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Theorem crngohomfo 26631
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 5679 . . . . . . . 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 5679 . . . . . . . 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 2707 . . . . . 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 2283 . . . . . . . . . . . . . 14  |-  ( 2nd `  R )  =  ( 2nd `  R )
12 crnghomfo.2 . . . . . . . . . . . . . 14  |-  X  =  ran  G
1310, 11, 12crngocom 26626 . . . . . . . . . . . . 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 5529 . . . . . . . . . 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 26625 . . . . . . . . . . 11  |-  ( R  e. CRingOps  ->  R  e.  RingOps )
18 eqid 2283 . . . . . . . . . . . 12  |-  ( 2nd `  S )  =  ( 2nd `  S )
1910, 12, 11, 18rngohommul 26601 . . . . . . . . . . 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 26601 . . . . . . . . . . . 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 2323 . . . . . . . . 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 5867 . . . . . . . . . 10  |-  ( ( y  =  ( F `
 w )  /\  z  =  ( F `  x ) )  -> 
( y ( 2nd `  S ) z )  =  ( ( F `
 w ) ( 2nd `  S ) ( F `  x
) ) )
26 oveq12 5867 . . . . . . . . . . 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 2297 . . . . . . . . 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 2673 . . . 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 2634 . 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 26623 . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   ran crn 4690   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   RingOpscrngo 21042    RngHom crnghom 26591  CRingOpsccring 26620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-rngo 21043  df-com2 21078  df-rngohom 26594  df-crngo 26621
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