Theorem mulgghm2 14943

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 14930	. . 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 13942	. . . . . 6 |- ( ( R e. Grp /\ n e. ZZ /\ .1. e. B ) -> ( n .x. .1. ) e. B )
7	6	3expa 1234	. . . . 5 |- ( ( ( R e. Grp /\ n e. ZZ ) /\ .1. e. B ) -> ( n .x. .1. ) e. B )
8	7	an32s 574	. . . 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 5862	. . 3 |- ( ( R e. Grp /\ .1. e. B ) -> F : ZZ --> B )
11		eqid 2238	. . . . . . . . 9 |- ( +g ` R ) = ( +g ` R )
12	4, 5, 11	mulgdir 13957	. . . . . . . 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 1256	. . . . . . 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 574	. . . . 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 6092	. . . . . 6 |- ( n = ( x + y ) -> ( n .x. .1. ) = ( ( x + y ) .x. .1. ) )
17		zaddcl 9684	. . . . . . 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 531	. . . . . . 7 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> R e. Grp )
20		simplr 533	. . . . . . 7 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> .1. e. B )
21	4, 5, 19, 18, 20	mulgcld 13947	. . . . . 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 5799	. . . . 5 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` ( x + y ) ) = ( ( x + y ) .x. .1. ) )
23		oveq1 6092	. . . . . . 7 |- ( n = x -> ( n .x. .1. ) = ( x .x. .1. ) )
24		simprl 535	. . . . . . 7 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> x e. ZZ )
25	4, 5, 19, 24, 20	mulgcld 13947	. . . . . . 7 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x .x. .1. ) e. B )
26	9, 23, 24, 25	fvmptd3 5799	. . . . . 6 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` x ) = ( x .x. .1. ) )
27		oveq1 6092	. . . . . . 7 |- ( n = y -> ( n .x. .1. ) = ( y .x. .1. ) )
28		simprr 537	. . . . . . 7 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> y e. ZZ )
29	4, 5, 19, 28, 20	mulgcld 13947	. . . . . . 7 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( y .x. .1. ) e. B )
30	9, 27, 28, 29	fvmptd3 5799	. . . . . 6 |- ( ( ( R e. Grp /\ .1. e. B ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` y ) = ( y .x. .1. ) )
31	26, 30	oveq12d 6103	. . . . 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 2281	. . . 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 2632	. . 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 14931	. . 3 |- ZZ = ( Base ` ℤring )
36		zringplusg 14932	. . 3 |- + = ( +g ` ℤring )
37	35, 4, 36, 11	isghm 14046	. 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 421	1 |- ( ( R e. Grp /\ .1. e. B ) -> F e. (ℤring GrpHom R ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   |-> cmpt 4192   -->wf 5373   ` cfv 5377  (class class class)co 6085   + caddc 8182   ZZcz 9644   Basecbs 13352   +g cplusg 13431   Grpcgrp 13805  ._gcmg 13922   GrpHom cghm 14043  ℤ_ringczring 14925

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-addf 8301  ax-mulf 8302

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-z 9645  df-dec 9778  df-uz 9922  df-rp 10055  df-fz 10412  df-seqfrec 10885  df-cj 11607  df-abs 11765  df-struct 13354  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360  df-plusg 13444  df-mulr 13445  df-starv 13446  df-tset 13450  df-ple 13451  df-ds 13453  df-unif 13454  df-0g 13612  df-topgen 13614  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-minusg 13809  df-mulg 13923  df-subg 13973  df-ghm 14044  df-cmn 14089  df-mgp 14218  df-ur 14263  df-ring 14302  df-cring 14303  df-subrg 14527  df-bl 14883  df-mopn 14884  df-fg 14886  df-metu 14887  df-cnfld 14894  df-zring 14926

This theorem is used by:  mulgrhm  14944