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Theorem isrhm 16745
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 16742 . . 3  |- RingHom  =  ( r  e.  Ring ,  s  e.  Ring  |->  ( ( r  GrpHom  s )  i^i  ( (mulGrp `  r
) MndHom  (mulGrp `  s )
) ) )
21elmpt2cl 6293 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  ( R  e.  Ring  /\  S  e.  Ring ) )
3 oveq12 6089 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( r  GrpHom  s )  =  ( R  GrpHom  S ) )
4 fveq2 5679 . . . . . . 7  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
5 fveq2 5679 . . . . . . 7  |-  ( s  =  S  ->  (mulGrp `  s )  =  (mulGrp `  S ) )
64, 5oveqan12d 6099 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( (mulGrp `  r
) MndHom  (mulGrp `  s )
)  =  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) ) )
73, 6ineq12d 3541 . . . . 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 6105 . . . . . 6  |-  ( R 
GrpHom  S )  e.  _V
98inex1 4421 . . . . 5  |-  ( ( R  GrpHom  S )  i^i  ( (mulGrp `  R
) MndHom  (mulGrp `  S )
) )  e.  _V
107, 1, 9ovmpt2a 6210 . . . 4  |-  ( ( R  e.  Ring  /\  S  e.  Ring )  ->  ( R RingHom  S )  =  ( ( R  GrpHom  S )  i^i  ( (mulGrp `  R ) MndHom  (mulGrp `  S
) ) ) )
1110eleq2d 2500 . . 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 3527 . . . 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 6092 . . . . . . 7  |-  ( M MndHom  N )  =  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) )
1615eqcomi 2437 . . . . . 6  |-  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) )  =  ( M MndHom  N )
1716eleq2i 2497 . . . . 5  |-  ( F  e.  ( (mulGrp `  R ) MndHom  (mulGrp `  S
) )  <->  F  e.  ( M MndHom  N ) )
1817anbi2i 687 . . . 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 249 . . 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 261 . 2  |-  ( ( R  e.  Ring  /\  S  e.  Ring )  ->  ( F  e.  ( R RingHom  S )  <->  ( F  e.  ( R  GrpHom  S )  /\  F  e.  ( M MndHom  N ) ) ) )
212, 20biadan2 635 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 setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1362    e. wcel 1755    i^i cin 3315   ` cfv 5406  (class class class)co 6080   MndHom cmhm 15445    GrpHom cghm 15724  mulGrpcmgp 16565   Ringcrg 16577   RingHom crh 16738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-recs 6818  df-rdg 6852  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-plusg 14234  df-0g 14363  df-mhm 15447  df-ghm 15725  df-mgp 16566  df-rng 16580  df-ur 16582  df-rnghom 16740
This theorem is referenced by:  rhmmhm  16746  rhmghm  16747  isrhm2d  16750  rhmco  16753  pwsco1rhm  16754  pwsco2rhm  16755  resrhm  16818  pwsdiagrhm  16822  rhmpropd  16824
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