Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngoisocnv Structured version   Unicode version

Theorem rngoisocnv 30302
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 5834 . . . . . . . 8  |-  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  ->  `' F : ran  ( 1st `  S
)
-1-1-onto-> ran  ( 1st `  R
) )
2 f1of 5822 . . . . . . . 8  |-  ( `' F : ran  ( 1st `  S ) -1-1-onto-> ran  ( 1st `  R )  ->  `' F : ran  ( 1st `  S ) --> ran  ( 1st `  R
) )
31, 2syl 16 . . . . . . 7  |-  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  ->  `' F : ran  ( 1st `  S
) --> ran  ( 1st `  R ) )
43ad2antll 728 . . . . . 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 2467 . . . . . . . . . 10  |-  ( 2nd `  R )  =  ( 2nd `  R )
6 eqid 2467 . . . . . . . . . 10  |-  (GId `  ( 2nd `  R ) )  =  (GId `  ( 2nd `  R ) )
7 eqid 2467 . . . . . . . . . 10  |-  ( 2nd `  S )  =  ( 2nd `  S )
8 eqid 2467 . . . . . . . . . 10  |-  (GId `  ( 2nd `  S ) )  =  (GId `  ( 2nd `  S ) )
95, 6, 7, 8rngohom1 30289 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  (GId `  ( 2nd `  R ) ) )  =  (GId `  ( 2nd `  S ) ) )
1093expa 1196 . . . . . . . 8  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  -> 
( F `  (GId `  ( 2nd `  R
) ) )  =  (GId `  ( 2nd `  S ) ) )
1110adantrr 716 . . . . . . 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 2467 . . . . . . . . . . 11  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
1312, 5, 6rngo1cl 25245 . . . . . . . . . 10  |-  ( R  e.  RingOps  ->  (GId `  ( 2nd `  R ) )  e.  ran  ( 1st `  R ) )
14 f1ocnvfv 6183 . . . . . . . . . 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 474 . . . . . . . . 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 453 . . . . . . . 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 748 . . . . . . 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 15 . . . . . 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 6182 . . . . . . . . . . . . . 14  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  x  e.  ran  ( 1st `  S ) )  -> 
( F `  ( `' F `  x ) )  =  x )
20 f1ocnvfv2 6182 . . . . . . . . . . . . . 14  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) )  -> 
( F `  ( `' F `  y ) )  =  y )
2119, 20anim12da 30119 . . . . . . . . . . . . 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 6304 . . . . . . . . . . . . 13  |-  ( ( ( F `  ( `' F `  x ) )  =  x  /\  ( F `  ( `' F `  y ) )  =  y )  ->  ( ( F `
 ( `' F `  x ) ) ( 1st `  S ) ( F `  ( `' F `  y ) ) )  =  ( x ( 1st `  S
) y ) )
2321, 22syl 16 . . . . . . . . . . . 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 713 . . . . . . . . . . 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 713 . . . . . . . . . 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 6187 . . . . . . . . . . . . . . . 16  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  x  e.  ran  ( 1st `  S ) )  -> 
( `' F `  x )  e.  ran  ( 1st `  R ) )
27 f1ocnvdm 6187 . . . . . . . . . . . . . . . 16  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) )  -> 
( `' F `  y )  e.  ran  ( 1st `  R ) )
2826, 27anim12da 30119 . . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . . . 16  |-  ( 1st `  R )  =  ( 1st `  R )
30 eqid 2467 . . . . . . . . . . . . . . . 16  |-  ( 1st `  S )  =  ( 1st `  S )
3129, 12, 30rngohomadd 30290 . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . 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 605 . . . . . . . . . . . . 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 1196 . . . . . . . . . . . 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 619 . . . . . . . . . . 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 429 . . . . . . . . . 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 2467 . . . . . . . . . . . . . . . 16  |-  ran  ( 1st `  S )  =  ran  ( 1st `  S
)
3830, 37rngogcl 25207 . . . . . . . . . . . . . . 15  |-  ( ( S  e.  RingOps  /\  x  e.  ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) )  ->  (
x ( 1st `  S
) y )  e. 
ran  ( 1st `  S
) )
39383expb 1197 . . . . . . . . . . . . . 14  |-  ( ( S  e.  RingOps  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( x ( 1st `  S ) y )  e.  ran  ( 1st `  S ) )
40 f1ocnvfv2 6182 . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . 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 802 . . . . . . . . . . . 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 717 . . . . . . . . . . 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 719 . . . . . . . . . 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 2519 . . . . . . . . 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 5821 . . . . . . . . . . . 12  |-  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  ->  F : ran  ( 1st `  R
) -1-1-> ran  ( 1st `  S
) )
4847ad2antlr 726 . . . . . . . . . . 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 6187 . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . 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 802 . . . . . . . . . . . 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 717 . . . . . . . . . . 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 25207 . . . . . . . . . . . . . . 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 1197 . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . 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 648 . . . . . . . . . . . 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 718 . . . . . . . . . . 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 6169 . . . . . . . . . . 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 1226 . . . . . . . . . 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 719 . . . . . . . . 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 210 . . . . . . . 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 6304 . . . . . . . . . . . . 13  |-  ( ( ( F `  ( `' F `  x ) )  =  x  /\  ( F `  ( `' F `  y ) )  =  y )  ->  ( ( F `
 ( `' F `  x ) ) ( 2nd `  S ) ( F `  ( `' F `  y ) ) )  =  ( x ( 2nd `  S
) y ) )
6421, 63syl 16 . . . . . . . . . . . 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 713 . . . . . . . . . . 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 713 . . . . . . . . . 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 30291 . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . 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 605 . . . . . . . . . . . . 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 1196 . . . . . . . . . . . 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 619 . . . . . . . . . . 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 429 . . . . . . . . . 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 25198 . . . . . . . . . . . . . . 15  |-  ( ( S  e.  RingOps  /\  x  e.  ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) )  ->  (
x ( 2nd `  S
) y )  e. 
ran  ( 1st `  S
) )
74733expb 1197 . . . . . . . . . . . . . 14  |-  ( ( S  e.  RingOps  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( x ( 2nd `  S ) y )  e.  ran  ( 1st `  S ) )
75 f1ocnvfv2 6182 . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . 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 802 . . . . . . . . . . . 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 717 . . . . . . . . . . 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 719 . . . . . . . . . 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 2519 . . . . . . . . 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 6187 . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . 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 802 . . . . . . . . . . . 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 717 . . . . . . . . . . 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 25198 . . . . . . . . . . . . . . 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 1197 . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . 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 648 . . . . . . . . . . . 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 718 . . . . . . . . . . 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 6169 . . . . . . . . . . 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 1226 . . . . . . . . . 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 719 . . . . . . . . 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 210 . . . . . . . 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 532 . . . . . . 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 2888 . . . . . 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 30286 . . . . . . . 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 453 . . . . . . 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 465 . . . . . 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 1179 . . . . 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 728 . . . . 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 532 . . . 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 434 . . 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 30299 . . 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 30299 . . . 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 453 . . 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 268 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngIso  S )  ->  `' F  e.  ( S  RngIso  R ) ) )
1091083impia 1193 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  `' F  e.  ( S  RngIso  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   `'ccnv 5004   ran crn 5006   -->wf 5590   -1-1->wf1 5591   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794  GIdcgi 25003   RingOpscrngo 25191    RngHom crnghom 30281    RngIso crngiso 30282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-map 7434  df-grpo 25007  df-gid 25008  df-ablo 25098  df-ass 25129  df-exid 25131  df-mgmOLD 25135  df-sgrOLD 25147  df-mndo 25154  df-rngo 25192  df-rngohom 30284  df-rngoiso 30297
This theorem is referenced by:  riscer  30309
  Copyright terms: Public domain W3C validator