eqtr3d 2475 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( F `  z
)  =  ( A  /  ( G `  z ) ) )
4933, 4syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  ->  F  Fn  RR )
50 fniniseg 5821 . . . . . . . . . 10  |-  ( F  Fn  RR  ->  (
z  e.  ( `' F " { ( A  /  ( G `
 z ) ) } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  ( A  /  ( G `  z )
) ) ) )
5149, 50syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( z  e.  ( `' F " { ( A  /  ( G `
 z ) ) } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  ( A  /  ( G `  z )
) ) ) )
5227, 48, 51mpbir2and 908 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
z  e.  ( `' F " { ( A  /  ( G `
 z ) ) } ) )
53 eqidd 2442 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( G `  z
)  =  ( G `
 z ) )
54 fniniseg 5821 . . . . . . . . . 10  |-  ( G  Fn  RR  ->  (
z  e.  ( `' G " { ( G `  z ) } )  <->  ( z  e.  RR  /\  ( G `
 z )  =  ( G `  z
) ) ) )
5526, 54syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( z  e.  ( `' G " { ( G `  z ) } )  <->  ( z  e.  RR  /\  ( G `
 z )  =  ( G `  z
) ) ) )
5627, 53, 55mpbir2and 908 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
z  e.  ( `' G " { ( G `  z ) } ) )
57 elin 3536 . . . . . . . 8  |-  ( z  e.  ( ( `' F " { ( A  /  ( G `
 z ) ) } )  i^i  ( `' G " { ( G `  z ) } ) )  <->  ( z  e.  ( `' F " { ( A  / 
( G `  z
) ) } )  /\  z  e.  ( `' G " { ( G `  z ) } ) ) )
5852, 56, 57sylanbrc 659 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
z  e.  ( ( `' F " { ( A  /  ( G `
 z ) ) } )  i^i  ( `' G " { ( G `  z ) } ) ) )
59 oveq2 6098 . . . . . . . . . . . 12  |-  ( y  =  ( G `  z )  ->  ( A  /  y )  =  ( A  /  ( G `  z )
) )
6059sneqd 3886 . . . . . . . . . . 11  |-  ( y  =  ( G `  z )  ->  { ( A  /  y ) }  =  { ( A  /  ( G `
 z ) ) } )
6160imaeq2d 5166 . . . . . . . . . 10  |-  ( y  =  ( G `  z )  ->  ( `' F " { ( A  /  y ) } )  =  ( `' F " { ( A  /  ( G `
 z ) ) } ) )
62 sneq 3884 . . . . . . . . . . 11  |-  ( y  =  ( G `  z )  ->  { y }  =  { ( G `  z ) } )
6362imaeq2d 5166 . . . . . . . . . 10  |-  ( y  =  ( G `  z )  ->  ( `' G " { y } )  =  ( `' G " { ( G `  z ) } ) )
6461, 63ineq12d 3550 . . . . . . . . 9  |-  ( y  =  ( G `  z )  ->  (
( `' F " { ( A  / 
y ) } )  i^i  ( `' G " { y } ) )  =  ( ( `' F " { ( A  /  ( G `
 z ) ) } )  i^i  ( `' G " { ( G `  z ) } ) ) )
6564eleq2d 2508 . . . . . . . 8  |-  ( y  =  ( G `  z )  ->  (
z  e.  ( ( `' F " { ( A  /  y ) } )  i^i  ( `' G " { y } ) )  <->  z  e.  ( ( `' F " { ( A  / 
( G `  z
) ) } )  i^i  ( `' G " { ( G `  z ) } ) ) ) )
6665rspcev 3070 . . . . . . 7  |-  ( ( ( G `  z
)  e.  ( ran 
G  \  { 0 } )  /\  z  e.  ( ( `' F " { ( A  / 
( G `  z
) ) } )  i^i  ( `' G " { ( G `  z ) } ) ) )  ->  E. y  e.  ( ran  G  \  { 0 } ) z  e.  ( ( `' F " { ( A  /  y ) } )  i^i  ( `' G " { y } ) ) )
6742, 58, 66syl2anc 656 . . . . . 6  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  ->  E. y  e.  ( ran  G  \  { 0 } ) z  e.  ( ( `' F " { ( A  / 
y ) } )  i^i  ( `' G " { y } ) ) )
6867ex 434 . . . . 5  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A )  ->  E. y  e.  ( ran  G  \  { 0 } ) z  e.  ( ( `' F " { ( A  /  y ) } )  i^i  ( `' G " { y } ) ) ) )
69 fniniseg 5821 . . . . . . . . . . 11  |-  ( F  Fn  RR  ->  (
z  e.  ( `' F " { ( A  /  y ) } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  ( A  /  y
) ) ) )
7018, 69syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( z  e.  ( `' F " { ( A  /  y ) } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  ( A  /  y
) ) ) )
71 fniniseg 5821 . . . . . . . . . . 11  |-  ( G  Fn  RR  ->  (
z  e.  ( `' G " { y } )  <->  ( z  e.  RR  /\  ( G `
 z )  =  y ) ) )
7219, 71syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( z  e.  ( `' G " { y } )  <->  ( z  e.  RR  /\  ( G `
 z )  =  y ) ) )
7370, 72anbi12d 705 . . . . . . . . 9  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( ( z  e.  ( `' F " { ( A  / 
y ) } )  /\  z  e.  ( `' G " { y } ) )  <->  ( (
z  e.  RR  /\  ( F `  z )  =  ( A  / 
y ) )  /\  ( z  e.  RR  /\  ( G `  z
)  =  y ) ) ) )
74 elin 3536 . . . . . . . . 9  |-  ( z  e.  ( ( `' F " { ( A  /  y ) } )  i^i  ( `' G " { y } ) )  <->  ( z  e.  ( `' F " { ( A  / 
y ) } )  /\  z  e.  ( `' G " { y } ) ) )
75 anandi 819 . . . . . . . . 9  |-  ( ( z  e.  RR  /\  ( ( F `  z )  =  ( A  /  y )  /\  ( G `  z )  =  y ) )  <->  ( (
z  e.  RR  /\  ( F `  z )  =  ( A  / 
y ) )  /\  ( z  e.  RR  /\  ( G `  z
)  =  y ) ) )
7673, 74, 753bitr4g 288 . . . . . . . 8  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( z  e.  ( ( `' F " { ( A  / 
y ) } )  i^i  ( `' G " { y } ) )  <->  ( z  e.  RR  /\  ( ( F `  z )  =  ( A  / 
y )  /\  ( G `  z )  =  y ) ) ) )
7776adantr 462 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  y  e.  ( ran  G  \  {
0 } ) )  ->  ( z  e.  ( ( `' F " { ( A  / 
y ) } )  i^i  ( `' G " { y } ) )  <->  ( z  e.  RR  /\  ( ( F `  z )  =  ( A  / 
y )  /\  ( G `  z )  =  y ) ) ) )
78 eldifi 3475 . . . . . . . . . . . 12  |-  ( A  e.  ( CC  \  { 0 } )  ->  A  e.  CC )
7978ad2antlr 721 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  A  e.  CC )
808ad2antrr 720 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  G : RR --> RR )
81 frn 5562 . . . . . . . . . . . . . 14  |-  ( G : RR --> RR  ->  ran 
G  C_  RR )
8280, 81syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  ran  G 
C_  RR )
83 simprl 750 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  y  e.  ( ran  G  \  { 0 } ) )
84 eldifsn 3997 . . . . . . . . . . . . . . 15  |-  ( y  e.  ( ran  G  \  { 0 } )  <-> 
( y  e.  ran  G  /\  y  =/=  0
) )
8583, 84sylib 196 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  (
y  e.  ran  G  /\  y  =/=  0
) )
8685simpld 456 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  y  e.  ran  G )
8782, 86sseldd 3354 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  y  e.  RR )
8887recnd 9408 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  y  e.  CC )
8985simprd 460 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  y  =/=  0 )
9079, 88, 89divcan1d 10104 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  (
( A  /  y
)  x.  y )  =  A )
91 oveq12 6099 . . . . . . . . . . 11  |-  ( ( ( F `  z
)  =  ( A  /  y )  /\  ( G `  z )  =  y )  -> 
( ( F `  z )  x.  ( G `  z )
)  =  ( ( A  /  y )  x.  y ) )
9291eqeq1d 2449 . . . . . . . . . 10  |-  ( ( ( F `  z
)  =  ( A  /  y )  /\  ( G `  z )  =  y )  -> 
( ( ( F `
 z )  x.  ( G `  z
) )  =  A  <-> 
( ( A  / 
y )  x.  y
)  =  A ) )
9390, 92syl5ibrcom 222 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  (
( ( F `  z )  =  ( A  /  y )  /\  ( G `  z )  =  y )  ->  ( ( F `  z )  x.  ( G `  z
) )  =  A ) )
9493anassrs 643 . . . . . . . 8  |-  ( ( ( ( ph  /\  A  e.  ( CC  \  { 0 } ) )  /\  y  e.  ( ran  G  \  { 0 } ) )  /\  z  e.  RR )  ->  (
( ( F `  z )  =  ( A  /  y )  /\  ( G `  z )  =  y )  ->  ( ( F `  z )  x.  ( G `  z
) )  =  A ) )
9594imdistanda 688 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  y  e.  ( ran  G  \  {
0 } ) )  ->  ( ( z  e.  RR  /\  (
( F `  z
)  =  ( A  /  y )  /\  ( G `  z )  =  y ) )  ->  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) ) )
9677, 95sylbid 215 . . . . . 6  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  y  e.  ( ran  G  \  {
0 } ) )  ->  ( z  e.  ( ( `' F " { ( A  / 
y ) } )  i^i  ( `' G " { y } ) )  ->  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) ) )
9796rexlimdva 2839 . . . . 5  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( E. y  e.  ( ran  G  \  { 0 } ) z  e.  ( ( `' F " { ( A  /  y ) } )  i^i  ( `' G " { y } ) )  -> 
( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z )
)  =  A ) ) )
9868, 97impbid 191 . . . 4  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A )  <->  E. y  e.  ( ran  G  \  { 0 } ) z  e.  ( ( `' F " { ( A  /  y ) } )  i^i  ( `' G " { y } ) ) ) )
9917, 25, 983bitrd 279 . . 3  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( z  e.  ( `' ( F  oF  x.  G ) " { A } )  <->  E. y  e.  ( ran  G  \  { 0 } ) z  e.  ( ( `' F " { ( A  / 
y ) } )  i^i  ( `' G " { y } ) ) ) )
100 eliun 4172 . . 3  |-  ( z  e.  U_ y  e.  ( ran  G  \  { 0 } ) ( ( `' F " { ( A  / 
y ) } )  i^i  ( `' G " { y } ) )  <->  E. y  e.  ( ran  G  \  {
0 } ) z  e.  ( ( `' F " { ( A  /  y ) } )  i^i  ( `' G " { y } ) ) )
10199, 100syl6bbr 263 . 2  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( z  e.  ( `' ( F  oF  x.  G ) " { A } )  <-> 
z  e.  U_ y  e.  ( ran  G  \  { 0 } ) ( ( `' F " { ( A  / 
y ) } )  i^i  ( `' G " { y } ) ) ) )
102101eqrdv 2439 1  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( `' ( F  oF  x.  G
) " { A } )  =  U_ y  e.  ( ran  G 
\  { 0 } ) ( ( `' F " { ( A  /  y ) } )  i^i  ( `' G " { y } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761    =/= wne 2604   E.wrex 2714   _Vcvv 2970    \ cdif 3322    i^i cin 3324    C_ wss 3325   {csn 3874   U_ciun 4168   `'ccnv 4835   dom cdm 4836   ran crn 4837   "cima 4839    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090    oFcof 6317   CCcc 9276   RRcr 9277   0cc0 9278    x. cmul 9283    / cdiv 9989   S.1citg1 21054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-sum 13160  df-itg1 21059
This theorem is referenced by:  i1fmul  21133
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