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Theorem isrhm 15816
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 15813 . . 3  |- RingHom  =  ( r  e.  Ring ,  s  e.  Ring  |->  ( ( r  GrpHom  s )  i^i  ( (mulGrp `  r
) MndHom  (mulGrp `  s )
) ) )
21elmpt2cl 6280 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  ( R  e.  Ring  /\  S  e.  Ring ) )
3 oveq12 6082 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( r  GrpHom  s )  =  ( R  GrpHom  S ) )
4 fveq2 5720 . . . . . . 7  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
5 fveq2 5720 . . . . . . 7  |-  ( s  =  S  ->  (mulGrp `  s )  =  (mulGrp `  S ) )
64, 5oveqan12d 6092 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( (mulGrp `  r
) MndHom  (mulGrp `  s )
)  =  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) ) )
73, 6ineq12d 3535 . . . . 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 6098 . . . . . 6  |-  ( R 
GrpHom  S )  e.  _V
98inex1 4336 . . . . 5  |-  ( ( R  GrpHom  S )  i^i  ( (mulGrp `  R
) MndHom  (mulGrp `  S )
) )  e.  _V
107, 1, 9ovmpt2a 6196 . . . 4  |-  ( ( R  e.  Ring  /\  S  e.  Ring )  ->  ( R RingHom  S )  =  ( ( R  GrpHom  S )  i^i  ( (mulGrp `  R ) MndHom  (mulGrp `  S
) ) ) )
1110eleq2d 2502 . . 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 3522 . . . 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 6085 . . . . . . 7  |-  ( M MndHom  N )  =  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) )
1615eqcomi 2439 . . . . . 6  |-  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) )  =  ( M MndHom  N )
1716eleq2i 2499 . . . . 5  |-  ( F  e.  ( (mulGrp `  R ) MndHom  (mulGrp `  S
) )  <->  F  e.  ( M MndHom  N ) )
1817anbi2i 676 . . . 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 241 . . 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 253 . 2  |-  ( ( R  e.  Ring  /\  S  e.  Ring )  ->  ( F  e.  ( R RingHom  S )  <->  ( F  e.  ( R  GrpHom  S )  /\  F  e.  ( M MndHom  N ) ) ) )
212, 20biadan2 624 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3311   ` cfv 5446  (class class class)co 6073   MndHom cmhm 14728    GrpHom cghm 14995  mulGrpcmgp 15640   Ringcrg 15652   RingHom crh 15809
This theorem is referenced by:  rhmmhm  15817  rhmghm  15818  isrhm2d  15821  rhmco  15824  pwsco1rhm  15825  pwsco2rhm  15826  resrhm  15889  pwsdiagrhm  15893  rhmpropd  15895
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-plusg 13534  df-0g 13719  df-mhm 14730  df-ghm 14996  df-mgp 15641  df-rng 15655  df-ur 15657  df-rnghom 15811
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