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Theorem i1fmullem 22227
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 22209 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
31, 2syl 16 . . . . . . . 8  |-  ( ph  ->  F : RR --> RR )
4 ffn 5737 . . . . . . . 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 22209 . . . . . . . . 9  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
86, 7syl 16 . . . . . . . 8  |-  ( ph  ->  G : RR --> RR )
9 ffn 5737 . . . . . . . 8  |-  ( G : RR --> RR  ->  G  Fn  RR )
108, 9syl 16 . . . . . . 7  |-  ( ph  ->  G  Fn  RR )
11 reex 9600 . . . . . . . 8  |-  RR  e.  _V
1211a1i 11 . . . . . . 7  |-  ( ph  ->  RR  e.  _V )
13 inidm 3703 . . . . . . 7  |-  ( RR 
i^i  RR )  =  RR
145, 10, 12, 12, 13offn 6550 . . . . . 6  |-  ( ph  ->  ( F  oF  x.  G )  Fn  RR )
1514adantr 465 . . . . 5  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( F  oF  x.  G )  Fn  RR )
16 fniniseg 6009 . . . . 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 465 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  ->  F  Fn  RR )
1910adantr 465 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  ->  G  Fn  RR )
2011a1i 11 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  ->  RR  e.  _V )
21 eqidd 2458 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  z  e.  RR )  ->  ( F `  z )  =  ( F `  z ) )
22 eqidd 2458 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  z  e.  RR )  ->  ( G `  z )  =  ( G `  z ) )
2318, 19, 20, 20, 13, 21, 22ofval 6548 . . . . . 6  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  z  e.  RR )  ->  ( ( F  oF  x.  G
) `  z )  =  ( ( F `
 z )  x.  ( G `  z
) ) )
2423eqeq1d 2459 . . . . 5  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  z  e.  RR )  ->  ( ( ( F  oF  x.  G ) `  z
)  =  A  <->  ( ( F `  z )  x.  ( G `  z
) )  =  A ) )
2524pm5.32da 641 . . . 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 725 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  ->  G  Fn  RR )
27 simprl 756 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
z  e.  RR )
28 fnfvelrn 6029 . . . . . . . . 9  |-  ( ( G  Fn  RR  /\  z  e.  RR )  ->  ( G `  z
)  e.  ran  G
)
2926, 27, 28syl2anc 661 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( G `  z
)  e.  ran  G
)
30 eldifsni 4158 . . . . . . . . . . 11  |-  ( A  e.  ( CC  \  { 0 } )  ->  A  =/=  0
)
3130ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  ->  A  =/=  0 )
32 simprr 757 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( ( F `  z )  x.  ( G `  z )
)  =  A )
333ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  ->  F : RR --> RR )
3433, 27ffvelrnd 6033 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( F `  z
)  e.  RR )
3534recnd 9639 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( F `  z
)  e.  CC )
3635mul01d 9796 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( ( F `  z )  x.  0 )  =  0 )
3731, 32, 363netr4d 2762 . . . . . . . . 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 6304 . . . . . . . . . 10  |-  ( ( G `  z )  =  0  ->  (
( F `  z
)  x.  ( G `
 z ) )  =  ( ( F `
 z )  x.  0 ) )
3938necon3i 2697 . . . . . . . . 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 4157 . . . . . . . 8  |-  ( ( G `  z )  e.  ( ran  G  \  { 0 } )  <-> 
( ( G `  z )  e.  ran  G  /\  ( G `  z )  =/=  0
) )
4229, 40, 41sylanbrc 664 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( G `  z
)  e.  ( ran 
G  \  { 0 } ) )
438ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  ->  G : RR --> RR )
4443, 27ffvelrnd 6033 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( G `  z
)  e.  RR )
4544recnd 9639 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( G `  z
)  e.  CC )
4635, 45, 40divcan4d 10347 . . . . . . . . . 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 6311 . . . . . . . . . 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 2500 . . . . . . . . 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 6009 . . . . . . . . . 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 922 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
z  e.  ( `' F " { ( A  /  ( G `
 z ) ) } ) )
53 eqidd 2458 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( G `  z
)  =  ( G `
 z ) )
54 fniniseg 6009 . . . . . . . . . 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 922 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
z  e.  ( `' G " { ( G `  z ) } ) )
57 elin 3683 . . . . . . . 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 664 . . . . . . 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 6304 . . . . . . . . . . . 12  |-  ( y  =  ( G `  z )  ->  ( A  /  y )  =  ( A  /  ( G `  z )
) )
6059sneqd 4044 . . . . . . . . . . 11  |-  ( y  =  ( G `  z )  ->  { ( A  /  y ) }  =  { ( A  /  ( G `
 z ) ) } )
6160imaeq2d 5347 . . . . . . . . . 10  |-  ( y  =  ( G `  z )  ->  ( `' F " { ( A  /  y ) } )  =  ( `' F " { ( A  /  ( G `
 z ) ) } ) )
62 sneq 4042 . . . . . . . . . . 11  |-  ( y  =  ( G `  z )  ->  { y }  =  { ( G `  z ) } )
6362imaeq2d 5347 . . . . . . . . . 10  |-  ( y  =  ( G `  z )  ->  ( `' G " { y } )  =  ( `' G " { ( G `  z ) } ) )
6461, 63ineq12d 3697 . . . . . . . . 9  |-  ( y  =  ( G `  z )  ->  (
( `' F " { ( A  / 
y ) } )  i^i  ( `' G " { y } ) )  =  ( ( `' F " { ( A  /  ( G `
 z ) ) } )  i^i  ( `' G " { ( G `  z ) } ) ) )
6564eleq2d 2527 . . . . . . . 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 3210 . . . . . . 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 661 . . . . . 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 6009 . . . . . . . . . . 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 6009 . . . . . . . . . . 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 710 . . . . . . . . 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 3683 . . . . . . . . 9  |-  ( z  e.  ( ( `' F " { ( A  /  y ) } )  i^i  ( `' G " { y } ) )  <->  ( z  e.  ( `' F " { ( A  / 
y ) } )  /\  z  e.  ( `' G " { y } ) ) )
75 anandi 828 . . . . . . . . 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 465 . . . . . . 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 3622 . . . . . . . . . . . 12  |-  ( A  e.  ( CC  \  { 0 } )  ->  A  e.  CC )
7978ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  A  e.  CC )
808ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  G : RR --> RR )
81 frn 5743 . . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  y  e.  ( ran  G  \  { 0 } ) )
84 eldifsn 4157 . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  y  e.  ran  G )
8782, 86sseldd 3500 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  y  e.  RR )
8887recnd 9639 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  y  e.  CC )
8985simprd 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  y  =/=  0 )
9079, 88, 89divcan1d 10342 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  (
( A  /  y
)  x.  y )  =  A )
91 oveq12 6305 . . . . . . . . . . 11  |-  ( ( ( F `  z
)  =  ( A  /  y )  /\  ( G `  z )  =  y )  -> 
( ( F `  z )  x.  ( G `  z )
)  =  ( ( A  /  y )  x.  y ) )
9291eqeq1d 2459 . . . . . . . . . 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 648 . . . . . . . 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 693 . . . . . . 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 2949 . . . . 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 4337 . . 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 2454 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 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808   _Vcvv 3109    \ cdif 3468    i^i cin 3470    C_ wss 3471   {csn 4032   U_ciun 4332   `'ccnv 5007   dom cdm 5008   ran crn 5009   "cima 5011    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296    oFcof 6537   CCcc 9507   RRcr 9508   0cc0 9509    x. cmul 9514    / cdiv 10227   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-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  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-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-sum 13521  df-itg1 22155
This theorem is referenced by:  i1fmul  22229
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