Theorem mulgghm2 14164

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 14151	. . 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 13269	. . . . . 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 5716	. . 3 |- ( ( R e. Grp /\ .1. e. B ) -> F : ZZ --> B )
11		eqid 2196	. . . . . . . . 9 |- ( +g ` R ) = ( +g ` R )
12	4, 5, 11	mulgdir 13284	. . . . . . . 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 5929	. . . . . 6 |- ( n = ( x + y ) -> ( n .x. .1. ) = ( ( x + y ) .x. .1. ) )
17		zaddcl 9366	. . . . . . 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 13274	. . . . . 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 5655	. . . . 5 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` ( x + y ) ) = ( ( x + y ) .x. .1. ) )
23		oveq1 5929	. . . . . . 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 13274	. . . . . . 7 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x .x. .1. ) e. B )
26	9, 23, 24, 25	fvmptd3 5655	. . . . . 6 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` x ) = ( x .x. .1. ) )
27		oveq1 5929	. . . . . . 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 13274	. . . . . . 7 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( y .x. .1. ) e. B )
30	9, 27, 28, 29	fvmptd3 5655	. . . . . 6 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` y ) = ( y .x. .1. ) )
31	26, 30	oveq12d 5940	. . . . 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 14152	. . 3 |- ZZ = ( Base ` ℤring )
36		zringplusg 14153	. . 3 |- + = ( +g ` ℤring )
37	35, 4, 36, 11	isghm 13373	. 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 4094   -->wf 5254   ` cfv 5258  (class class class)co 5922   + caddc 7882   ZZcz 9326   Basecbs 12678   +g cplusg 12755   Grpcgrp 13132  ._gcmg 13249   GrpHom cghm 13370  ℤ_ringczring 14146

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 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-addf 8001  ax-mulf 8002

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 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-tp 3630  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-5 9052  df-6 9053  df-7 9054  df-8 9055  df-9 9056  df-n0 9250  df-z 9327  df-dec 9458  df-uz 9602  df-rp 9729  df-fz 10084  df-seqfrec 10540  df-cj 11007  df-abs 11164  df-struct 12680  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-starv 12770  df-tset 12774  df-ple 12775  df-ds 12777  df-unif 12778  df-0g 12929  df-topgen 12931  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-mulg 13250  df-subg 13300  df-ghm 13371  df-cmn 13416  df-mgp 13477  df-ur 13516  df-ring 13554  df-cring 13555  df-subrg 13775  df-bl 14102  df-mopn 14103  df-fg 14105  df-metu 14106  df-cnfld 14113  df-zring 14147

This theorem is referenced by:  mulgrhm  14165