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Theorem rhmpropd 15596
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 15388 . . . . 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 15388 . . . . 5  |-  ( ph  ->  ( K  e.  Ring  <->  M  e.  Ring ) )
115, 10anbi12d 691 . . . 4  |-  ( ph  ->  ( ( J  e. 
Ring  /\  K  e.  Ring ) 
<->  ( L  e.  Ring  /\  M  e.  Ring )
) )
121, 6, 2, 7, 3, 8ghmpropd 14736 . . . . . 6  |-  ( ph  ->  ( J  GrpHom  K )  =  ( L  GrpHom  M ) )
1312eleq2d 2363 . . . . 5  |-  ( ph  ->  ( f  e.  ( J  GrpHom  K )  <->  f  e.  ( L  GrpHom  M ) ) )
14 eqid 2296 . . . . . . . . 9  |-  (mulGrp `  J )  =  (mulGrp `  J )
15 eqid 2296 . . . . . . . . 9  |-  ( Base `  J )  =  (
Base `  J )
1614, 15mgpbas 15347 . . . . . . . 8  |-  ( Base `  J )  =  (
Base `  (mulGrp `  J
) )
171, 16syl6eq 2344 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  (mulGrp `  J )
) )
18 eqid 2296 . . . . . . . . 9  |-  (mulGrp `  K )  =  (mulGrp `  K )
19 eqid 2296 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
2018, 19mgpbas 15347 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  (mulGrp `  K
) )
216, 20syl6eq 2344 . . . . . . 7  |-  ( ph  ->  C  =  ( Base `  (mulGrp `  K )
) )
22 eqid 2296 . . . . . . . . 9  |-  (mulGrp `  L )  =  (mulGrp `  L )
23 eqid 2296 . . . . . . . . 9  |-  ( Base `  L )  =  (
Base `  L )
2422, 23mgpbas 15347 . . . . . . . 8  |-  ( Base `  L )  =  (
Base `  (mulGrp `  L
) )
252, 24syl6eq 2344 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  (mulGrp `  L )
) )
26 eqid 2296 . . . . . . . . 9  |-  (mulGrp `  M )  =  (mulGrp `  M )
27 eqid 2296 . . . . . . . . 9  |-  ( Base `  M )  =  (
Base `  M )
2826, 27mgpbas 15347 . . . . . . . 8  |-  ( Base `  M )  =  (
Base `  (mulGrp `  M
) )
297, 28syl6eq 2344 . . . . . . 7  |-  ( ph  ->  C  =  ( Base `  (mulGrp `  M )
) )
30 eqid 2296 . . . . . . . . . 10  |-  ( .r
`  J )  =  ( .r `  J
)
3114, 30mgpplusg 15345 . . . . . . . . 9  |-  ( .r
`  J )  =  ( +g  `  (mulGrp `  J ) )
3231oveqi 5887 . . . . . . . 8  |-  ( x ( .r `  J
) y )  =  ( x ( +g  `  (mulGrp `  J )
) y )
33 eqid 2296 . . . . . . . . . 10  |-  ( .r
`  L )  =  ( .r `  L
)
3422, 33mgpplusg 15345 . . . . . . . . 9  |-  ( .r
`  L )  =  ( +g  `  (mulGrp `  L ) )
3534oveqi 5887 . . . . . . . 8  |-  ( x ( .r `  L
) y )  =  ( x ( +g  `  (mulGrp `  L )
) y )
364, 32, 353eqtr3g 2351 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  (mulGrp `  J )
) y )  =  ( x ( +g  `  (mulGrp `  L )
) y ) )
37 eqid 2296 . . . . . . . . . 10  |-  ( .r
`  K )  =  ( .r `  K
)
3818, 37mgpplusg 15345 . . . . . . . . 9  |-  ( .r
`  K )  =  ( +g  `  (mulGrp `  K ) )
3938oveqi 5887 . . . . . . . 8  |-  ( x ( .r `  K
) y )  =  ( x ( +g  `  (mulGrp `  K )
) y )
40 eqid 2296 . . . . . . . . . 10  |-  ( .r
`  M )  =  ( .r `  M
)
4126, 40mgpplusg 15345 . . . . . . . . 9  |-  ( .r
`  M )  =  ( +g  `  (mulGrp `  M ) )
4241oveqi 5887 . . . . . . . 8  |-  ( x ( .r `  M
) y )  =  ( x ( +g  `  (mulGrp `  M )
) y )
439, 39, 423eqtr3g 2351 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  C ) )  -> 
( x ( +g  `  (mulGrp `  K )
) y )  =  ( x ( +g  `  (mulGrp `  M )
) y ) )
4417, 21, 25, 29, 36, 43mhmpropd 14437 . . . . . 6  |-  ( ph  ->  ( (mulGrp `  J
) MndHom  (mulGrp `  K )
)  =  ( (mulGrp `  L ) MndHom  (mulGrp `  M ) ) )
4544eleq2d 2363 . . . . 5  |-  ( ph  ->  ( f  e.  ( (mulGrp `  J ) MndHom  (mulGrp `  K ) )  <->  f  e.  ( (mulGrp `  L ) MndHom  (mulGrp `  M ) ) ) )
4613, 45anbi12d 691 . . . 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 691 . . 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 15517 . . 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 15517 . . 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 279 . 2  |-  ( ph  ->  ( f  e.  ( J RingHom  K )  <->  f  e.  ( L RingHom  M ) ) )
5150eqrdv 2294 1  |-  ( ph  ->  ( J RingHom  K )  =  ( L RingHom  M
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   .rcmulr 13225   MndHom cmhm 14429    GrpHom cghm 14696  mulGrpcmgp 15341   Ringcrg 15353   RingHom crh 15510
This theorem is referenced by:  zrhpropd  16485  evl1rhm  19428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-0g 13420  df-mnd 14383  df-mhm 14431  df-grp 14505  df-ghm 14697  df-mgp 15342  df-rng 15356  df-ur 15358  df-rnghom 15512
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