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Theorem rhmpropd 16900
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 16676 . . . . 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 16676 . . . . 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 15784 . . . . . 6  |-  ( ph  ->  ( J  GrpHom  K )  =  ( L  GrpHom  M ) )
1312eleq2d 2510 . . . . 5  |-  ( ph  ->  ( f  e.  ( J  GrpHom  K )  <->  f  e.  ( L  GrpHom  M ) ) )
14 eqid 2443 . . . . . . . . 9  |-  (mulGrp `  J )  =  (mulGrp `  J )
15 eqid 2443 . . . . . . . . 9  |-  ( Base `  J )  =  (
Base `  J )
1614, 15mgpbas 16597 . . . . . . . 8  |-  ( Base `  J )  =  (
Base `  (mulGrp `  J
) )
171, 16syl6eq 2491 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  (mulGrp `  J )
) )
18 eqid 2443 . . . . . . . . 9  |-  (mulGrp `  K )  =  (mulGrp `  K )
19 eqid 2443 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2018, 19mgpbas 16597 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  (mulGrp `  K
) )
216, 20syl6eq 2491 . . . . . . 7  |-  ( ph  ->  C  =  ( Base `  (mulGrp `  K )
) )
22 eqid 2443 . . . . . . . . 9  |-  (mulGrp `  L )  =  (mulGrp `  L )
23 eqid 2443 . . . . . . . . 9  |-  ( Base `  L )  =  (
Base `  L )
2422, 23mgpbas 16597 . . . . . . . 8  |-  ( Base `  L )  =  (
Base `  (mulGrp `  L
) )
252, 24syl6eq 2491 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  (mulGrp `  L )
) )
26 eqid 2443 . . . . . . . . 9  |-  (mulGrp `  M )  =  (mulGrp `  M )
27 eqid 2443 . . . . . . . . 9  |-  ( Base `  M )  =  (
Base `  M )
2826, 27mgpbas 16597 . . . . . . . 8  |-  ( Base `  M )  =  (
Base `  (mulGrp `  M
) )
297, 28syl6eq 2491 . . . . . . 7  |-  ( ph  ->  C  =  ( Base `  (mulGrp `  M )
) )
30 eqid 2443 . . . . . . . . . 10  |-  ( .r
`  J )  =  ( .r `  J
)
3114, 30mgpplusg 16595 . . . . . . . . 9  |-  ( .r
`  J )  =  ( +g  `  (mulGrp `  J ) )
3231oveqi 6104 . . . . . . . 8  |-  ( x ( .r `  J
) y )  =  ( x ( +g  `  (mulGrp `  J )
) y )
33 eqid 2443 . . . . . . . . . 10  |-  ( .r
`  L )  =  ( .r `  L
)
3422, 33mgpplusg 16595 . . . . . . . . 9  |-  ( .r
`  L )  =  ( +g  `  (mulGrp `  L ) )
3534oveqi 6104 . . . . . . . 8  |-  ( x ( .r `  L
) y )  =  ( x ( +g  `  (mulGrp `  L )
) y )
364, 32, 353eqtr3g 2498 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  (mulGrp `  J )
) y )  =  ( x ( +g  `  (mulGrp `  L )
) y ) )
37 eqid 2443 . . . . . . . . . 10  |-  ( .r
`  K )  =  ( .r `  K
)
3818, 37mgpplusg 16595 . . . . . . . . 9  |-  ( .r
`  K )  =  ( +g  `  (mulGrp `  K ) )
3938oveqi 6104 . . . . . . . 8  |-  ( x ( .r `  K
) y )  =  ( x ( +g  `  (mulGrp `  K )
) y )
40 eqid 2443 . . . . . . . . . 10  |-  ( .r
`  M )  =  ( .r `  M
)
4126, 40mgpplusg 16595 . . . . . . . . 9  |-  ( .r
`  M )  =  ( +g  `  (mulGrp `  M ) )
4241oveqi 6104 . . . . . . . 8  |-  ( x ( .r `  M
) y )  =  ( x ( +g  `  (mulGrp `  M )
) y )
439, 39, 423eqtr3g 2498 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  C ) )  -> 
( x ( +g  `  (mulGrp `  K )
) y )  =  ( x ( +g  `  (mulGrp `  M )
) y ) )
4417, 21, 25, 29, 36, 43mhmpropd 15470 . . . . . 6  |-  ( ph  ->  ( (mulGrp `  J
) MndHom  (mulGrp `  K )
)  =  ( (mulGrp `  L ) MndHom  (mulGrp `  M ) ) )
4544eleq2d 2510 . . . . 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 16811 . . 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 16811 . . 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 2441 1  |-  ( ph  ->  ( J RingHom  K )  =  ( L RingHom  M
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   ` cfv 5418  (class class class)co 6091   Basecbs 14174   +g cplusg 14238   .rcmulr 14239   MndHom cmhm 15462    GrpHom cghm 15744  mulGrpcmgp 16591   Ringcrg 16645   RingHom crh 16804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-recs 6832  df-rdg 6866  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-plusg 14251  df-0g 14380  df-mnd 15415  df-mhm 15464  df-grp 15545  df-ghm 15745  df-mgp 16592  df-ur 16604  df-rng 16647  df-rnghom 16806
This theorem is referenced by:  evls1rhm  17757  evl1rhm  17766  zrhpropd  17946
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