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Theorem isrhm 15517
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 15514 . . 3  |- RingHom  =  ( r  e.  Ring ,  s  e.  Ring  |->  ( ( r  GrpHom  s )  i^i  ( (mulGrp `  r
) MndHom  (mulGrp `  s )
) ) )
21elmpt2cl 6077 . 2  |-  ( F  e.  ( R RingHom  S
)  ->  ( R  e.  Ring  /\  S  e.  Ring ) )
3 oveq12 5883 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( r  GrpHom  s )  =  ( R  GrpHom  S ) )
4 fveq2 5541 . . . . . . 7  |-  ( r  =  R  ->  (mulGrp `  r )  =  (mulGrp `  R ) )
5 fveq2 5541 . . . . . . 7  |-  ( s  =  S  ->  (mulGrp `  s )  =  (mulGrp `  S ) )
64, 5oveqan12d 5893 . . . . . 6  |-  ( ( r  =  R  /\  s  =  S )  ->  ( (mulGrp `  r
) MndHom  (mulGrp `  s )
)  =  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) ) )
73, 6ineq12d 3384 . . . . 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 5899 . . . . . 6  |-  ( R 
GrpHom  S )  e.  _V
98inex1 4171 . . . . 5  |-  ( ( R  GrpHom  S )  i^i  ( (mulGrp `  R
) MndHom  (mulGrp `  S )
) )  e.  _V
107, 1, 9ovmpt2a 5994 . . . 4  |-  ( ( R  e.  Ring  /\  S  e.  Ring )  ->  ( R RingHom  S )  =  ( ( R  GrpHom  S )  i^i  ( (mulGrp `  R ) MndHom  (mulGrp `  S
) ) ) )
1110eleq2d 2363 . . 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 3371 . . . 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 5886 . . . . . . 7  |-  ( M MndHom  N )  =  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) )
1615eqcomi 2300 . . . . . 6  |-  ( (mulGrp `  R ) MndHom  (mulGrp `  S ) )  =  ( M MndHom  N )
1716eleq2i 2360 . . . . 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 1632    e. wcel 1696    i^i cin 3164   ` cfv 5271  (class class class)co 5874   MndHom cmhm 14429    GrpHom cghm 14696  mulGrpcmgp 15341   Ringcrg 15353   RingHom crh 15510
This theorem is referenced by:  rhmmhm  15518  rhmghm  15519  isrhm2d  15522  rhmco  15525  pwsco1rhm  15526  pwsco2rhm  15527  resrhm  15590  pwsdiagrhm  15594  rhmpropd  15596
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-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-mhm 14431  df-ghm 14697  df-mgp 15342  df-rng 15356  df-ur 15358  df-rnghom 15512
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