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Theorem isrhm 15501
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 15498 . . 3  |- RingHom  =  ( r  e.  Ring ,  s  e.  Ring  |->  ( ( r  GrpHom  s )  i^i  ( (mulGrp `  r
) MndHom  (mulGrp `  s )
) ) )
21elmpt2cl 6061 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  ( R  e.  Ring  /\  S  e.  Ring ) )
3 oveq12 5867 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( r  GrpHom  s )  =  ( R  GrpHom  S ) )
4 fveq2 5525 . . . . . . 7  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
5 fveq2 5525 . . . . . . 7  |-  ( s  =  S  ->  (mulGrp `  s )  =  (mulGrp `  S ) )
64, 5oveqan12d 5877 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( (mulGrp `  r
) MndHom  (mulGrp `  s )
)  =  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) ) )
73, 6ineq12d 3371 . . . . 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 5883 . . . . . 6  |-  ( R 
GrpHom  S )  e.  _V
98inex1 4155 . . . . 5  |-  ( ( R  GrpHom  S )  i^i  ( (mulGrp `  R
) MndHom  (mulGrp `  S )
) )  e.  _V
107, 1, 9ovmpt2a 5978 . . . 4  |-  ( ( R  e.  Ring  /\  S  e.  Ring )  ->  ( R RingHom  S )  =  ( ( R  GrpHom  S )  i^i  ( (mulGrp `  R ) MndHom  (mulGrp `  S
) ) ) )
1110eleq2d 2350 . . 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 3358 . . . 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 5870 . . . . . . 7  |-  ( M MndHom  N )  =  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) )
1615eqcomi 2287 . . . . . 6  |-  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) )  =  ( M MndHom  N )
1716eleq2i 2347 . . . . 5  |-  ( F  e.  ( (mulGrp `  R ) MndHom  (mulGrp `  S
) )  <->  F  e.  ( M MndHom  N ) )
1817anbi2i 675 . . . 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 240 . . 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 252 . 2  |-  ( ( R  e.  Ring  /\  S  e.  Ring )  ->  ( F  e.  ( R RingHom  S )  <->  ( F  e.  ( R  GrpHom  S )  /\  F  e.  ( M MndHom  N ) ) ) )
212, 20biadan2 623 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 176    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151   ` cfv 5255  (class class class)co 5858   MndHom cmhm 14413    GrpHom cghm 14680  mulGrpcmgp 15325   Ringcrg 15337   RingHom crh 15494
This theorem is referenced by:  rhmmhm  15502  rhmghm  15503  isrhm2d  15506  rhmco  15509  pwsco1rhm  15510  pwsco2rhm  15511  resrhm  15574  pwsdiagrhm  15578  rhmpropd  15580
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-0g 13404  df-mhm 14415  df-ghm 14681  df-mgp 15326  df-rng 15340  df-ur 15342  df-rnghom 15496
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