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Theorem rhmpropd 17591
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, 4ringpropd 17357 . . . . 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, 9ringpropd 17357 . . . . 5  |-  ( ph  ->  ( K  e.  Ring  <->  M  e.  Ring ) )
115, 10anbi12d 710 . . . 4  |-  ( ph  ->  ( ( J  e. 
Ring  /\  K  e.  Ring ) 
<->  ( L  e.  Ring  /\  M  e.  Ring )
) )
121, 6, 2, 7, 3, 8ghmpropd 16431 . . . . . 6  |-  ( ph  ->  ( J  GrpHom  K )  =  ( L  GrpHom  M ) )
1312eleq2d 2527 . . . . 5  |-  ( ph  ->  ( f  e.  ( J  GrpHom  K )  <->  f  e.  ( L  GrpHom  M ) ) )
14 eqid 2457 . . . . . . . . 9  |-  (mulGrp `  J )  =  (mulGrp `  J )
15 eqid 2457 . . . . . . . . 9  |-  ( Base `  J )  =  (
Base `  J )
1614, 15mgpbas 17274 . . . . . . . 8  |-  ( Base `  J )  =  (
Base `  (mulGrp `  J
) )
171, 16syl6eq 2514 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  (mulGrp `  J )
) )
18 eqid 2457 . . . . . . . . 9  |-  (mulGrp `  K )  =  (mulGrp `  K )
19 eqid 2457 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2018, 19mgpbas 17274 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  (mulGrp `  K
) )
216, 20syl6eq 2514 . . . . . . 7  |-  ( ph  ->  C  =  ( Base `  (mulGrp `  K )
) )
22 eqid 2457 . . . . . . . . 9  |-  (mulGrp `  L )  =  (mulGrp `  L )
23 eqid 2457 . . . . . . . . 9  |-  ( Base `  L )  =  (
Base `  L )
2422, 23mgpbas 17274 . . . . . . . 8  |-  ( Base `  L )  =  (
Base `  (mulGrp `  L
) )
252, 24syl6eq 2514 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  (mulGrp `  L )
) )
26 eqid 2457 . . . . . . . . 9  |-  (mulGrp `  M )  =  (mulGrp `  M )
27 eqid 2457 . . . . . . . . 9  |-  ( Base `  M )  =  (
Base `  M )
2826, 27mgpbas 17274 . . . . . . . 8  |-  ( Base `  M )  =  (
Base `  (mulGrp `  M
) )
297, 28syl6eq 2514 . . . . . . 7  |-  ( ph  ->  C  =  ( Base `  (mulGrp `  M )
) )
30 eqid 2457 . . . . . . . . . 10  |-  ( .r
`  J )  =  ( .r `  J
)
3114, 30mgpplusg 17272 . . . . . . . . 9  |-  ( .r
`  J )  =  ( +g  `  (mulGrp `  J ) )
3231oveqi 6309 . . . . . . . 8  |-  ( x ( .r `  J
) y )  =  ( x ( +g  `  (mulGrp `  J )
) y )
33 eqid 2457 . . . . . . . . . 10  |-  ( .r
`  L )  =  ( .r `  L
)
3422, 33mgpplusg 17272 . . . . . . . . 9  |-  ( .r
`  L )  =  ( +g  `  (mulGrp `  L ) )
3534oveqi 6309 . . . . . . . 8  |-  ( x ( .r `  L
) y )  =  ( x ( +g  `  (mulGrp `  L )
) y )
364, 32, 353eqtr3g 2521 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  (mulGrp `  J )
) y )  =  ( x ( +g  `  (mulGrp `  L )
) y ) )
37 eqid 2457 . . . . . . . . . 10  |-  ( .r
`  K )  =  ( .r `  K
)
3818, 37mgpplusg 17272 . . . . . . . . 9  |-  ( .r
`  K )  =  ( +g  `  (mulGrp `  K ) )
3938oveqi 6309 . . . . . . . 8  |-  ( x ( .r `  K
) y )  =  ( x ( +g  `  (mulGrp `  K )
) y )
40 eqid 2457 . . . . . . . . . 10  |-  ( .r
`  M )  =  ( .r `  M
)
4126, 40mgpplusg 17272 . . . . . . . . 9  |-  ( .r
`  M )  =  ( +g  `  (mulGrp `  M ) )
4241oveqi 6309 . . . . . . . 8  |-  ( x ( .r `  M
) y )  =  ( x ( +g  `  (mulGrp `  M )
) y )
439, 39, 423eqtr3g 2521 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  C ) )  -> 
( x ( +g  `  (mulGrp `  K )
) y )  =  ( x ( +g  `  (mulGrp `  M )
) y ) )
4417, 21, 25, 29, 36, 43mhmpropd 16099 . . . . . 6  |-  ( ph  ->  ( (mulGrp `  J
) MndHom  (mulGrp `  K )
)  =  ( (mulGrp `  L ) MndHom  (mulGrp `  M ) ) )
4544eleq2d 2527 . . . . 5  |-  ( ph  ->  ( f  e.  ( (mulGrp `  J ) MndHom  (mulGrp `  K ) )  <->  f  e.  ( (mulGrp `  L ) MndHom  (mulGrp `  M ) ) ) )
4613, 45anbi12d 710 . . . 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 710 . . 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 17497 . . 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 17497 . . 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 2454 1  |-  ( ph  ->  ( J RingHom  K )  =  ( L RingHom  M
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   ` cfv 5594  (class class class)co 6296   Basecbs 14644   +g cplusg 14712   .rcmulr 14713   MndHom cmhm 16091    GrpHom cghm 16391  mulGrpcmgp 17268   Ringcrg 17325   RingHom crh 17488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-plusg 14725  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mhm 16093  df-grp 16184  df-ghm 16392  df-mgp 17269  df-ur 17281  df-ring 17327  df-rnghom 17491
This theorem is referenced by:  evls1rhm  18486  evl1rhm  18495  zrhpropd  18679
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