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Theorem crngohomfo 26734
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 731 . 2  |-  ( ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  ->  S  e.  RingOps )
2 foelrn 5695 . . . . . . . 8  |-  ( ( F : X -onto-> Y  /\  y  e.  Y
)  ->  E. w  e.  X  y  =  ( F `  w ) )
32ex 423 . . . . . . 7  |-  ( F : X -onto-> Y  -> 
( y  e.  Y  ->  E. w  e.  X  y  =  ( F `  w ) ) )
4 foelrn 5695 . . . . . . . 8  |-  ( ( F : X -onto-> Y  /\  z  e.  Y
)  ->  E. x  e.  X  z  =  ( F `  x ) )
54ex 423 . . . . . . 7  |-  ( F : X -onto-> Y  -> 
( z  e.  Y  ->  E. x  e.  X  z  =  ( F `  x ) ) )
63, 5anim12d 546 . . . . . 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 2720 . . . . . 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 218 . . . . 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 709 . . . 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 2296 . . . . . . . . . . . . . 14  |-  ( 2nd `  R )  =  ( 2nd `  R )
12 crnghomfo.2 . . . . . . . . . . . . . 14  |-  X  =  ran  G
1310, 11, 12crngocom 26729 . . . . . . . . . . . . 13  |-  ( ( R  e. CRingOps  /\  w  e.  X  /\  x  e.  X )  ->  (
w ( 2nd `  R
) x )  =  ( x ( 2nd `  R ) w ) )
14133expb 1152 . . . . . . . . . . . 12  |-  ( ( R  e. CRingOps  /\  (
w  e.  X  /\  x  e.  X )
)  ->  ( w
( 2nd `  R
) x )  =  ( x ( 2nd `  R ) w ) )
15143ad2antl1 1117 . . . . . . . . . . 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 5545 . . . . . . . . . 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 26728 . . . . . . . . . . 11  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
18 eqid 2296 . . . . . . . . . . . 12  |-  ( 2nd `  S )  =  ( 2nd `  S )
1910, 12, 11, 18rngohommul 26704 . . . . . . . . . . 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 1230 . . . . . . . . . 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 26704 . . . . . . . . . . . 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 777 . . . . . . . . . . 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 1230 . . . . . . . . . 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 2336 . . . . . . . . 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 5883 . . . . . . . . . 10  |-  ( ( y  =  ( F `
 w )  /\  z  =  ( F `  x ) )  -> 
( y ( 2nd `  S ) z )  =  ( ( F `
 w ) ( 2nd `  S ) ( F `  x
) ) )
26 oveq12 5883 . . . . . . . . . . 11  |-  ( ( z  =  ( F `
 x )  /\  y  =  ( F `  w ) )  -> 
( z ( 2nd `  S ) y )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  w
) ) )
2726ancoms 439 . . . . . . . . . 10  |-  ( ( y  =  ( F `
 w )  /\  z  =  ( F `  x ) )  -> 
( z ( 2nd `  S ) y )  =  ( ( F `
 x ) ( 2nd `  S ) ( F `  w
) ) )
2825, 27eqeq12d 2310 . . . . . . . . 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 213 . . . . . . . 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 423 . . . . . . 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 1151 . . . . . 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 697 . . . . 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 2686 . . . 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 40 . . 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 2647 . 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 26726 . 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 645 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   ran crn 4706   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   RingOpscrngo 21058    RngHom crnghom 26694  CRingOpsccring 26723
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-rab 2565  df-v 2803  df-sbc 3005  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-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-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-rngo 21059  df-com2 21094  df-rngohom 26697  df-crngo 26724
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