Theorem mulgghm2 14704

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 14691	. . 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 13806	. . . . . 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 5809	. . 3 |- ( ( R e. Grp /\ .1. e. B ) -> F : ZZ --> B )
11		eqid 2231	. . . . . . . . 9 |- ( +g ` R ) = ( +g ` R )
12	4, 5, 11	mulgdir 13821	. . . . . . . 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 6035	. . . . . 6 |- ( n = ( x + y ) -> ( n .x. .1. ) = ( ( x + y ) .x. .1. ) )
17		zaddcl 9580	. . . . . . 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 13811	. . . . . 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 5749	. . . . 5 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` ( x + y ) ) = ( ( x + y ) .x. .1. ) )
23		oveq1 6035	. . . . . . 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 13811	. . . . . . 7 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x .x. .1. ) e. B )
26	9, 23, 24, 25	fvmptd3 5749	. . . . . 6 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` x ) = ( x .x. .1. ) )
27		oveq1 6035	. . . . . . 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 13811	. . . . . . 7 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( y .x. .1. ) e. B )
30	9, 27, 28, 29	fvmptd3 5749	. . . . . 6 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` y ) = ( y .x. .1. ) )
31	26, 30	oveq12d 6046	. . . . 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 2274	. . . 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 2615	. . 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 14692	. . 3 |- ZZ = ( Base ` ℤring )
36		zringplusg 14693	. . 3 |- + = ( +g ` ℤring )
37	35, 4, 36, 11	isghm 13910	. 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 2202   A.wral 2511   |-> cmpt 4155   -->wf 5329   ` cfv 5333  (class class class)co 6028   + caddc 8095   ZZcz 9540   Basecbs 13162   +g cplusg 13240   Grpcgrp 13663  ._gcmg 13786   GrpHom cghm 13907  ℤ_ringczring 14686

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-addf 8214  ax-mulf 8215

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-tp 3681  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-5 9264  df-6 9265  df-7 9266  df-8 9267  df-9 9268  df-n0 9462  df-z 9541  df-dec 9673  df-uz 9817  df-rp 9950  df-fz 10306  df-seqfrec 10773  df-cj 11482  df-abs 11639  df-struct 13164  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-iress 13170  df-plusg 13253  df-mulr 13254  df-starv 13255  df-tset 13259  df-ple 13260  df-ds 13262  df-unif 13263  df-0g 13421  df-topgen 13423  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-mulg 13787  df-subg 13837  df-ghm 13908  df-cmn 13953  df-mgp 14015  df-ur 14054  df-ring 14092  df-cring 14093  df-subrg 14314  df-bl 14642  df-mopn 14643  df-fg 14645  df-metu 14646  df-cnfld 14653  df-zring 14687

This theorem is referenced by:  mulgrhm  14705