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Theorem crngohomfo 26309
Description: The image of a homomorphism from a commutative ring is commutative. (Contributed by Jeff Madsen, 4-Jan-2011.)
Hypotheses
Ref Expression
crnghomfo.1  |-  G  =  ( 1st `  R
)
crnghomfo.2  |-  X  =  ran  G
crnghomfo.3  |-  J  =  ( 1st `  S
)
crnghomfo.4  |-  Y  =  ran  J
Assertion
Ref Expression
crngohomfo  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  ->  S  e. CRingOps )

Proof of Theorem crngohomfo
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 732 . 2  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  ->  S  e.  RingOps )
2 foelrn 5829 . . . . . . . 8  |-  ( ( F : X -onto-> Y  /\  y  e.  Y
)  ->  E. w  e.  X  y  =  ( F `  w ) )
32ex 424 . . . . . . 7  |-  ( F : X -onto-> Y  -> 
( y  e.  Y  ->  E. w  e.  X  y  =  ( F `  w ) ) )
4 foelrn 5829 . . . . . . . 8  |-  ( ( F : X -onto-> Y  /\  z  e.  Y
)  ->  E. x  e.  X  z  =  ( F `  x ) )
54ex 424 . . . . . . 7  |-  ( F : X -onto-> Y  -> 
( z  e.  Y  ->  E. x  e.  X  z  =  ( F `  x ) ) )
63, 5anim12d 547 . . . . . 6  |-  ( F : X -onto-> Y  -> 
( ( y  e.  Y  /\  z  e.  Y )  ->  ( E. w  e.  X  y  =  ( F `  w )  /\  E. x  e.  X  z  =  ( F `  x ) ) ) )
7 reeanv 2820 . . . . . 6  |-  ( E. w  e.  X  E. x  e.  X  (
y  =  ( F `
 w )  /\  z  =  ( F `  x ) )  <->  ( E. w  e.  X  y  =  ( F `  w )  /\  E. x  e.  X  z  =  ( F `  x ) ) )
86, 7syl6ibr 219 . . . . 5  |-  ( F : X -onto-> Y  -> 
( ( y  e.  Y  /\  z  e.  Y )  ->  E. w  e.  X  E. x  e.  X  ( y  =  ( F `  w )  /\  z  =  ( F `  x ) ) ) )
98ad2antll 710 . . . 4  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  -> 
( ( y  e.  Y  /\  z  e.  Y )  ->  E. w  e.  X  E. x  e.  X  ( y  =  ( F `  w )  /\  z  =  ( F `  x ) ) ) )
10 crnghomfo.1 . . . . . . . . . . . . . 14  |-  G  =  ( 1st `  R
)
11 eqid 2389 . . . . . . . . . . . . . 14  |-  ( 2nd `  R )  =  ( 2nd `  R )
12 crnghomfo.2 . . . . . . . . . . . . . 14  |-  X  =  ran  G
1310, 11, 12crngocom 26304 . . . . . . . . . . . . 13  |-  ( ( R  e. CRingOps  /\  w  e.  X  /\  x  e.  X )  ->  (
w ( 2nd `  R
) x )  =  ( x ( 2nd `  R ) w ) )
14133expb 1154 . . . . . . . . . . . 12  |-  ( ( R  e. CRingOps  /\  (
w  e.  X  /\  x  e.  X )
)  ->  ( w
( 2nd `  R
) x )  =  ( x ( 2nd `  R ) w ) )
15143ad2antl1 1119 . . . . . . . . . . 11  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( w ( 2nd `  R ) x )  =  ( x ( 2nd `  R ) w ) )
1615fveq2d 5674 . . . . . . . . . 10  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( F `  (
w ( 2nd `  R
) x ) )  =  ( F `  ( x ( 2nd `  R ) w ) ) )
17 crngorngo 26303 . . . . . . . . . . 11  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
18 eqid 2389 . . . . . . . . . . . 12  |-  ( 2nd `  S )  =  ( 2nd `  S )
1910, 12, 11, 18rngohommul 26279 . . . . . . . . . . 11  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( F `  (
w ( 2nd `  R
) x ) )  =  ( ( F `
 w ) ( 2nd `  S ) ( F `  x
) ) )
2017, 19syl3anl1 1232 . . . . . . . . . 10  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( F `  (
w ( 2nd `  R
) x ) )  =  ( ( F `
 w ) ( 2nd `  S ) ( F `  x
) ) )
2110, 12, 11, 18rngohommul 26279 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( x  e.  X  /\  w  e.  X ) )  -> 
( F `  (
x ( 2nd `  R
) w ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  w
) ) )
2221ancom2s 778 . . . . . . . . . . 11  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( F `  (
x ( 2nd `  R
) w ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  w
) ) )
2317, 22syl3anl1 1232 . . . . . . . . . 10  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( F `  (
x ( 2nd `  R
) w ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  w
) ) )
2416, 20, 233eqtr3d 2429 . . . . . . . . 9  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( ( F `  w ) ( 2nd `  S ) ( F `
 x ) )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  w
) ) )
25 oveq12 6031 . . . . . . . . . 10  |-  ( ( y  =  ( F `
 w )  /\  z  =  ( F `  x ) )  -> 
( y ( 2nd `  S ) z )  =  ( ( F `
 w ) ( 2nd `  S ) ( F `  x
) ) )
26 oveq12 6031 . . . . . . . . . . 11  |-  ( ( z  =  ( F `
 x )  /\  y  =  ( F `  w ) )  -> 
( z ( 2nd `  S ) y )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  w
) ) )
2726ancoms 440 . . . . . . . . . 10  |-  ( ( y  =  ( F `
 w )  /\  z  =  ( F `  x ) )  -> 
( z ( 2nd `  S ) y )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  w
) ) )
2825, 27eqeq12d 2403 . . . . . . . . 9  |-  ( ( y  =  ( F `
 w )  /\  z  =  ( F `  x ) )  -> 
( ( y ( 2nd `  S ) z )  =  ( z ( 2nd `  S
) y )  <->  ( ( F `  w )
( 2nd `  S
) ( F `  x ) )  =  ( ( F `  x ) ( 2nd `  S ) ( F `
 w ) ) ) )
2924, 28syl5ibrcom 214 . . . . . . . 8  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( w  e.  X  /\  x  e.  X ) )  -> 
( ( y  =  ( F `  w
)  /\  z  =  ( F `  x ) )  ->  ( y
( 2nd `  S
) z )  =  ( z ( 2nd `  S ) y ) ) )
3029ex 424 . . . . . . 7  |-  ( ( R  e. CRingOps  /\  S  e.  RingOps 
/\  F  e.  ( R  RngHom  S ) )  ->  ( ( w  e.  X  /\  x  e.  X )  ->  (
( y  =  ( F `  w )  /\  z  =  ( F `  x ) )  ->  ( y
( 2nd `  S
) z )  =  ( z ( 2nd `  S ) y ) ) ) )
31303expa 1153 . . . . . 6  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  ->  ( (
w  e.  X  /\  x  e.  X )  ->  ( ( y  =  ( F `  w
)  /\  z  =  ( F `  x ) )  ->  ( y
( 2nd `  S
) z )  =  ( z ( 2nd `  S ) y ) ) ) )
3231adantrr 698 . . . . 5  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  -> 
( ( w  e.  X  /\  x  e.  X )  ->  (
( y  =  ( F `  w )  /\  z  =  ( F `  x ) )  ->  ( y
( 2nd `  S
) z )  =  ( z ( 2nd `  S ) y ) ) ) )
3332rexlimdvv 2781 . . . 4  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  -> 
( E. w  e.  X  E. x  e.  X  ( y  =  ( F `  w
)  /\  z  =  ( F `  x ) )  ->  ( y
( 2nd `  S
) z )  =  ( z ( 2nd `  S ) y ) ) )
349, 33syld 42 . . 3  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  -> 
( ( y  e.  Y  /\  z  e.  Y )  ->  (
y ( 2nd `  S
) z )  =  ( z ( 2nd `  S ) y ) ) )
3534ralrimivv 2742 . 2  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  ->  A. y  e.  Y  A. z  e.  Y  ( y ( 2nd `  S ) z )  =  ( z ( 2nd `  S ) y ) )
36 crnghomfo.3 . . 3  |-  J  =  ( 1st `  S
)
37 crnghomfo.4 . . 3  |-  Y  =  ran  J
3836, 18, 37iscrngo2 26301 . 2  |-  ( S  e. CRingOps 
<->  ( S  e.  RingOps  /\  A. y  e.  Y  A. z  e.  Y  (
y ( 2nd `  S
) z )  =  ( z ( 2nd `  S ) y ) ) )
391, 35, 38sylanbrc 646 1  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  ->  S  e. CRingOps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2651   E.wrex 2652   ran crn 4821   -onto->wfo 5394   ` cfv 5396  (class class class)co 6022   1stc1st 6288   2ndc2nd 6289   RingOpscrngo 21813    RngHom crnghom 26269  CRingOpsccring 26298
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 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fo 5402  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-map 6958  df-rngo 21814  df-com2 21849  df-rngohom 26272  df-crngo 26299
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