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Theorem rngoisocnv 26715
Description: The inverse of a ring isomorphism is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
Assertion
Ref Expression
rngoisocnv  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  `' F  e.  ( S  RngIso  R ) )

Proof of Theorem rngoisocnv
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1ocnv 5501 . . . . . . . 8  |-  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  ->  `' F : ran  ( 1st `  S
)
-1-1-onto-> ran  ( 1st `  R
) )
2 f1of 5488 . . . . . . . 8  |-  ( `' F : ran  ( 1st `  S ) -1-1-onto-> ran  ( 1st `  R )  ->  `' F : ran  ( 1st `  S ) --> ran  ( 1st `  R
) )
31, 2syl 15 . . . . . . 7  |-  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  ->  `' F : ran  ( 1st `  S
) --> ran  ( 1st `  R ) )
43ad2antll 709 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  ->  `' F : ran  ( 1st `  S ) --> ran  ( 1st `  R
) )
5 eqid 2296 . . . . . . . . . 10  |-  ( 2nd `  R )  =  ( 2nd `  R )
6 eqid 2296 . . . . . . . . . 10  |-  (GId `  ( 2nd `  R ) )  =  (GId `  ( 2nd `  R ) )
7 eqid 2296 . . . . . . . . . 10  |-  ( 2nd `  S )  =  ( 2nd `  S )
8 eqid 2296 . . . . . . . . . 10  |-  (GId `  ( 2nd `  S ) )  =  (GId `  ( 2nd `  S ) )
95, 6, 7, 8rngohom1 26702 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  (GId `  ( 2nd `  R ) ) )  =  (GId `  ( 2nd `  S ) ) )
1093expa 1151 . . . . . . . 8  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  -> 
( F `  (GId `  ( 2nd `  R
) ) )  =  (GId `  ( 2nd `  S ) ) )
1110adantrr 697 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  -> 
( F `  (GId `  ( 2nd `  R
) ) )  =  (GId `  ( 2nd `  S ) ) )
12 eqid 2296 . . . . . . . . . . 11  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
1312, 5, 6rngo1cl 21112 . . . . . . . . . 10  |-  ( R  e.  RingOps  ->  (GId `  ( 2nd `  R ) )  e.  ran  ( 1st `  R ) )
14 f1ocnvfv 5810 . . . . . . . . . 10  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  (GId `  ( 2nd `  R
) )  e.  ran  ( 1st `  R ) )  ->  ( ( F `  (GId `  ( 2nd `  R ) ) )  =  (GId `  ( 2nd `  S ) )  ->  ( `' F `  (GId `  ( 2nd `  S ) ) )  =  (GId `  ( 2nd `  R ) ) ) )
1513, 14sylan2 460 . . . . . . . . 9  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  R  e.  RingOps )  -> 
( ( F `  (GId `  ( 2nd `  R
) ) )  =  (GId `  ( 2nd `  S ) )  -> 
( `' F `  (GId `  ( 2nd `  S
) ) )  =  (GId `  ( 2nd `  R ) ) ) )
1615ancoms 439 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  (
( F `  (GId `  ( 2nd `  R
) ) )  =  (GId `  ( 2nd `  S ) )  -> 
( `' F `  (GId `  ( 2nd `  S
) ) )  =  (GId `  ( 2nd `  R ) ) ) )
1716ad2ant2rl 729 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  -> 
( ( F `  (GId `  ( 2nd `  R
) ) )  =  (GId `  ( 2nd `  S ) )  -> 
( `' F `  (GId `  ( 2nd `  S
) ) )  =  (GId `  ( 2nd `  R ) ) ) )
1811, 17mpd 14 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  -> 
( `' F `  (GId `  ( 2nd `  S
) ) )  =  (GId `  ( 2nd `  R ) ) )
19 f1ocnvfv2 5809 . . . . . . . . . . . . . 14  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  x  e.  ran  ( 1st `  S ) )  -> 
( F `  ( `' F `  x ) )  =  x )
20 f1ocnvfv2 5809 . . . . . . . . . . . . . 14  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) )  -> 
( F `  ( `' F `  y ) )  =  y )
2119, 20anim12da 26435 . . . . . . . . . . . . 13  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( F `  ( `' F `  x ) )  =  x  /\  ( F `  ( `' F `  y ) )  =  y ) )
22 oveq12 5883 . . . . . . . . . . . . 13  |-  ( ( ( F `  ( `' F `  x ) )  =  x  /\  ( F `  ( `' F `  y ) )  =  y )  ->  ( ( F `
 ( `' F `  x ) ) ( 1st `  S ) ( F `  ( `' F `  y ) ) )  =  ( x ( 1st `  S
) y ) )
2321, 22syl 15 . . . . . . . . . . . 12  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( F `  ( `' F `  x ) ) ( 1st `  S
) ( F `  ( `' F `  y ) ) )  =  ( x ( 1st `  S
) y ) )
2423adantll 694 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( F `
 ( `' F `  x ) ) ( 1st `  S ) ( F `  ( `' F `  y ) ) )  =  ( x ( 1st `  S
) y ) )
2524adantll 694 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( F `  ( `' F `  x ) ) ( 1st `  S
) ( F `  ( `' F `  y ) ) )  =  ( x ( 1st `  S
) y ) )
26 f1ocnvdm 5812 . . . . . . . . . . . . . . . 16  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  x  e.  ran  ( 1st `  S ) )  -> 
( `' F `  x )  e.  ran  ( 1st `  R ) )
27 f1ocnvdm 5812 . . . . . . . . . . . . . . . 16  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) )  -> 
( `' F `  y )  e.  ran  ( 1st `  R ) )
2826, 27anim12da 26435 . . . . . . . . . . . . . . 15  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( `' F `  x )  e.  ran  ( 1st `  R )  /\  ( `' F `  y )  e.  ran  ( 1st `  R ) ) )
29 eqid 2296 . . . . . . . . . . . . . . . 16  |-  ( 1st `  R )  =  ( 1st `  R )
30 eqid 2296 . . . . . . . . . . . . . . . 16  |-  ( 1st `  S )  =  ( 1st `  S )
3129, 12, 30rngohomadd 26703 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( ( `' F `  x )  e.  ran  ( 1st `  R )  /\  ( `' F `  y )  e.  ran  ( 1st `  R ) ) )  ->  ( F `  ( ( `' F `  x ) ( 1st `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 1st `  S
) ( F `  ( `' F `  y ) ) ) )
3228, 31sylan2 460 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  /\  ( x  e.  ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) ) ) )  ->  ( F `  ( ( `' F `  x ) ( 1st `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 1st `  S
) ( F `  ( `' F `  y ) ) ) )
3332exp32 588 . . . . . . . . . . . . 13  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
)  ->  ( (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) )  -> 
( F `  (
( `' F `  x ) ( 1st `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 1st `  S
) ( F `  ( `' F `  y ) ) ) ) ) )
34333expa 1151 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  -> 
( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  ->  ( (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) )  -> 
( F `  (
( `' F `  x ) ( 1st `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 1st `  S
) ( F `  ( `' F `  y ) ) ) ) ) )
3534impr 602 . . . . . . . . . . 11  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  -> 
( ( x  e. 
ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) )  ->  ( F `  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) )  =  ( ( F `
 ( `' F `  x ) ) ( 1st `  S ) ( F `  ( `' F `  y ) ) ) ) )
3635imp 418 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) )  =  ( ( F `
 ( `' F `  x ) ) ( 1st `  S ) ( F `  ( `' F `  y ) ) ) )
37 eqid 2296 . . . . . . . . . . . . . . . 16  |-  ran  ( 1st `  S )  =  ran  ( 1st `  S
)
3830, 37rngogcl 21074 . . . . . . . . . . . . . . 15  |-  ( ( S  e.  RingOps  /\  x  e.  ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) )  ->  (
x ( 1st `  S
) y )  e. 
ran  ( 1st `  S
) )
39383expb 1152 . . . . . . . . . . . . . 14  |-  ( ( S  e.  RingOps  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( x ( 1st `  S ) y )  e.  ran  ( 1st `  S ) )
40 f1ocnvfv2 5809 . . . . . . . . . . . . . . 15  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x ( 1st `  S ) y )  e.  ran  ( 1st `  S ) )  -> 
( F `  ( `' F `  ( x ( 1st `  S
) y ) ) )  =  ( x ( 1st `  S
) y ) )
4140ancoms 439 . . . . . . . . . . . . . 14  |-  ( ( ( x ( 1st `  S ) y )  e.  ran  ( 1st `  S )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( F `  ( `' F `  ( x
( 1st `  S
) y ) ) )  =  ( x ( 1st `  S
) y ) )
4239, 41sylan 457 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  RingOps  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( F `  ( `' F `  ( x
( 1st `  S
) y ) ) )  =  ( x ( 1st `  S
) y ) )
4342an32s 779 . . . . . . . . . . . 12  |-  ( ( ( S  e.  RingOps  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x ( 1st `  S
) y ) ) )  =  ( x ( 1st `  S
) y ) )
4443adantlll 698 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x ( 1st `  S
) y ) ) )  =  ( x ( 1st `  S
) y ) )
4544adantlrl 700 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x
( 1st `  S
) y ) ) )  =  ( x ( 1st `  S
) y ) )
4625, 36, 453eqtr4rd 2339 . . . . . . . . 9  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x
( 1st `  S
) y ) ) )  =  ( F `
 ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) ) )
47 f1of1 5487 . . . . . . . . . . . 12  |-  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  ->  F : ran  ( 1st `  R
) -1-1-> ran  ( 1st `  S
) )
4847ad2antlr 707 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  F : ran  ( 1st `  R )
-1-1-> ran  ( 1st `  S
) )
49 f1ocnvdm 5812 . . . . . . . . . . . . . . 15  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x ( 1st `  S ) y )  e.  ran  ( 1st `  S ) )  -> 
( `' F `  ( x ( 1st `  S ) y ) )  e.  ran  ( 1st `  R ) )
5049ancoms 439 . . . . . . . . . . . . . 14  |-  ( ( ( x ( 1st `  S ) y )  e.  ran  ( 1st `  S )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( `' F `  ( x ( 1st `  S
) y ) )  e.  ran  ( 1st `  R ) )
5139, 50sylan 457 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  RingOps  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( `' F `  ( x ( 1st `  S
) y ) )  e.  ran  ( 1st `  R ) )
5251an32s 779 . . . . . . . . . . . 12  |-  ( ( ( S  e.  RingOps  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( `' F `  ( x ( 1st `  S ) y ) )  e.  ran  ( 1st `  R ) )
5352adantlll 698 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( `' F `  ( x ( 1st `  S ) y ) )  e.  ran  ( 1st `  R ) )
5429, 12rngogcl 21074 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RingOps  /\  ( `' F `  x )  e.  ran  ( 1st `  R )  /\  ( `' F `  y )  e.  ran  ( 1st `  R ) )  -> 
( ( `' F `  x ) ( 1st `  R ) ( `' F `  y ) )  e.  ran  ( 1st `  R ) )
55543expb 1152 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RingOps  /\  (
( `' F `  x )  e.  ran  ( 1st `  R )  /\  ( `' F `  y )  e.  ran  ( 1st `  R ) ) )  ->  (
( `' F `  x ) ( 1st `  R ) ( `' F `  y ) )  e.  ran  ( 1st `  R ) )
5628, 55sylan2 460 . . . . . . . . . . . . 13  |-  ( ( R  e.  RingOps  /\  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  /\  ( x  e.  ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) ) ) )  ->  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  e. 
ran  ( 1st `  R
) )
5756anassrs 629 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RingOps  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  e. 
ran  ( 1st `  R
) )
5857adantllr 699 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  e. 
ran  ( 1st `  R
) )
59 f1fveq 5802 . . . . . . . . . . 11  |-  ( ( F : ran  ( 1st `  R ) -1-1-> ran  ( 1st `  S )  /\  ( ( `' F `  ( x ( 1st `  S
) y ) )  e.  ran  ( 1st `  R )  /\  (
( `' F `  x ) ( 1st `  R ) ( `' F `  y ) )  e.  ran  ( 1st `  R ) ) )  ->  ( ( F `  ( `' F `  ( x
( 1st `  S
) y ) ) )  =  ( F `
 ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) )  <-> 
( `' F `  ( x ( 1st `  S ) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) ) )
6048, 53, 58, 59syl12anc 1180 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( F `
 ( `' F `  ( x ( 1st `  S ) y ) ) )  =  ( F `  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) )  <-> 
( `' F `  ( x ( 1st `  S ) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) ) )
6160adantlrl 700 . . . . . . . . 9  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( F `  ( `' F `  ( x ( 1st `  S
) y ) ) )  =  ( F `
 ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) )  <-> 
( `' F `  ( x ( 1st `  S ) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) ) )
6246, 61mpbid 201 . . . . . . . 8  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( `' F `  ( x ( 1st `  S
) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) )
63 oveq12 5883 . . . . . . . . . . . . 13  |-  ( ( ( F `  ( `' F `  x ) )  =  x  /\  ( F `  ( `' F `  y ) )  =  y )  ->  ( ( F `
 ( `' F `  x ) ) ( 2nd `  S ) ( F `  ( `' F `  y ) ) )  =  ( x ( 2nd `  S
) y ) )
6421, 63syl 15 . . . . . . . . . . . 12  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( F `  ( `' F `  x ) ) ( 2nd `  S
) ( F `  ( `' F `  y ) ) )  =  ( x ( 2nd `  S
) y ) )
6564adantll 694 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( F `
 ( `' F `  x ) ) ( 2nd `  S ) ( F `  ( `' F `  y ) ) )  =  ( x ( 2nd `  S
) y ) )
6665adantll 694 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( F `  ( `' F `  x ) ) ( 2nd `  S
) ( F `  ( `' F `  y ) ) )  =  ( x ( 2nd `  S
) y ) )
6729, 12, 5, 7rngohommul 26704 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( ( `' F `  x )  e.  ran  ( 1st `  R )  /\  ( `' F `  y )  e.  ran  ( 1st `  R ) ) )  ->  ( F `  ( ( `' F `  x ) ( 2nd `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 2nd `  S
) ( F `  ( `' F `  y ) ) ) )
6828, 67sylan2 460 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  /\  ( x  e.  ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) ) ) )  ->  ( F `  ( ( `' F `  x ) ( 2nd `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 2nd `  S
) ( F `  ( `' F `  y ) ) ) )
6968exp32 588 . . . . . . . . . . . . 13  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
)  ->  ( (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) )  -> 
( F `  (
( `' F `  x ) ( 2nd `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 2nd `  S
) ( F `  ( `' F `  y ) ) ) ) ) )
70693expa 1151 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  -> 
( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  ->  ( (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) )  -> 
( F `  (
( `' F `  x ) ( 2nd `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 2nd `  S
) ( F `  ( `' F `  y ) ) ) ) ) )
7170impr 602 . . . . . . . . . . 11  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  -> 
( ( x  e. 
ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) )  ->  ( F `  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) )  =  ( ( F `
 ( `' F `  x ) ) ( 2nd `  S ) ( F `  ( `' F `  y ) ) ) ) )
7271imp 418 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) )  =  ( ( F `
 ( `' F `  x ) ) ( 2nd `  S ) ( F `  ( `' F `  y ) ) ) )
7330, 7, 37rngocl 21065 . . . . . . . . . . . . . . 15  |-  ( ( S  e.  RingOps  /\  x  e.  ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) )  ->  (
x ( 2nd `  S
) y )  e. 
ran  ( 1st `  S
) )
74733expb 1152 . . . . . . . . . . . . . 14  |-  ( ( S  e.  RingOps  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( x ( 2nd `  S ) y )  e.  ran  ( 1st `  S ) )
75 f1ocnvfv2 5809 . . . . . . . . . . . . . . 15  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x ( 2nd `  S ) y )  e.  ran  ( 1st `  S ) )  -> 
( F `  ( `' F `  ( x ( 2nd `  S
) y ) ) )  =  ( x ( 2nd `  S
) y ) )
7675ancoms 439 . . . . . . . . . . . . . 14  |-  ( ( ( x ( 2nd `  S ) y )  e.  ran  ( 1st `  S )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( F `  ( `' F `  ( x
( 2nd `  S
) y ) ) )  =  ( x ( 2nd `  S
) y ) )
7774, 76sylan 457 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  RingOps  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( F `  ( `' F `  ( x
( 2nd `  S
) y ) ) )  =  ( x ( 2nd `  S
) y ) )
7877an32s 779 . . . . . . . . . . . 12  |-  ( ( ( S  e.  RingOps  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x ( 2nd `  S
) y ) ) )  =  ( x ( 2nd `  S
) y ) )
7978adantlll 698 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x ( 2nd `  S
) y ) ) )  =  ( x ( 2nd `  S
) y ) )
8079adantlrl 700 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x
( 2nd `  S
) y ) ) )  =  ( x ( 2nd `  S
) y ) )
8166, 72, 803eqtr4rd 2339 . . . . . . . . 9  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x
( 2nd `  S
) y ) ) )  =  ( F `
 ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) )
82 f1ocnvdm 5812 . . . . . . . . . . . . . . 15  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x ( 2nd `  S ) y )  e.  ran  ( 1st `  S ) )  -> 
( `' F `  ( x ( 2nd `  S ) y ) )  e.  ran  ( 1st `  R ) )
8382ancoms 439 . . . . . . . . . . . . . 14  |-  ( ( ( x ( 2nd `  S ) y )  e.  ran  ( 1st `  S )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( `' F `  ( x ( 2nd `  S
) y ) )  e.  ran  ( 1st `  R ) )
8474, 83sylan 457 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  RingOps  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( `' F `  ( x ( 2nd `  S
) y ) )  e.  ran  ( 1st `  R ) )
8584an32s 779 . . . . . . . . . . . 12  |-  ( ( ( S  e.  RingOps  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( `' F `  ( x ( 2nd `  S ) y ) )  e.  ran  ( 1st `  R ) )
8685adantlll 698 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( `' F `  ( x ( 2nd `  S ) y ) )  e.  ran  ( 1st `  R ) )
8729, 5, 12rngocl 21065 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RingOps  /\  ( `' F `  x )  e.  ran  ( 1st `  R )  /\  ( `' F `  y )  e.  ran  ( 1st `  R ) )  -> 
( ( `' F `  x ) ( 2nd `  R ) ( `' F `  y ) )  e.  ran  ( 1st `  R ) )
88873expb 1152 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RingOps  /\  (
( `' F `  x )  e.  ran  ( 1st `  R )  /\  ( `' F `  y )  e.  ran  ( 1st `  R ) ) )  ->  (
( `' F `  x ) ( 2nd `  R ) ( `' F `  y ) )  e.  ran  ( 1st `  R ) )
8928, 88sylan2 460 . . . . . . . . . . . . 13  |-  ( ( R  e.  RingOps  /\  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  /\  ( x  e.  ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) ) ) )  ->  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) )  e. 
ran  ( 1st `  R
) )
9089anassrs 629 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RingOps  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) )  e. 
ran  ( 1st `  R
) )
9190adantllr 699 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) )  e. 
ran  ( 1st `  R
) )
92 f1fveq 5802 . . . . . . . . . . 11  |-  ( ( F : ran  ( 1st `  R ) -1-1-> ran  ( 1st `  S )  /\  ( ( `' F `  ( x ( 2nd `  S
) y ) )  e.  ran  ( 1st `  R )  /\  (
( `' F `  x ) ( 2nd `  R ) ( `' F `  y ) )  e.  ran  ( 1st `  R ) ) )  ->  ( ( F `  ( `' F `  ( x
( 2nd `  S
) y ) ) )  =  ( F `
 ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) )  <-> 
( `' F `  ( x ( 2nd `  S ) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) )
9348, 86, 91, 92syl12anc 1180 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( F `
 ( `' F `  ( x ( 2nd `  S ) y ) ) )  =  ( F `  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) )  <-> 
( `' F `  ( x ( 2nd `  S ) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) )
9493adantlrl 700 . . . . . . . . 9  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( F `  ( `' F `  ( x ( 2nd `  S
) y ) ) )  =  ( F `
 ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) )  <-> 
( `' F `  ( x ( 2nd `  S ) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) )
9581, 94mpbid 201 . . . . . . . 8  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( `' F `  ( x ( 2nd `  S
) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) )
9662, 95jca 518 . . . . . . 7  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( `' F `  ( x ( 1st `  S ) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  /\  ( `' F `  ( x ( 2nd `  S
) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) )
9796ralrimivva 2648 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  ->  A. x  e.  ran  ( 1st `  S ) A. y  e.  ran  ( 1st `  S ) ( ( `' F `  ( x ( 1st `  S ) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  /\  ( `' F `  ( x ( 2nd `  S
) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) )
9830, 7, 37, 8, 29, 5, 12, 6isrngohom 26699 . . . . . . . 8  |-  ( ( S  e.  RingOps  /\  R  e.  RingOps )  ->  ( `' F  e.  ( S  RngHom  R )  <->  ( `' F : ran  ( 1st `  S ) --> ran  ( 1st `  R )  /\  ( `' F `  (GId `  ( 2nd `  S ) ) )  =  (GId
`  ( 2nd `  R
) )  /\  A. x  e.  ran  ( 1st `  S ) A. y  e.  ran  ( 1st `  S
) ( ( `' F `  ( x ( 1st `  S
) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  /\  ( `' F `  ( x ( 2nd `  S
) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) ) ) )
9998ancoms 439 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( `' F  e.  ( S  RngHom  R )  <->  ( `' F : ran  ( 1st `  S ) --> ran  ( 1st `  R )  /\  ( `' F `  (GId `  ( 2nd `  S ) ) )  =  (GId
`  ( 2nd `  R
) )  /\  A. x  e.  ran  ( 1st `  S ) A. y  e.  ran  ( 1st `  S
) ( ( `' F `  ( x ( 1st `  S
) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  /\  ( `' F `  ( x ( 2nd `  S
) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) ) ) )
10099adantr 451 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  -> 
( `' F  e.  ( S  RngHom  R )  <-> 
( `' F : ran  ( 1st `  S
) --> ran  ( 1st `  R )  /\  ( `' F `  (GId `  ( 2nd `  S ) ) )  =  (GId
`  ( 2nd `  R
) )  /\  A. x  e.  ran  ( 1st `  S ) A. y  e.  ran  ( 1st `  S
) ( ( `' F `  ( x ( 1st `  S
) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  /\  ( `' F `  ( x ( 2nd `  S
) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) ) ) )
1014, 18, 97, 100mpbir3and 1135 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  ->  `' F  e.  ( S  RngHom  R ) )
1021ad2antll 709 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  ->  `' F : ran  ( 1st `  S ) -1-1-onto-> ran  ( 1st `  R ) )
103101, 102jca 518 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  -> 
( `' F  e.  ( S  RngHom  R )  /\  `' F : ran  ( 1st `  S
)
-1-1-onto-> ran  ( 1st `  R
) ) )
104103ex 423 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  (
( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  ->  ( `' F  e.  ( S  RngHom  R )  /\  `' F : ran  ( 1st `  S ) -1-1-onto-> ran  ( 1st `  R ) ) ) )
10529, 12, 30, 37isrngoiso 26712 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngIso  S )  <->  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) ) )
10630, 37, 29, 12isrngoiso 26712 . . . 4  |-  ( ( S  e.  RingOps  /\  R  e.  RingOps )  ->  ( `' F  e.  ( S  RngIso  R )  <->  ( `' F  e.  ( S  RngHom  R )  /\  `' F : ran  ( 1st `  S ) -1-1-onto-> ran  ( 1st `  R
) ) ) )
107106ancoms 439 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( `' F  e.  ( S  RngIso  R )  <->  ( `' F  e.  ( S  RngHom  R )  /\  `' F : ran  ( 1st `  S ) -1-1-onto-> ran  ( 1st `  R
) ) ) )
108104, 105, 1073imtr4d 259 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngIso  S )  ->  `' F  e.  ( S  RngIso  R ) ) )
1091083impia 1148 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  `' F  e.  ( S  RngIso  R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   `'ccnv 4704   ran crn 4706   -->wf 5267   -1-1->wf1 5268   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137  GIdcgi 20870   RingOpscrngo 21058    RngHom crnghom 26694    RngIso crngiso 26695
This theorem is referenced by:  riscer  26722
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-map 6790  df-grpo 20874  df-gid 20875  df-ablo 20965  df-ass 20996  df-exid 20998  df-mgm 21002  df-sgr 21014  df-mndo 21021  df-rngo 21059  df-rngohom 26697  df-rngoiso 26710
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