Theorem mulgghm2 14240

Description: The powers of a group element give a homomorphism from ZZ to a group. The name .1. should not be taken as a constraint as it may be any group element. (Contributed by Mario Carneiro, 13-Jun-2015.) (Revised by AV, 12-Jun-2019.)

Hypotheses

Ref	Expression
mulgghm2.m	|- .x. = (.g ` R )
mulgghm2.f	|- F = ( n e. ZZ |-> ( n .x. .1. ) )
mulgghm2.b	|- B = ( Base ` R )

Assertion

Ref	Expression
mulgghm2	|- ( ( R e. Grp /\ .1. e. B ) -> F e. (ℤring GrpHom R ) )

Distinct variable groups:   B, n   R, n   .x. , n   .1. , n

Allowed substitution hint:   F( n)

Proof of Theorem mulgghm2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . 3 |- ( ( R e. Grp /\ .1. e. B ) -> R e. Grp )
2		zringgrp 14227	. . 3 |- ℤring e. Grp
3	1, 2	jctil 312	. 2 |- ( ( R e. Grp /\ .1. e. B ) -> (ℤring e. Grp /\ R e. Grp ) )
4		mulgghm2.b	. . . . . . 7 |- B = ( Base ` R )
5		mulgghm2.m	. . . . . . 7 |- .x. = (.g ` R )
6	4, 5	mulgcl 13345	. . . . . 6 |- ( ( R e. Grp /\ n e. ZZ /\ .1. e. B ) -> ( n .x. .1. ) e. B )
7	6	3expa 1205	. . . . 5 |- ( ( ( R e. Grp /\ n e. ZZ ) /\ .1. e. B ) -> ( n .x. .1. ) e. B )
8	7	an32s 568	. . . 4 |- ( ( ( R e. Grp /\ .1. e. B ) /\ n e. ZZ ) -> ( n .x. .1. ) e. B )
9		mulgghm2.f	. . . 4 |- F = ( n e. ZZ |-> ( n .x. .1. ) )
10	8, 9	fmptd 5719	. . 3 |- ( ( R e. Grp /\ .1. e. B ) -> F : ZZ --> B )
11		eqid 2196	. . . . . . . . 9 |- ( +g ` R ) = ( +g ` R )
12	4, 5, 11	mulgdir 13360	. . . . . . . 8 |- ( ( R e. Grp /\ ( x e. ZZ /\ y e. ZZ /\ .1. e. B ) ) -> ( ( x + y ) .x. .1. ) = ( ( x .x. .1. ) ( +g ` R ) ( y .x. .1. ) ) )
13	12	3exp2 1227	. . . . . . 7 |- ( R e. Grp -> ( x e. ZZ -> ( y e. ZZ -> ( .1. e. B -> ( ( x + y ) .x. .1. ) = ( ( x .x. .1. ) ( +g ` R ) ( y .x. .1. ) ) ) ) ) )
14	13	imp42 354	. . . . . 6 |- ( ( ( R e. Grp /\ ( x e. ZZ /\ y e. ZZ ) ) /\ .1. e. B ) -> ( ( x + y ) .x. .1. ) = ( ( x .x. .1. ) ( +g ` R ) ( y .x. .1. ) ) )
15	14	an32s 568	. . . . 5 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( x + y ) .x. .1. ) = ( ( x .x. .1. ) ( +g ` R ) ( y .x. .1. ) ) )
16		oveq1 5932	. . . . . 6 |- ( n = ( x + y ) -> ( n .x. .1. ) = ( ( x + y ) .x. .1. ) )
17		zaddcl 9383	. . . . . . 7 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( x + y ) e. ZZ )
18	17	adantl 277	. . . . . 6 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x + y ) e. ZZ )
19		simpll 527	. . . . . . 7 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> R e. Grp )
20		simplr 528	. . . . . . 7 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> .1. e. B )
21	4, 5, 19, 18, 20	mulgcld 13350	. . . . . 6 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( x + y ) .x. .1. ) e. B )
22	9, 16, 18, 21	fvmptd3 5658	. . . . 5 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` ( x + y ) ) = ( ( x + y ) .x. .1. ) )
23		oveq1 5932	. . . . . . 7 |- ( n = x -> ( n .x. .1. ) = ( x .x. .1. ) )
24		simprl 529	. . . . . . 7 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> x e. ZZ )
25	4, 5, 19, 24, 20	mulgcld 13350	. . . . . . 7 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x .x. .1. ) e. B )
26	9, 23, 24, 25	fvmptd3 5658	. . . . . 6 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` x ) = ( x .x. .1. ) )
27		oveq1 5932	. . . . . . 7 |- ( n = y -> ( n .x. .1. ) = ( y .x. .1. ) )
28		simprr 531	. . . . . . 7 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> y e. ZZ )
29	4, 5, 19, 28, 20	mulgcld 13350	. . . . . . 7 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( y .x. .1. ) e. B )
30	9, 27, 28, 29	fvmptd3 5658	. . . . . 6 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` y ) = ( y .x. .1. ) )
31	26, 30	oveq12d 5943	. . . . 5 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( F ` x ) ( +g ` R ) ( F ` y ) ) = ( ( x .x. .1. ) ( +g ` R ) ( y .x. .1. ) ) )
32	15, 22, 31	3eqtr4d 2239	. . . 4 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` ( x + y ) ) = ( ( F ` x ) ( +g ` R ) ( F ` y ) ) )
33	32	ralrimivva 2579	. . 3 |- ( ( R e. Grp /\ .1. e. B ) -> A. x e. ZZ A. y e. ZZ ( F ` ( x + y ) ) = ( ( F ` x ) ( +g ` R ) ( F ` y ) ) )
34	10, 33	jca 306	. 2 |- ( ( R e. Grp /\ .1. e. B ) -> ( F : ZZ --> B /\ A. x e. ZZ A. y e. ZZ ( F ` ( x + y ) ) = ( ( F ` x ) ( +g ` R ) ( F ` y ) ) ) )
35		zringbas 14228	. . 3 |- ZZ = ( Base ` ℤring )
36		zringplusg 14229	. . 3 |- + = ( +g ` ℤring )
37	35, 4, 36, 11	isghm 13449	. 2 |- ( F e. (ℤring GrpHom R ) <-> ( (ℤring e. Grp /\ R e. Grp ) /\ ( F : ZZ --> B /\ A. x e. ZZ A. y e. ZZ ( F ` ( x + y ) ) = ( ( F ` x ) ( +g ` R ) ( F ` y ) ) ) ) )
38	3, 34, 37	sylanbrc 417	1 |- ( ( R e. Grp /\ .1. e. B ) -> F e. (ℤring GrpHom R ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   A.wral 2475   |-> cmpt 4095   -->wf 5255   ` cfv 5259  (class class class)co 5925   + caddc 7899   ZZcz 9343   Basecbs 12703   +g cplusg 12780   Grpcgrp 13202  ._gcmg 13325   GrpHom cghm 13446  ℤ_ringczring 14222

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-addf 8018  ax-mulf 8019

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-tp 3631  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-6 9070  df-7 9071  df-8 9072  df-9 9073  df-n0 9267  df-z 9344  df-dec 9475  df-uz 9619  df-rp 9746  df-fz 10101  df-seqfrec 10557  df-cj 11024  df-abs 11181  df-struct 12705  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-mulr 12794  df-starv 12795  df-tset 12799  df-ple 12800  df-ds 12802  df-unif 12803  df-0g 12960  df-topgen 12962  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205  df-minusg 13206  df-mulg 13326  df-subg 13376  df-ghm 13447  df-cmn 13492  df-mgp 13553  df-ur 13592  df-ring 13630  df-cring 13631  df-subrg 13851  df-bl 14178  df-mopn 14179  df-fg 14181  df-metu 14182  df-cnfld 14189  df-zring 14223

This theorem is referenced by:  mulgrhm  14241