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Theorem i1fmullem 19453
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  o F  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 19435 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
31, 2syl 16 . . . . . . . 8  |-  ( ph  ->  F : RR --> RR )
4 ffn 5531 . . . . . . . 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 19435 . . . . . . . . 9  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
86, 7syl 16 . . . . . . . 8  |-  ( ph  ->  G : RR --> RR )
9 ffn 5531 . . . . . . . 8  |-  ( G : RR --> RR  ->  G  Fn  RR )
108, 9syl 16 . . . . . . 7  |-  ( ph  ->  G  Fn  RR )
11 reex 9014 . . . . . . . 8  |-  RR  e.  _V
1211a1i 11 . . . . . . 7  |-  ( ph  ->  RR  e.  _V )
13 inidm 3493 . . . . . . 7  |-  ( RR 
i^i  RR )  =  RR
145, 10, 12, 12, 13offn 6255 . . . . . 6  |-  ( ph  ->  ( F  o F  x.  G )  Fn  RR )
1514adantr 452 . . . . 5  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( F  o F  x.  G )  Fn  RR )
16 fniniseg 5790 . . . . 5  |-  ( ( F  o F  x.  G )  Fn  RR  ->  ( z  e.  ( `' ( F  o F  x.  G ) " { A } )  <-> 
( z  e.  RR  /\  ( ( F  o F  x.  G ) `  z )  =  A ) ) )
1715, 16syl 16 . . . 4  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( z  e.  ( `' ( F  o F  x.  G ) " { A } )  <-> 
( z  e.  RR  /\  ( ( F  o F  x.  G ) `  z )  =  A ) ) )
185adantr 452 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  ->  F  Fn  RR )
1910adantr 452 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  ->  G  Fn  RR )
2011a1i 11 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  ->  RR  e.  _V )
21 eqidd 2388 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  z  e.  RR )  ->  ( F `  z )  =  ( F `  z ) )
22 eqidd 2388 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  z  e.  RR )  ->  ( G `  z )  =  ( G `  z ) )
2318, 19, 20, 20, 13, 21, 22ofval 6253 . . . . . 6  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  z  e.  RR )  ->  ( ( F  o F  x.  G
) `  z )  =  ( ( F `
 z )  x.  ( G `  z
) ) )
2423eqeq1d 2395 . . . . 5  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  z  e.  RR )  ->  ( ( ( F  o F  x.  G ) `  z
)  =  A  <->  ( ( F `  z )  x.  ( G `  z
) )  =  A ) )
2524pm5.32da 623 . . . 4  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( ( z  e.  RR  /\  ( ( F  o F  x.  G ) `  z
)  =  A )  <-> 
( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z )
)  =  A ) ) )
2610ad2antrr 707 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  ->  G  Fn  RR )
27 simprl 733 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
z  e.  RR )
28 fnfvelrn 5806 . . . . . . . . 9  |-  ( ( G  Fn  RR  /\  z  e.  RR )  ->  ( G `  z
)  e.  ran  G
)
2926, 27, 28syl2anc 643 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( G `  z
)  e.  ran  G
)
30 eldifsni 3871 . . . . . . . . . . 11  |-  ( A  e.  ( CC  \  { 0 } )  ->  A  =/=  0
)
3130ad2antlr 708 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  ->  A  =/=  0 )
32 simprr 734 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( ( F `  z )  x.  ( G `  z )
)  =  A )
333ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  ->  F : RR --> RR )
3433, 27ffvelrnd 5810 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( F `  z
)  e.  RR )
3534recnd 9047 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( F `  z
)  e.  CC )
3635mul01d 9197 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( ( F `  z )  x.  0 )  =  0 )
3731, 32, 363netr4d 2577 . . . . . . . . 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 6028 . . . . . . . . . 10  |-  ( ( G `  z )  =  0  ->  (
( F `  z
)  x.  ( G `
 z ) )  =  ( ( F `
 z )  x.  0 ) )
3938necon3i 2589 . . . . . . . . 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 3870 . . . . . . . 8  |-  ( ( G `  z )  e.  ( ran  G  \  { 0 } )  <-> 
( ( G `  z )  e.  ran  G  /\  ( G `  z )  =/=  0
) )
4229, 40, 41sylanbrc 646 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( G `  z
)  e.  ( ran 
G  \  { 0 } ) )
438ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  ->  G : RR --> RR )
4443, 27ffvelrnd 5810 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( G `  z
)  e.  RR )
4544recnd 9047 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( G `  z
)  e.  CC )
4635, 45, 40divcan4d 9728 . . . . . . . . . 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 6035 . . . . . . . . . 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 2421 . . . . . . . . 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 5790 . . . . . . . . . 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 889 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
z  e.  ( `' F " { ( A  /  ( G `
 z ) ) } ) )
53 eqidd 2388 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( G `  z
)  =  ( G `
 z ) )
54 fniniseg 5790 . . . . . . . . . 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 889 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
z  e.  ( `' G " { ( G `  z ) } ) )
57 elin 3473 . . . . . . . 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 646 . . . . . . 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 6028 . . . . . . . . . . . 12  |-  ( y  =  ( G `  z )  ->  ( A  /  y )  =  ( A  /  ( G `  z )
) )
6059sneqd 3770 . . . . . . . . . . 11  |-  ( y  =  ( G `  z )  ->  { ( A  /  y ) }  =  { ( A  /  ( G `
 z ) ) } )
6160imaeq2d 5143 . . . . . . . . . 10  |-  ( y  =  ( G `  z )  ->  ( `' F " { ( A  /  y ) } )  =  ( `' F " { ( A  /  ( G `
 z ) ) } ) )
62 sneq 3768 . . . . . . . . . . 11  |-  ( y  =  ( G `  z )  ->  { y }  =  { ( G `  z ) } )
6362imaeq2d 5143 . . . . . . . . . 10  |-  ( y  =  ( G `  z )  ->  ( `' G " { y } )  =  ( `' G " { ( G `  z ) } ) )
6461, 63ineq12d 3486 . . . . . . . . 9  |-  ( y  =  ( G `  z )  ->  (
( `' F " { ( A  / 
y ) } )  i^i  ( `' G " { y } ) )  =  ( ( `' F " { ( A  /  ( G `
 z ) ) } )  i^i  ( `' G " { ( G `  z ) } ) ) )
6564eleq2d 2454 . . . . . . . 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 2995 . . . . . . 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 643 . . . . . 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 424 . . . . 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 5790 . . . . . . . . . . 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 5790 . . . . . . . . . . 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 692 . . . . . . . . 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 3473 . . . . . . . . 9  |-  ( z  e.  ( ( `' F " { ( A  /  y ) } )  i^i  ( `' G " { y } ) )  <->  ( z  e.  ( `' F " { ( A  / 
y ) } )  /\  z  e.  ( `' G " { y } ) ) )
75 anandi 802 . . . . . . . . 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 280 . . . . . . . 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 452 . . . . . . 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 3412 . . . . . . . . . . . 12  |-  ( A  e.  ( CC  \  { 0 } )  ->  A  e.  CC )
7978ad2antlr 708 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  A  e.  CC )
808ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  G : RR --> RR )
81 frn 5537 . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  y  e.  ( ran  G  \  { 0 } ) )
84 eldifsn 3870 . . . . . . . . . . . . . . 15  |-  ( y  e.  ( ran  G  \  { 0 } )  <-> 
( y  e.  ran  G  /\  y  =/=  0
) )
8583, 84sylib 189 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  (
y  e.  ran  G  /\  y  =/=  0
) )
8685simpld 446 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  y  e.  ran  G )
8782, 86sseldd 3292 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  y  e.  RR )
8887recnd 9047 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  y  e.  CC )
8985simprd 450 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  y  =/=  0 )
9079, 88, 89divcan1d 9723 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  (
( A  /  y
)  x.  y )  =  A )
91 oveq12 6029 . . . . . . . . . . 11  |-  ( ( ( F `  z
)  =  ( A  /  y )  /\  ( G `  z )  =  y )  -> 
( ( F `  z )  x.  ( G `  z )
)  =  ( ( A  /  y )  x.  y ) )
9291eqeq1d 2395 . . . . . . . . . 10  |-  ( ( ( F `  z
)  =  ( A  /  y )  /\  ( G `  z )  =  y )  -> 
( ( ( F `
 z )  x.  ( G `  z
) )  =  A  <-> 
( ( A  / 
y )  x.  y
)  =  A ) )
9390, 92syl5ibrcom 214 . . . . . . . . 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 630 . . . . . . . 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 675 . . . . . . 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 207 . . . . . 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 2773 . . . . 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 184 . . . 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 271 . . 3  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( z  e.  ( `' ( F  o F  x.  G ) " { A } )  <->  E. y  e.  ( ran  G  \  { 0 } ) z  e.  ( ( `' F " { ( A  / 
y ) } )  i^i  ( `' G " { y } ) ) ) )
100 eliun 4039 . . 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 255 . 2  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( z  e.  ( `' ( F  o F  x.  G ) " { A } )  <-> 
z  e.  U_ y  e.  ( ran  G  \  { 0 } ) ( ( `' F " { ( A  / 
y ) } )  i^i  ( `' G " { y } ) ) ) )
102101eqrdv 2385 1  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( `' ( F  o F  x.  G
) " { A } )  =  U_ y  e.  ( ran  G 
\  { 0 } ) ( ( `' F " { ( A  /  y ) } )  i^i  ( `' G " { y } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   E.wrex 2650   _Vcvv 2899    \ cdif 3260    i^i cin 3262    C_ wss 3263   {csn 3757   U_ciun 4035   `'ccnv 4817   dom cdm 4818   ran crn 4819   "cima 4821    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020    o Fcof 6242   CCcc 8921   RRcr 8922   0cc0 8923    x. cmul 8928    / cdiv 9609   S.1citg1 19374
This theorem is referenced by:  i1fmul  19455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-riota 6485  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-sum 12407  df-itg1 19380
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