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Theorem i1fmullem 22652
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 22634 . . . . . . . . 9  |-  ( F  e.  dom  S.1  ->  F : RR --> RR )
31, 2syl 17 . . . . . . . 8  |-  ( ph  ->  F : RR --> RR )
4 ffn 5728 . . . . . . . 8  |-  ( F : RR --> RR  ->  F  Fn  RR )
53, 4syl 17 . . . . . . 7  |-  ( ph  ->  F  Fn  RR )
6 i1fadd.2 . . . . . . . . 9  |-  ( ph  ->  G  e.  dom  S.1 )
7 i1ff 22634 . . . . . . . . 9  |-  ( G  e.  dom  S.1  ->  G : RR --> RR )
86, 7syl 17 . . . . . . . 8  |-  ( ph  ->  G : RR --> RR )
9 ffn 5728 . . . . . . . 8  |-  ( G : RR --> RR  ->  G  Fn  RR )
108, 9syl 17 . . . . . . 7  |-  ( ph  ->  G  Fn  RR )
11 reex 9630 . . . . . . . 8  |-  RR  e.  _V
1211a1i 11 . . . . . . 7  |-  ( ph  ->  RR  e.  _V )
13 inidm 3641 . . . . . . 7  |-  ( RR 
i^i  RR )  =  RR
145, 10, 12, 12, 13offn 6542 . . . . . 6  |-  ( ph  ->  ( F  oF  x.  G )  Fn  RR )
1514adantr 467 . . . . 5  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( F  oF  x.  G )  Fn  RR )
16 fniniseg 6003 . . . . 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 17 . . . 4  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( z  e.  ( `' ( F  oF  x.  G ) " { A } )  <-> 
( z  e.  RR  /\  ( ( F  oF  x.  G ) `  z )  =  A ) ) )
185adantr 467 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  ->  F  Fn  RR )
1910adantr 467 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  ->  G  Fn  RR )
2011a1i 11 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  ->  RR  e.  _V )
21 eqidd 2452 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  z  e.  RR )  ->  ( F `  z )  =  ( F `  z ) )
22 eqidd 2452 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  z  e.  RR )  ->  ( G `  z )  =  ( G `  z ) )
2318, 19, 20, 20, 13, 21, 22ofval 6540 . . . . . 6  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  z  e.  RR )  ->  ( ( F  oF  x.  G
) `  z )  =  ( ( F `
 z )  x.  ( G `  z
) ) )
2423eqeq1d 2453 . . . . 5  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  z  e.  RR )  ->  ( ( ( F  oF  x.  G ) `  z
)  =  A  <->  ( ( F `  z )  x.  ( G `  z
) )  =  A ) )
2524pm5.32da 647 . . . 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 732 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  ->  G  Fn  RR )
27 simprl 764 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
z  e.  RR )
28 fnfvelrn 6019 . . . . . . . . 9  |-  ( ( G  Fn  RR  /\  z  e.  RR )  ->  ( G `  z
)  e.  ran  G
)
2926, 27, 28syl2anc 667 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( G `  z
)  e.  ran  G
)
30 eldifsni 4098 . . . . . . . . . . 11  |-  ( A  e.  ( CC  \  { 0 } )  ->  A  =/=  0
)
3130ad2antlr 733 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  ->  A  =/=  0 )
32 simprr 766 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( ( F `  z )  x.  ( G `  z )
)  =  A )
333ad2antrr 732 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  ->  F : RR --> RR )
3433, 27ffvelrnd 6023 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( F `  z
)  e.  RR )
3534recnd 9669 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( F `  z
)  e.  CC )
3635mul01d 9832 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( ( F `  z )  x.  0 )  =  0 )
3731, 32, 363netr4d 2701 . . . . . . . . 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 6298 . . . . . . . . . 10  |-  ( ( G `  z )  =  0  ->  (
( F `  z
)  x.  ( G `
 z ) )  =  ( ( F `
 z )  x.  0 ) )
3938necon3i 2656 . . . . . . . . 9  |-  ( ( ( F `  z
)  x.  ( G `
 z ) )  =/=  ( ( F `
 z )  x.  0 )  ->  ( G `  z )  =/=  0 )
4037, 39syl 17 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( G `  z
)  =/=  0 )
41 eldifsn 4097 . . . . . . . 8  |-  ( ( G `  z )  e.  ( ran  G  \  { 0 } )  <-> 
( ( G `  z )  e.  ran  G  /\  ( G `  z )  =/=  0
) )
4229, 40, 41sylanbrc 670 . . . . . . 7  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( G `  z
)  e.  ( ran 
G  \  { 0 } ) )
438ad2antrr 732 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  ->  G : RR --> RR )
4443, 27ffvelrnd 6023 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( G `  z
)  e.  RR )
4544recnd 9669 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( G `  z
)  e.  CC )
4635, 45, 40divcan4d 10389 . . . . . . . . . 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 6305 . . . . . . . . . 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 2487 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( F `  z
)  =  ( A  /  ( G `  z ) ) )
4933, 4syl 17 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  ->  F  Fn  RR )
50 fniniseg 6003 . . . . . . . . . 10  |-  ( F  Fn  RR  ->  (
z  e.  ( `' F " { ( A  /  ( G `
 z ) ) } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  ( A  /  ( G `  z )
) ) ) )
5149, 50syl 17 . . . . . . . . 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 933 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
z  e.  ( `' F " { ( A  /  ( G `
 z ) ) } ) )
53 eqidd 2452 . . . . . . . . 9  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
( G `  z
)  =  ( G `
 z ) )
54 fniniseg 6003 . . . . . . . . . 10  |-  ( G  Fn  RR  ->  (
z  e.  ( `' G " { ( G `  z ) } )  <->  ( z  e.  RR  /\  ( G `
 z )  =  ( G `  z
) ) ) )
5526, 54syl 17 . . . . . . . . 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 933 . . . . . . . 8  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( z  e.  RR  /\  ( ( F `  z )  x.  ( G `  z ) )  =  A ) )  -> 
z  e.  ( `' G " { ( G `  z ) } ) )
57 elin 3617 . . . . . . . 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 670 . . . . . . 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 6298 . . . . . . . . . . . 12  |-  ( y  =  ( G `  z )  ->  ( A  /  y )  =  ( A  /  ( G `  z )
) )
6059sneqd 3980 . . . . . . . . . . 11  |-  ( y  =  ( G `  z )  ->  { ( A  /  y ) }  =  { ( A  /  ( G `
 z ) ) } )
6160imaeq2d 5168 . . . . . . . . . 10  |-  ( y  =  ( G `  z )  ->  ( `' F " { ( A  /  y ) } )  =  ( `' F " { ( A  /  ( G `
 z ) ) } ) )
62 sneq 3978 . . . . . . . . . . 11  |-  ( y  =  ( G `  z )  ->  { y }  =  { ( G `  z ) } )
6362imaeq2d 5168 . . . . . . . . . 10  |-  ( y  =  ( G `  z )  ->  ( `' G " { y } )  =  ( `' G " { ( G `  z ) } ) )
6461, 63ineq12d 3635 . . . . . . . . 9  |-  ( y  =  ( G `  z )  ->  (
( `' F " { ( A  / 
y ) } )  i^i  ( `' G " { y } ) )  =  ( ( `' F " { ( A  /  ( G `
 z ) ) } )  i^i  ( `' G " { ( G `  z ) } ) ) )
6564eleq2d 2514 . . . . . . . 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 3150 . . . . . . 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 667 . . . . . 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 436 . . . . 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 6003 . . . . . . . . . . 11  |-  ( F  Fn  RR  ->  (
z  e.  ( `' F " { ( A  /  y ) } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  ( A  /  y
) ) ) )
7018, 69syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( z  e.  ( `' F " { ( A  /  y ) } )  <->  ( z  e.  RR  /\  ( F `
 z )  =  ( A  /  y
) ) ) )
71 fniniseg 6003 . . . . . . . . . . 11  |-  ( G  Fn  RR  ->  (
z  e.  ( `' G " { y } )  <->  ( z  e.  RR  /\  ( G `
 z )  =  y ) ) )
7219, 71syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  A  e.  ( CC  \  { 0 } ) )  -> 
( z  e.  ( `' G " { y } )  <->  ( z  e.  RR  /\  ( G `
 z )  =  y ) ) )
7370, 72anbi12d 717 . . . . . . . . 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 3617 . . . . . . . . 9  |-  ( z  e.  ( ( `' F " { ( A  /  y ) } )  i^i  ( `' G " { y } ) )  <->  ( z  e.  ( `' F " { ( A  / 
y ) } )  /\  z  e.  ( `' G " { y } ) ) )
75 anandi 837 . . . . . . . . 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 292 . . . . . . . 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 467 . . . . . . 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 3555 . . . . . . . . . . . 12  |-  ( A  e.  ( CC  \  { 0 } )  ->  A  e.  CC )
7978ad2antlr 733 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  A  e.  CC )
808ad2antrr 732 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  G : RR --> RR )
81 frn 5735 . . . . . . . . . . . . . 14  |-  ( G : RR --> RR  ->  ran 
G  C_  RR )
8280, 81syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  ran  G 
C_  RR )
83 simprl 764 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  y  e.  ( ran  G  \  { 0 } ) )
84 eldifsn 4097 . . . . . . . . . . . . . . 15  |-  ( y  e.  ( ran  G  \  { 0 } )  <-> 
( y  e.  ran  G  /\  y  =/=  0
) )
8583, 84sylib 200 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  (
y  e.  ran  G  /\  y  =/=  0
) )
8685simpld 461 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  y  e.  ran  G )
8782, 86sseldd 3433 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  y  e.  RR )
8887recnd 9669 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  y  e.  CC )
8985simprd 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  y  =/=  0 )
9079, 88, 89divcan1d 10384 . . . . . . . . . 10  |-  ( ( ( ph  /\  A  e.  ( CC  \  {
0 } ) )  /\  ( y  e.  ( ran  G  \  { 0 } )  /\  z  e.  RR ) )  ->  (
( A  /  y
)  x.  y )  =  A )
91 oveq12 6299 . . . . . . . . . . 11  |-  ( ( ( F `  z
)  =  ( A  /  y )  /\  ( G `  z )  =  y )  -> 
( ( F `  z )  x.  ( G `  z )
)  =  ( ( A  /  y )  x.  y ) )
9291eqeq1d 2453 . . . . . . . . . 10  |-  ( ( ( F `  z
)  =  ( A  /  y )  /\  ( G `  z )  =  y )  -> 
( ( ( F `
 z )  x.  ( G `  z
) )  =  A  <-> 
( ( A  / 
y )  x.  y
)  =  A ) )
9390, 92syl5ibrcom 226 . . . . . . . . 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 654 . . . . . . . 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 699 . . . . . . 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 219 . . . . . 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 2879 . . . . 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 194 . . . 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 283 . . 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 4283 . . 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 267 . 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 2449 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 188    /\ wa 371    = wceq 1444    e. wcel 1887    =/= wne 2622   E.wrex 2738   _Vcvv 3045    \ cdif 3401    i^i cin 3403    C_ wss 3404   {csn 3968   U_ciun 4278   `'ccnv 4833   dom cdm 4834   ran crn 4835   "cima 4837    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290    oFcof 6529   CCcc 9537   RRcr 9538   0cc0 9539    x. cmul 9544    / cdiv 10269   S.1citg1 22573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-sum 13753  df-itg1 22578
This theorem is referenced by:  i1fmul  22654
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