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Theorem isrnghmmul 39946
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 2451 . . 3  |-  ( Base `  R )  =  (
Base `  R )
2 eqid 2451 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
3 eqid 2451 . . 3  |-  ( .r
`  S )  =  ( .r `  S
)
41, 2, 3isrnghm 39945 . 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 39931 . . . . . . . . . 10  |-  ( R  e. Rng  ->  M  e. SGrp )
7 sgrpmgm 16532 . . . . . . . . . 10  |-  ( M  e. SGrp  ->  M  e. Mgm )
86, 7syl 17 . . . . . . . . 9  |-  ( R  e. Rng  ->  M  e. Mgm )
9 isrnghmmul.n . . . . . . . . . . 11  |-  N  =  (mulGrp `  S )
109rngmgp 39931 . . . . . . . . . 10  |-  ( S  e. Rng  ->  N  e. SGrp )
11 sgrpmgm 16532 . . . . . . . . . 10  |-  ( N  e. SGrp  ->  N  e. Mgm )
1210, 11syl 17 . . . . . . . . 9  |-  ( S  e. Rng  ->  N  e. Mgm )
138, 12anim12i 570 . . . . . . . 8  |-  ( ( R  e. Rng  /\  S  e. Rng )  ->  ( M  e. Mgm  /\  N  e. Mgm )
)
14 eqid 2451 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
151, 14ghmf 16887 . . . . . . . 8  |-  ( F  e.  ( R  GrpHom  S )  ->  F :
( Base `  R ) --> ( Base `  S )
)
1613, 15anim12i 570 . . . . . . 7  |-  ( ( ( R  e. Rng  /\  S  e. Rng )  /\  F  e.  ( R  GrpHom  S ) )  -> 
( ( M  e. Mgm  /\  N  e. Mgm )  /\  F : ( Base `  R ) --> ( Base `  S ) ) )
1716biantrurd 511 . . . . . 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 655 . . . . . 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 265 . . . . 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 17729 . . . . . 6  |-  ( Base `  R )  =  (
Base `  M )
219, 14mgpbas 17729 . . . . . 6  |-  ( Base `  S )  =  (
Base `  N )
225, 2mgpplusg 17727 . . . . . 6  |-  ( .r
`  R )  =  ( +g  `  M
)
239, 3mgpplusg 17727 . . . . . 6  |-  ( .r
`  S )  =  ( +g  `  N
)
2420, 21, 22, 23ismgmhm 39836 . . . . 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 267 . . . 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 647 . . 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 643 . 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 253 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 188    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   -->wf 5578   ` cfv 5582  (class class class)co 6290   Basecbs 15121   .rcmulr 15191  Mgmcmgm 16486  SGrpcsgrp 16526    GrpHom cghm 16880  mulGrpcmgp 17723   MgmHom cmgmhm 39830  Rngcrng 39927   RngHomo crngh 39938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-plusg 15203  df-sgrp 16527  df-ghm 16881  df-abl 17433  df-mgp 17724  df-mgmhm 39832  df-rng0 39928  df-rnghomo 39940
This theorem is referenced by:  rnghmmgmhm  39947  rnghmval2  39948  rnghmf1o  39956  rnghmco  39960  idrnghm  39961  c0rnghm  39966  rhmisrnghm  39973
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