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Theorem rhmpropd 15823
Description: Ring homomorphism depends only on the ring attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.)
Hypotheses
Ref Expression
rhmpropd.a  |-  ( ph  ->  B  =  ( Base `  J ) )
rhmpropd.b  |-  ( ph  ->  C  =  ( Base `  K ) )
rhmpropd.c  |-  ( ph  ->  B  =  ( Base `  L ) )
rhmpropd.d  |-  ( ph  ->  C  =  ( Base `  M ) )
rhmpropd.e  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  J ) y )  =  ( x ( +g  `  L ) y ) )
rhmpropd.f  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  C ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  M ) y ) )
rhmpropd.g  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  J ) y )  =  ( x ( .r `  L
) y ) )
rhmpropd.h  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  C ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  M
) y ) )
Assertion
Ref Expression
rhmpropd  |-  ( ph  ->  ( J RingHom  K )  =  ( L RingHom  M
) )
Distinct variable groups:    x, y, J    x, K, y    x, L, y    x, M, y    ph, x, y    x, B, y    x, C, y

Proof of Theorem rhmpropd
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 rhmpropd.a . . . . . 6  |-  ( ph  ->  B  =  ( Base `  J ) )
2 rhmpropd.c . . . . . 6  |-  ( ph  ->  B  =  ( Base `  L ) )
3 rhmpropd.e . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  J ) y )  =  ( x ( +g  `  L ) y ) )
4 rhmpropd.g . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  J ) y )  =  ( x ( .r `  L
) y ) )
51, 2, 3, 4rngpropd 15615 . . . . 5  |-  ( ph  ->  ( J  e.  Ring  <->  L  e.  Ring ) )
6 rhmpropd.b . . . . . 6  |-  ( ph  ->  C  =  ( Base `  K ) )
7 rhmpropd.d . . . . . 6  |-  ( ph  ->  C  =  ( Base `  M ) )
8 rhmpropd.f . . . . . 6  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  C ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  M ) y ) )
9 rhmpropd.h . . . . . 6  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  C ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  M
) y ) )
106, 7, 8, 9rngpropd 15615 . . . . 5  |-  ( ph  ->  ( K  e.  Ring  <->  M  e.  Ring ) )
115, 10anbi12d 692 . . . 4  |-  ( ph  ->  ( ( J  e. 
Ring  /\  K  e.  Ring ) 
<->  ( L  e.  Ring  /\  M  e.  Ring )
) )
121, 6, 2, 7, 3, 8ghmpropd 14963 . . . . . 6  |-  ( ph  ->  ( J  GrpHom  K )  =  ( L  GrpHom  M ) )
1312eleq2d 2447 . . . . 5  |-  ( ph  ->  ( f  e.  ( J  GrpHom  K )  <->  f  e.  ( L  GrpHom  M ) ) )
14 eqid 2380 . . . . . . . . 9  |-  (mulGrp `  J )  =  (mulGrp `  J )
15 eqid 2380 . . . . . . . . 9  |-  ( Base `  J )  =  (
Base `  J )
1614, 15mgpbas 15574 . . . . . . . 8  |-  ( Base `  J )  =  (
Base `  (mulGrp `  J
) )
171, 16syl6eq 2428 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  (mulGrp `  J )
) )
18 eqid 2380 . . . . . . . . 9  |-  (mulGrp `  K )  =  (mulGrp `  K )
19 eqid 2380 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2018, 19mgpbas 15574 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  (mulGrp `  K
) )
216, 20syl6eq 2428 . . . . . . 7  |-  ( ph  ->  C  =  ( Base `  (mulGrp `  K )
) )
22 eqid 2380 . . . . . . . . 9  |-  (mulGrp `  L )  =  (mulGrp `  L )
23 eqid 2380 . . . . . . . . 9  |-  ( Base `  L )  =  (
Base `  L )
2422, 23mgpbas 15574 . . . . . . . 8  |-  ( Base `  L )  =  (
Base `  (mulGrp `  L
) )
252, 24syl6eq 2428 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  (mulGrp `  L )
) )
26 eqid 2380 . . . . . . . . 9  |-  (mulGrp `  M )  =  (mulGrp `  M )
27 eqid 2380 . . . . . . . . 9  |-  ( Base `  M )  =  (
Base `  M )
2826, 27mgpbas 15574 . . . . . . . 8  |-  ( Base `  M )  =  (
Base `  (mulGrp `  M
) )
297, 28syl6eq 2428 . . . . . . 7  |-  ( ph  ->  C  =  ( Base `  (mulGrp `  M )
) )
30 eqid 2380 . . . . . . . . . 10  |-  ( .r
`  J )  =  ( .r `  J
)
3114, 30mgpplusg 15572 . . . . . . . . 9  |-  ( .r
`  J )  =  ( +g  `  (mulGrp `  J ) )
3231oveqi 6026 . . . . . . . 8  |-  ( x ( .r `  J
) y )  =  ( x ( +g  `  (mulGrp `  J )
) y )
33 eqid 2380 . . . . . . . . . 10  |-  ( .r
`  L )  =  ( .r `  L
)
3422, 33mgpplusg 15572 . . . . . . . . 9  |-  ( .r
`  L )  =  ( +g  `  (mulGrp `  L ) )
3534oveqi 6026 . . . . . . . 8  |-  ( x ( .r `  L
) y )  =  ( x ( +g  `  (mulGrp `  L )
) y )
364, 32, 353eqtr3g 2435 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  (mulGrp `  J )
) y )  =  ( x ( +g  `  (mulGrp `  L )
) y ) )
37 eqid 2380 . . . . . . . . . 10  |-  ( .r
`  K )  =  ( .r `  K
)
3818, 37mgpplusg 15572 . . . . . . . . 9  |-  ( .r
`  K )  =  ( +g  `  (mulGrp `  K ) )
3938oveqi 6026 . . . . . . . 8  |-  ( x ( .r `  K
) y )  =  ( x ( +g  `  (mulGrp `  K )
) y )
40 eqid 2380 . . . . . . . . . 10  |-  ( .r
`  M )  =  ( .r `  M
)
4126, 40mgpplusg 15572 . . . . . . . . 9  |-  ( .r
`  M )  =  ( +g  `  (mulGrp `  M ) )
4241oveqi 6026 . . . . . . . 8  |-  ( x ( .r `  M
) y )  =  ( x ( +g  `  (mulGrp `  M )
) y )
439, 39, 423eqtr3g 2435 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  C ) )  -> 
( x ( +g  `  (mulGrp `  K )
) y )  =  ( x ( +g  `  (mulGrp `  M )
) y ) )
4417, 21, 25, 29, 36, 43mhmpropd 14664 . . . . . 6  |-  ( ph  ->  ( (mulGrp `  J
) MndHom  (mulGrp `  K )
)  =  ( (mulGrp `  L ) MndHom  (mulGrp `  M ) ) )
4544eleq2d 2447 . . . . 5  |-  ( ph  ->  ( f  e.  ( (mulGrp `  J ) MndHom  (mulGrp `  K ) )  <->  f  e.  ( (mulGrp `  L ) MndHom  (mulGrp `  M ) ) ) )
4613, 45anbi12d 692 . . . 4  |-  ( ph  ->  ( ( f  e.  ( J  GrpHom  K )  /\  f  e.  ( (mulGrp `  J ) MndHom  (mulGrp `  K ) ) )  <-> 
( f  e.  ( L  GrpHom  M )  /\  f  e.  ( (mulGrp `  L ) MndHom  (mulGrp `  M ) ) ) ) )
4711, 46anbi12d 692 . . 3  |-  ( ph  ->  ( ( ( J  e.  Ring  /\  K  e. 
Ring )  /\  (
f  e.  ( J 
GrpHom  K )  /\  f  e.  ( (mulGrp `  J
) MndHom  (mulGrp `  K )
) ) )  <->  ( ( L  e.  Ring  /\  M  e.  Ring )  /\  (
f  e.  ( L 
GrpHom  M )  /\  f  e.  ( (mulGrp `  L
) MndHom  (mulGrp `  M )
) ) ) ) )
4814, 18isrhm 15744 . . 3  |-  ( f  e.  ( J RingHom  K
)  <->  ( ( J  e.  Ring  /\  K  e. 
Ring )  /\  (
f  e.  ( J 
GrpHom  K )  /\  f  e.  ( (mulGrp `  J
) MndHom  (mulGrp `  K )
) ) ) )
4922, 26isrhm 15744 . . 3  |-  ( f  e.  ( L RingHom  M
)  <->  ( ( L  e.  Ring  /\  M  e. 
Ring )  /\  (
f  e.  ( L 
GrpHom  M )  /\  f  e.  ( (mulGrp `  L
) MndHom  (mulGrp `  M )
) ) ) )
5047, 48, 493bitr4g 280 . 2  |-  ( ph  ->  ( f  e.  ( J RingHom  K )  <->  f  e.  ( L RingHom  M ) ) )
5150eqrdv 2378 1  |-  ( ph  ->  ( J RingHom  K )  =  ( L RingHom  M
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   ` cfv 5387  (class class class)co 6013   Basecbs 13389   +g cplusg 13449   .rcmulr 13450   MndHom cmhm 14656    GrpHom cghm 14923  mulGrpcmgp 15568   Ringcrg 15580   RingHom crh 15737
This theorem is referenced by:  zrhpropd  16712  evl1rhm  19809
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-plusg 13462  df-0g 13647  df-mnd 14610  df-mhm 14658  df-grp 14732  df-ghm 14924  df-mgp 15569  df-rng 15583  df-ur 15585  df-rnghom 15739
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