Theorem mulgghm2 14773

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 14760	. . 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 13873	. . . . . 6 |- ( ( R e. Grp /\ n e. ZZ /\ .1. e. B ) -> ( n .x. .1. ) e. B )
7	6	3expa 1230	. . . . 5 |- ( ( ( R e. Grp /\ n e. ZZ ) /\ .1. e. B ) -> ( n .x. .1. ) e. B )
8	7	an32s 570	. . . 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 5833	. . 3 |- ( ( R e. Grp /\ .1. e. B ) -> F : ZZ --> B )
11		eqid 2234	. . . . . . . . 9 |- ( +g ` R ) = ( +g ` R )
12	4, 5, 11	mulgdir 13888	. . . . . . . 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 1252	. . . . . . 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 570	. . . . 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 6059	. . . . . 6 |- ( n = ( x + y ) -> ( n .x. .1. ) = ( ( x + y ) .x. .1. ) )
17		zaddcl 9619	. . . . . . 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 529	. . . . . . 7 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> .1. e. B )
21	4, 5, 19, 18, 20	mulgcld 13878	. . . . . 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 5773	. . . . 5 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` ( x + y ) ) = ( ( x + y ) .x. .1. ) )
23		oveq1 6059	. . . . . . 7 |- ( n = x -> ( n .x. .1. ) = ( x .x. .1. ) )
24		simprl 531	. . . . . . 7 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> x e. ZZ )
25	4, 5, 19, 24, 20	mulgcld 13878	. . . . . . 7 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x .x. .1. ) e. B )
26	9, 23, 24, 25	fvmptd3 5773	. . . . . 6 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` x ) = ( x .x. .1. ) )
27		oveq1 6059	. . . . . . 7 |- ( n = y -> ( n .x. .1. ) = ( y .x. .1. ) )
28		simprr 533	. . . . . . 7 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> y e. ZZ )
29	4, 5, 19, 28, 20	mulgcld 13878	. . . . . . 7 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( y .x. .1. ) e. B )
30	9, 27, 28, 29	fvmptd3 5773	. . . . . 6 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` y ) = ( y .x. .1. ) )
31	26, 30	oveq12d 6070	. . . . 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 2277	. . . 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 2626	. . 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 14761	. . 3 |- ZZ = ( Base ` ℤring )
36		zringplusg 14762	. . 3 |- + = ( +g ` ℤring )
37	35, 4, 36, 11	isghm 13977	. 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 1398   e. wcel 2205   A.wral 2522   |-> cmpt 4173   -->wf 5350   ` cfv 5354  (class class class)co 6052   + caddc 8132   ZZcz 9579   Basecbs 13229   +g cplusg 13307   Grpcgrp 13730  ._gcmg 13853   GrpHom cghm 13974  ℤ_ringczring 14755

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-addf 8251  ax-mulf 8252

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-tp 3699  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302  df-7 9303  df-8 9304  df-9 9305  df-n0 9499  df-z 9580  df-dec 9713  df-uz 9857  df-rp 9990  df-fz 10346  df-seqfrec 10814  df-cj 11531  df-abs 11688  df-struct 13231  df-ndx 13232  df-slot 13233  df-base 13235  df-sets 13236  df-iress 13237  df-plusg 13320  df-mulr 13321  df-starv 13322  df-tset 13326  df-ple 13327  df-ds 13329  df-unif 13330  df-0g 13488  df-topgen 13490  df-mgm 13586  df-sgrp 13632  df-mnd 13647  df-grp 13733  df-minusg 13734  df-mulg 13854  df-subg 13904  df-ghm 13975  df-cmn 14020  df-mgp 14082  df-ur 14121  df-ring 14159  df-cring 14160  df-subrg 14381  df-bl 14711  df-mopn 14712  df-fg 14714  df-metu 14715  df-cnfld 14722  df-zring 14756

This theorem is referenced by:  mulgrhm  14774