Theorem rhmval 14131

Description: The ring homomorphisms between two rings. (Contributed by AV, 1-Mar-2020.)

Assertion

Ref	Expression
rhmval	|- ( ( R e. Ring /\ S e. Ring ) -> ( R RingHom S ) = ( ( R GrpHom S ) i^i ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) ) )

Proof of Theorem rhmval

Dummy variables s r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfrhm2 14112	. . 3 |- RingHom = ( r e. Ring , s e. Ring |-> ( ( r GrpHom s ) i^i ( (mulGrp ` r ) MndHom (mulGrp ` s ) ) ) )
2	1	a1i 9	. 2 |- ( ( R e. Ring /\ S e. Ring ) -> RingHom = ( r e. Ring , s e. Ring |-> ( ( r GrpHom s ) i^i ( (mulGrp ` r ) MndHom (mulGrp ` s ) ) ) ) )
3		oveq12 6009	. . . 4 |- ( ( r = R /\ s = S ) -> ( r GrpHom s ) = ( R GrpHom S ) )
4		fveq2 5626	. . . . 5 |- ( r = R -> (mulGrp ` r ) = (mulGrp ` R ) )
5		fveq2 5626	. . . . 5 |- ( s = S -> (mulGrp ` s ) = (mulGrp ` S ) )
6	4, 5	oveqan12d 6019	. . . 4 |- ( ( r = R /\ s = S ) -> ( (mulGrp ` r ) MndHom (mulGrp ` s ) ) = ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) )
7	3, 6	ineq12d 3406	. . 3 |- ( ( 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	7	adantl 277	. 2 |- ( ( ( R e. Ring /\ S e. Ring ) /\ ( r = R /\ s = S ) ) -> ( ( r GrpHom s ) i^i ( (mulGrp ` r ) MndHom (mulGrp ` s ) ) ) = ( ( R GrpHom S ) i^i ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) ) )
9		simpl 109	. 2 |- ( ( R e. Ring /\ S e. Ring ) -> R e. Ring )
10		simpr 110	. 2 |- ( ( R e. Ring /\ S e. Ring ) -> S e. Ring )
11		ringgrp 13959	. . . 4 |- ( R e. Ring -> R e. Grp )
12		ringgrp 13959	. . . 4 |- ( S e. Ring -> S e. Grp )
13		ghmex 13787	. . . 4 |- ( ( R e. Grp /\ S e. Grp ) -> ( R GrpHom S ) e. _V )
14	11, 12, 13	syl2an 289	. . 3 |- ( ( R e. Ring /\ S e. Ring ) -> ( R GrpHom S ) e. _V )
15		inex1g 4219	. . 3 |- ( ( R GrpHom S ) e. _V -> ( ( R GrpHom S ) i^i ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) ) e. _V )
16	14, 15	syl 14	. 2 |- ( ( R e. Ring /\ S e. Ring ) -> ( ( R GrpHom S ) i^i ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) ) e. _V )
17	2, 8, 9, 10, 16	ovmpod 6131	1 |- ( ( R e. Ring /\ S e. Ring ) -> ( R RingHom S ) = ( ( R GrpHom S ) i^i ( (mulGrp ` R ) MndHom (mulGrp ` S ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2799   i^i cin 3196   ` cfv 5317  (class class class)co 6000   e. cmpo 6002   MndHom cmhm 13485   Grpcgrp 13528   GrpHom cghm 13772  mulGrpcmgp 13878   Ringcrg 13954   RingHom crh 14108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-pre-ltirr 8107  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-map 6795  df-pnf 8179  df-mnf 8180  df-ltxr 8182  df-inn 9107  df-2 9165  df-3 9166  df-ndx 13030  df-slot 13031  df-base 13033  df-sets 13034  df-plusg 13118  df-mulr 13119  df-0g 13286  df-mgm 13384  df-sgrp 13430  df-mnd 13445  df-mhm 13487  df-grp 13531  df-ghm 13773  df-mgp 13879  df-ur 13918  df-ring 13956  df-rhm 14110

This theorem is referenced by: (None)