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Theorem rhmpropd 16880
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 16666 . . . . 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 16666 . . . . 5  |-  ( ph  ->  ( K  e.  Ring  <->  M  e.  Ring ) )
115, 10anbi12d 705 . . . 4  |-  ( ph  ->  ( ( J  e. 
Ring  /\  K  e.  Ring ) 
<->  ( L  e.  Ring  /\  M  e.  Ring )
) )
121, 6, 2, 7, 3, 8ghmpropd 15777 . . . . . 6  |-  ( ph  ->  ( J  GrpHom  K )  =  ( L  GrpHom  M ) )
1312eleq2d 2508 . . . . 5  |-  ( ph  ->  ( f  e.  ( J  GrpHom  K )  <->  f  e.  ( L  GrpHom  M ) ) )
14 eqid 2441 . . . . . . . . 9  |-  (mulGrp `  J )  =  (mulGrp `  J )
15 eqid 2441 . . . . . . . . 9  |-  ( Base `  J )  =  (
Base `  J )
1614, 15mgpbas 16587 . . . . . . . 8  |-  ( Base `  J )  =  (
Base `  (mulGrp `  J
) )
171, 16syl6eq 2489 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  (mulGrp `  J )
) )
18 eqid 2441 . . . . . . . . 9  |-  (mulGrp `  K )  =  (mulGrp `  K )
19 eqid 2441 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2018, 19mgpbas 16587 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  (mulGrp `  K
) )
216, 20syl6eq 2489 . . . . . . 7  |-  ( ph  ->  C  =  ( Base `  (mulGrp `  K )
) )
22 eqid 2441 . . . . . . . . 9  |-  (mulGrp `  L )  =  (mulGrp `  L )
23 eqid 2441 . . . . . . . . 9  |-  ( Base `  L )  =  (
Base `  L )
2422, 23mgpbas 16587 . . . . . . . 8  |-  ( Base `  L )  =  (
Base `  (mulGrp `  L
) )
252, 24syl6eq 2489 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  (mulGrp `  L )
) )
26 eqid 2441 . . . . . . . . 9  |-  (mulGrp `  M )  =  (mulGrp `  M )
27 eqid 2441 . . . . . . . . 9  |-  ( Base `  M )  =  (
Base `  M )
2826, 27mgpbas 16587 . . . . . . . 8  |-  ( Base `  M )  =  (
Base `  (mulGrp `  M
) )
297, 28syl6eq 2489 . . . . . . 7  |-  ( ph  ->  C  =  ( Base `  (mulGrp `  M )
) )
30 eqid 2441 . . . . . . . . . 10  |-  ( .r
`  J )  =  ( .r `  J
)
3114, 30mgpplusg 16585 . . . . . . . . 9  |-  ( .r
`  J )  =  ( +g  `  (mulGrp `  J ) )
3231oveqi 6103 . . . . . . . 8  |-  ( x ( .r `  J
) y )  =  ( x ( +g  `  (mulGrp `  J )
) y )
33 eqid 2441 . . . . . . . . . 10  |-  ( .r
`  L )  =  ( .r `  L
)
3422, 33mgpplusg 16585 . . . . . . . . 9  |-  ( .r
`  L )  =  ( +g  `  (mulGrp `  L ) )
3534oveqi 6103 . . . . . . . 8  |-  ( x ( .r `  L
) y )  =  ( x ( +g  `  (mulGrp `  L )
) y )
364, 32, 353eqtr3g 2496 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  (mulGrp `  J )
) y )  =  ( x ( +g  `  (mulGrp `  L )
) y ) )
37 eqid 2441 . . . . . . . . . 10  |-  ( .r
`  K )  =  ( .r `  K
)
3818, 37mgpplusg 16585 . . . . . . . . 9  |-  ( .r
`  K )  =  ( +g  `  (mulGrp `  K ) )
3938oveqi 6103 . . . . . . . 8  |-  ( x ( .r `  K
) y )  =  ( x ( +g  `  (mulGrp `  K )
) y )
40 eqid 2441 . . . . . . . . . 10  |-  ( .r
`  M )  =  ( .r `  M
)
4126, 40mgpplusg 16585 . . . . . . . . 9  |-  ( .r
`  M )  =  ( +g  `  (mulGrp `  M ) )
4241oveqi 6103 . . . . . . . 8  |-  ( x ( .r `  M
) y )  =  ( x ( +g  `  (mulGrp `  M )
) y )
439, 39, 423eqtr3g 2496 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  C ) )  -> 
( x ( +g  `  (mulGrp `  K )
) y )  =  ( x ( +g  `  (mulGrp `  M )
) y ) )
4417, 21, 25, 29, 36, 43mhmpropd 15466 . . . . . 6  |-  ( ph  ->  ( (mulGrp `  J
) MndHom  (mulGrp `  K )
)  =  ( (mulGrp `  L ) MndHom  (mulGrp `  M ) ) )
4544eleq2d 2508 . . . . 5  |-  ( ph  ->  ( f  e.  ( (mulGrp `  J ) MndHom  (mulGrp `  K ) )  <->  f  e.  ( (mulGrp `  L ) MndHom  (mulGrp `  M ) ) ) )
4613, 45anbi12d 705 . . . 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 705 . . 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 16801 . . 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 16801 . . 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 288 . 2  |-  ( ph  ->  ( f  e.  ( J RingHom  K )  <->  f  e.  ( L RingHom  M ) ) )
5150eqrdv 2439 1  |-  ( ph  ->  ( J RingHom  K )  =  ( L RingHom  M
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   ` cfv 5415  (class class class)co 6090   Basecbs 14170   +g cplusg 14234   .rcmulr 14235   MndHom cmhm 15458    GrpHom cghm 15737  mulGrpcmgp 16581   Ringcrg 16635   RingHom crh 16794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-plusg 14247  df-0g 14376  df-mnd 15411  df-mhm 15460  df-grp 15538  df-ghm 15738  df-mgp 16582  df-ur 16594  df-rng 16637  df-rnghom 16796
This theorem is referenced by:  zrhpropd  17846  evl1rhm  21438
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