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Theorem isrnghmmul 32843
Description: A function is a non-unital ring homomorphism iff it preserves both addition and multiplication. (Contributed by AV, 27-Feb-2020.)
Hypotheses
Ref Expression
isrnghmmul.m  |-  M  =  (mulGrp `  R )
isrnghmmul.n  |-  N  =  (mulGrp `  S )
Assertion
Ref Expression
isrnghmmul  |-  ( F  e.  ( R RngHomo  S
)  <->  ( ( R  e. Rng  /\  S  e. Rng )  /\  ( F  e.  ( R  GrpHom  S )  /\  F  e.  ( M MgmHom  N ) ) ) )

Proof of Theorem isrnghmmul
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2457 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2457 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
3 eqid 2457 . . 3  |-  ( .r
`  S )  =  ( .r `  S
)
41, 2, 3isrnghm 32842 . 2  |-  ( F  e.  ( R RngHomo  S
)  <->  ( ( R  e. Rng  /\  S  e. Rng )  /\  ( F  e.  ( R  GrpHom  S )  /\  A. x  e.  ( Base `  R
) A. y  e.  ( Base `  R
) ( F `  ( x ( .r
`  R ) y ) )  =  ( ( F `  x
) ( .r `  S ) ( F `
 y ) ) ) ) )
5 isrnghmmul.m . . . . . . . . . . 11  |-  M  =  (mulGrp `  R )
65rngmgp 32828 . . . . . . . . . 10  |-  ( R  e. Rng  ->  M  e. SGrp )
7 sgrpmgm 16043 . . . . . . . . . 10  |-  ( M  e. SGrp  ->  M  e. Mgm )
86, 7syl 16 . . . . . . . . 9  |-  ( R  e. Rng  ->  M  e. Mgm )
9 isrnghmmul.n . . . . . . . . . . 11  |-  N  =  (mulGrp `  S )
109rngmgp 32828 . . . . . . . . . 10  |-  ( S  e. Rng  ->  N  e. SGrp )
11 sgrpmgm 16043 . . . . . . . . . 10  |-  ( N  e. SGrp  ->  N  e. Mgm )
1210, 11syl 16 . . . . . . . . 9  |-  ( S  e. Rng  ->  N  e. Mgm )
138, 12anim12i 566 . . . . . . . 8  |-  ( ( R  e. Rng  /\  S  e. Rng )  ->  ( M  e. Mgm  /\  N  e. Mgm )
)
14 eqid 2457 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
151, 14ghmf 16398 . . . . . . . 8  |-  ( F  e.  ( R  GrpHom  S )  ->  F :
( Base `  R ) --> ( Base `  S )
)
1613, 15anim12i 566 . . . . . . 7  |-  ( ( ( R  e. Rng  /\  S  e. Rng )  /\  F  e.  ( R  GrpHom  S ) )  -> 
( ( M  e. Mgm  /\  N  e. Mgm )  /\  F : ( Base `  R ) --> ( Base `  S ) ) )
1716biantrurd 508 . . . . . 6  |-  ( ( ( R  e. Rng  /\  S  e. Rng )  /\  F  e.  ( R  GrpHom  S ) )  -> 
( A. x  e.  ( Base `  R
) A. y  e.  ( Base `  R
) ( F `  ( x ( .r
`  R ) y ) )  =  ( ( F `  x
) ( .r `  S ) ( F `
 y ) )  <-> 
( ( ( M  e. Mgm  /\  N  e. Mgm )  /\  F : (
Base `  R ) --> ( Base `  S )
)  /\  A. x  e.  ( Base `  R
) A. y  e.  ( Base `  R
) ( F `  ( x ( .r
`  R ) y ) )  =  ( ( F `  x
) ( .r `  S ) ( F `
 y ) ) ) ) )
18 anass 649 . . . . . 6  |-  ( ( ( ( M  e. Mgm  /\  N  e. Mgm )  /\  F : ( Base `  R ) --> ( Base `  S ) )  /\  A. x  e.  ( Base `  R ) A. y  e.  ( Base `  R
) ( F `  ( x ( .r
`  R ) y ) )  =  ( ( F `  x
) ( .r `  S ) ( F `
 y ) ) )  <->  ( ( M  e. Mgm  /\  N  e. Mgm )  /\  ( F :
( Base `  R ) --> ( Base `  S )  /\  A. x  e.  (
Base `  R ) A. y  e.  ( Base `  R ) ( F `  ( x ( .r `  R
) y ) )  =  ( ( F `
 x ) ( .r `  S ) ( F `  y
) ) ) ) )
1917, 18syl6bb 261 . . . . 5  |-  ( ( ( R  e. Rng  /\  S  e. Rng )  /\  F  e.  ( R  GrpHom  S ) )  -> 
( A. x  e.  ( Base `  R
) A. y  e.  ( Base `  R
) ( F `  ( x ( .r
`  R ) y ) )  =  ( ( F `  x
) ( .r `  S ) ( F `
 y ) )  <-> 
( ( M  e. Mgm  /\  N  e. Mgm )  /\  ( F : (
Base `  R ) --> ( Base `  S )  /\  A. x  e.  (
Base `  R ) A. y  e.  ( Base `  R ) ( F `  ( x ( .r `  R
) y ) )  =  ( ( F `
 x ) ( .r `  S ) ( F `  y
) ) ) ) ) )
205, 1mgpbas 17274 . . . . . 6  |-  ( Base `  R )  =  (
Base `  M )
219, 14mgpbas 17274 . . . . . 6  |-  ( Base `  S )  =  (
Base `  N )
225, 2mgpplusg 17272 . . . . . 6  |-  ( .r
`  R )  =  ( +g  `  M
)
239, 3mgpplusg 17272 . . . . . 6  |-  ( .r
`  S )  =  ( +g  `  N
)
2420, 21, 22, 23ismgmhm 32733 . . . . 5  |-  ( F  e.  ( M MgmHom  N
)  <->  ( ( M  e. Mgm  /\  N  e. Mgm )  /\  ( F :
( Base `  R ) --> ( Base `  S )  /\  A. x  e.  (
Base `  R ) A. y  e.  ( Base `  R ) ( F `  ( x ( .r `  R
) y ) )  =  ( ( F `
 x ) ( .r `  S ) ( F `  y
) ) ) ) )
2519, 24syl6bbr 263 . . . 4  |-  ( ( ( R  e. Rng  /\  S  e. Rng )  /\  F  e.  ( R  GrpHom  S ) )  -> 
( A. x  e.  ( Base `  R
) A. y  e.  ( Base `  R
) ( F `  ( x ( .r
`  R ) y ) )  =  ( ( F `  x
) ( .r `  S ) ( F `
 y ) )  <-> 
F  e.  ( M MgmHom  N ) ) )
2625pm5.32da 641 . . 3  |-  ( ( R  e. Rng  /\  S  e. Rng )  ->  ( ( F  e.  ( R  GrpHom  S )  /\  A. x  e.  ( Base `  R ) A. y  e.  ( Base `  R
) ( F `  ( x ( .r
`  R ) y ) )  =  ( ( F `  x
) ( .r `  S ) ( F `
 y ) ) )  <->  ( F  e.  ( R  GrpHom  S )  /\  F  e.  ( M MgmHom  N ) ) ) )
2726pm5.32i 637 . 2  |-  ( ( ( R  e. Rng  /\  S  e. Rng )  /\  ( F  e.  ( R  GrpHom  S )  /\  A. x  e.  ( Base `  R ) A. y  e.  ( Base `  R
) ( F `  ( x ( .r
`  R ) y ) )  =  ( ( F `  x
) ( .r `  S ) ( F `
 y ) ) ) )  <->  ( ( R  e. Rng  /\  S  e. Rng )  /\  ( F  e.  ( R  GrpHom  S )  /\  F  e.  ( M MgmHom  N ) ) ) )
284, 27bitri 249 1  |-  ( F  e.  ( R RngHomo  S
)  <->  ( ( R  e. Rng  /\  S  e. Rng )  /\  ( F  e.  ( R  GrpHom  S )  /\  F  e.  ( M MgmHom  N ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   -->wf 5590   ` cfv 5594  (class class class)co 6296   Basecbs 14644   .rcmulr 14713  Mgmcmgm 15997  SGrpcsgrp 16037    GrpHom cghm 16391  mulGrpcmgp 17268   MgmHom cmgmhm 32727  Rngcrng 32824   RngHomo crngh 32835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-plusg 14725  df-sgrp 16038  df-ghm 16392  df-abl 16928  df-mgp 17269  df-mgmhm 32729  df-rng0 32825  df-rnghomo 32837
This theorem is referenced by:  rnghmmgmhm  32844  rnghmval2  32845  rnghmf1o  32853  rnghmco  32857  idrnghm  32858  c0rnghm  32863  rhmisrnghm  32870
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