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Theorem isrhm 15829
Description: A function is a ring homomorphism iff it preserves both addition and multiplication. (Contributed by Stefan O'Rear, 7-Mar-2015.)
Hypotheses
Ref Expression
isrhm.m  |-  M  =  (mulGrp `  R )
isrhm.n  |-  N  =  (mulGrp `  S )
Assertion
Ref Expression
isrhm  |-  ( F  e.  ( R RingHom  S
)  <->  ( ( R  e.  Ring  /\  S  e. 
Ring )  /\  ( F  e.  ( R  GrpHom  S )  /\  F  e.  ( M MndHom  N ) ) ) )

Proof of Theorem isrhm
Dummy variables  r 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfrhm2 15826 . . 3  |- RingHom  =  ( r  e.  Ring ,  s  e.  Ring  |->  ( ( r  GrpHom  s )  i^i  ( (mulGrp `  r
) MndHom  (mulGrp `  s )
) ) )
21elmpt2cl 6291 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  ( R  e.  Ring  /\  S  e.  Ring ) )
3 oveq12 6093 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( r  GrpHom  s )  =  ( R  GrpHom  S ) )
4 fveq2 5731 . . . . . . 7  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
5 fveq2 5731 . . . . . . 7  |-  ( s  =  S  ->  (mulGrp `  s )  =  (mulGrp `  S ) )
64, 5oveqan12d 6103 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( (mulGrp `  r
) MndHom  (mulGrp `  s )
)  =  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) ) )
73, 6ineq12d 3545 . . . . 5  |-  ( ( r  =  R  /\  s  =  S )  ->  ( ( r  GrpHom  s )  i^i  ( (mulGrp `  r ) MndHom  (mulGrp `  s ) ) )  =  ( ( R 
GrpHom  S )  i^i  (
(mulGrp `  R ) MndHom  (mulGrp `  S ) ) ) )
8 ovex 6109 . . . . . 6  |-  ( R 
GrpHom  S )  e.  _V
98inex1 4347 . . . . 5  |-  ( ( R  GrpHom  S )  i^i  ( (mulGrp `  R
) MndHom  (mulGrp `  S )
) )  e.  _V
107, 1, 9ovmpt2a 6207 . . . 4  |-  ( ( R  e.  Ring  /\  S  e.  Ring )  ->  ( R RingHom  S )  =  ( ( R  GrpHom  S )  i^i  ( (mulGrp `  R ) MndHom  (mulGrp `  S
) ) ) )
1110eleq2d 2505 . . 3  |-  ( ( R  e.  Ring  /\  S  e.  Ring )  ->  ( F  e.  ( R RingHom  S )  <->  F  e.  (
( R  GrpHom  S )  i^i  ( (mulGrp `  R ) MndHom  (mulGrp `  S
) ) ) ) )
12 elin 3532 . . . 4  |-  ( F  e.  ( ( R 
GrpHom  S )  i^i  (
(mulGrp `  R ) MndHom  (mulGrp `  S ) ) )  <-> 
( F  e.  ( R  GrpHom  S )  /\  F  e.  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) ) ) )
13 isrhm.m . . . . . . . 8  |-  M  =  (mulGrp `  R )
14 isrhm.n . . . . . . . 8  |-  N  =  (mulGrp `  S )
1513, 14oveq12i 6096 . . . . . . 7  |-  ( M MndHom  N )  =  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) )
1615eqcomi 2442 . . . . . 6  |-  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) )  =  ( M MndHom  N )
1716eleq2i 2502 . . . . 5  |-  ( F  e.  ( (mulGrp `  R ) MndHom  (mulGrp `  S
) )  <->  F  e.  ( M MndHom  N ) )
1817anbi2i 677 . . . 4  |-  ( ( F  e.  ( R 
GrpHom  S )  /\  F  e.  ( (mulGrp `  R
) MndHom  (mulGrp `  S )
) )  <->  ( F  e.  ( R  GrpHom  S )  /\  F  e.  ( M MndHom  N ) ) )
1912, 18bitri 242 . . 3  |-  ( F  e.  ( ( R 
GrpHom  S )  i^i  (
(mulGrp `  R ) MndHom  (mulGrp `  S ) ) )  <-> 
( F  e.  ( R  GrpHom  S )  /\  F  e.  ( M MndHom  N ) ) )
2011, 19syl6bb 254 . 2  |-  ( ( R  e.  Ring  /\  S  e.  Ring )  ->  ( F  e.  ( R RingHom  S )  <->  ( F  e.  ( R  GrpHom  S )  /\  F  e.  ( M MndHom  N ) ) ) )
212, 20biadan2 625 1  |-  ( F  e.  ( R RingHom  S
)  <->  ( ( R  e.  Ring  /\  S  e. 
Ring )  /\  ( F  e.  ( R  GrpHom  S )  /\  F  e.  ( M MndHom  N ) ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    i^i cin 3321   ` cfv 5457  (class class class)co 6084   MndHom cmhm 14741    GrpHom cghm 15008  mulGrpcmgp 15653   Ringcrg 15665   RingHom crh 15822
This theorem is referenced by:  rhmmhm  15830  rhmghm  15831  isrhm2d  15834  rhmco  15837  pwsco1rhm  15838  pwsco2rhm  15839  resrhm  15902  pwsdiagrhm  15906  rhmpropd  15908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-plusg 13547  df-0g 13732  df-mhm 14743  df-ghm 15009  df-mgp 15654  df-rng 15668  df-ur 15670  df-rnghom 15824
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