Theorem mulgrhm 13932

Description: The powers of the element 1 give a ring homomorphism from ZZ to a ring. (Contributed by Mario Carneiro, 14-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. ) )
mulgrhm.1	|- .1. = ( 1r ` R )

Assertion

Ref	Expression
mulgrhm	|- ( R e. Ring -> F e. (ℤring RingHom R ) )

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

Allowed substitution hint:   F( n)

Proof of Theorem mulgrhm

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

Step	Hyp	Ref	Expression
1		zringbas 13920	. 2 |- ZZ = ( Base ` ℤring )
2		zring1 13925	. 2 |- 1 = ( 1r ` ℤring )
3		mulgrhm.1	. 2 |- .1. = ( 1r ` R )
4		zringmulr 13923	. 2 |- x. = ( .r ` ℤring )
5		eqid 2189	. 2 |- ( .r ` R ) = ( .r ` R )
6		zringring 13917	. . 3 |- ℤring e. Ring
7	6	a1i 9	. 2 |- ( R e. Ring -> ℤring e. Ring )
8		id 19	. 2 |- ( R e. Ring -> R e. Ring )
9		mulgghm2.f	. . . 4 |- F = ( n e. ZZ |-> ( n .x. .1. ) )
10		oveq1 5907	. . . 4 |- ( n = 1 -> ( n .x. .1. ) = ( 1 .x. .1. ) )
11		1zzd 9315	. . . 4 |- ( R e. Ring -> 1 e. ZZ )
12		eqid 2189	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
13	12, 3	ringidcl 13399	. . . . . 6 |- ( R e. Ring -> .1. e. ( Base ` R ) )
14		mulgghm2.m	. . . . . . 7 |- .x. = (.g ` R )
15	12, 14	mulg1 13094	. . . . . 6 |- ( .1. e. ( Base ` R ) -> ( 1 .x. .1. ) = .1. )
16	13, 15	syl 14	. . . . 5 |- ( R e. Ring -> ( 1 .x. .1. ) = .1. )
17	16, 13	eqeltrd 2266	. . . 4 |- ( R e. Ring -> ( 1 .x. .1. ) e. ( Base ` R ) )
18	9, 10, 11, 17	fvmptd3 5633	. . 3 |- ( R e. Ring -> ( F ` 1 ) = ( 1 .x. .1. ) )
19	18, 16	eqtrd 2222	. 2 |- ( R e. Ring -> ( F ` 1 ) = .1. )
20		ringgrp 13380	. . . . . . . 8 |- ( R e. Ring -> R e. Grp )
21	20	adantr 276	. . . . . . 7 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> R e. Grp )
22		simprr 531	. . . . . . 7 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> y e. ZZ )
23	13	adantr 276	. . . . . . 7 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> .1. e. ( Base ` R ) )
24	12, 14, 21, 22, 23	mulgcld 13109	. . . . . 6 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( y .x. .1. ) e. ( Base ` R ) )
25	12, 5, 3	ringlidm 13402	. . . . . 6 |- ( ( R e. Ring /\ ( y .x. .1. ) e. ( Base ` R ) ) -> ( .1. ( .r ` R ) ( y .x. .1. ) ) = ( y .x. .1. ) )
26	24, 25	syldan 282	. . . . 5 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( .1. ( .r ` R ) ( y .x. .1. ) ) = ( y .x. .1. ) )
27	26	oveq2d 5916	. . . 4 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x .x. ( .1. ( .r ` R ) ( y .x. .1. ) ) ) = ( x .x. ( y .x. .1. ) ) )
28		simpl 109	. . . . 5 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> R e. Ring )
29		simprl 529	. . . . 5 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> x e. ZZ )
30	12, 14, 5	mulgass2 13435	. . . . 5 |- ( ( R e. Ring /\ ( x e. ZZ /\ .1. e. ( Base ` R ) /\ ( y .x. .1. ) e. ( Base ` R ) ) ) -> ( ( x .x. .1. ) ( .r ` R ) ( y .x. .1. ) ) = ( x .x. ( .1. ( .r ` R ) ( y .x. .1. ) ) ) )
31	28, 29, 23, 24, 30	syl13anc 1251	. . . 4 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( x .x. .1. ) ( .r ` R ) ( y .x. .1. ) ) = ( x .x. ( .1. ( .r ` R ) ( y .x. .1. ) ) ) )
32	12, 14	mulgass 13124	. . . . 5 |- ( ( R e. Grp /\ ( x e. ZZ /\ y e. ZZ /\ .1. e. ( Base ` R ) ) ) -> ( ( x x. y ) .x. .1. ) = ( x .x. ( y .x. .1. ) ) )
33	21, 29, 22, 23, 32	syl13anc 1251	. . . 4 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( x x. y ) .x. .1. ) = ( x .x. ( y .x. .1. ) ) )
34	27, 31, 33	3eqtr4rd 2233	. . 3 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( x x. y ) .x. .1. ) = ( ( x .x. .1. ) ( .r ` R ) ( y .x. .1. ) ) )
35		oveq1 5907	. . . 4 |- ( n = ( x x. y ) -> ( n .x. .1. ) = ( ( x x. y ) .x. .1. ) )
36		zmulcl 9341	. . . . 5 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( x x. y ) e. ZZ )
37	36	adantl 277	. . . 4 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x x. y ) e. ZZ )
38	12, 14, 21, 37, 23	mulgcld 13109	. . . 4 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( x x. y ) .x. .1. ) e. ( Base ` R ) )
39	9, 35, 37, 38	fvmptd3 5633	. . 3 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` ( x x. y ) ) = ( ( x x. y ) .x. .1. ) )
40		oveq1 5907	. . . . 5 |- ( n = x -> ( n .x. .1. ) = ( x .x. .1. ) )
41	12, 14, 21, 29, 23	mulgcld 13109	. . . . 5 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x .x. .1. ) e. ( Base ` R ) )
42	9, 40, 29, 41	fvmptd3 5633	. . . 4 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` x ) = ( x .x. .1. ) )
43		oveq1 5907	. . . . 5 |- ( n = y -> ( n .x. .1. ) = ( y .x. .1. ) )
44	9, 43, 22, 24	fvmptd3 5633	. . . 4 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` y ) = ( y .x. .1. ) )
45	42, 44	oveq12d 5918	. . 3 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( F ` x ) ( .r ` R ) ( F ` y ) ) = ( ( x .x. .1. ) ( .r ` R ) ( y .x. .1. ) ) )
46	34, 39, 45	3eqtr4d 2232	. 2 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` ( x x. y ) ) = ( ( F ` x ) ( .r ` R ) ( F ` y ) ) )
47	14, 9, 12	mulgghm2 13931	. . 3 |- ( ( R e. Grp /\ .1. e. ( Base ` R ) ) -> F e. (ℤring GrpHom R ) )
48	20, 13, 47	syl2anc 411	. 2 |- ( R e. Ring -> F e. (ℤring GrpHom R ) )
49	1, 2, 3, 4, 5, 7, 8, 19, 46, 48	isrhm2d 13540	1 |- ( R e. Ring -> F e. (ℤring RingHom R ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   |-> cmpt 4082   ` cfv 5238  (class class class)co 5900   1c1 7847   x. cmul 7851   ZZcz 9288   Basecbs 12523   .rcmulr 12601   Grpcgrp 12968  ._gcmg 13084   GrpHom cghm 13204   1rcur 13338   Ringcrg 13375   RingHom crh 13525  ℤ_ringczring 13914

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-iinf 4608  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-mulrcl 7945  ax-addcom 7946  ax-mulcom 7947  ax-addass 7948  ax-mulass 7949  ax-distr 7950  ax-i2m1 7951  ax-0lt1 7952  ax-1rid 7953  ax-0id 7954  ax-rnegex 7955  ax-precex 7956  ax-cnre 7957  ax-pre-ltirr 7958  ax-pre-ltwlin 7959  ax-pre-lttrn 7960  ax-pre-apti 7961  ax-pre-ltadd 7962  ax-pre-mulgt0 7963  ax-addf 7968  ax-mulf 7969

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 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3595  df-sn 3616  df-pr 3617  df-tp 3618  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-id 4314  df-iord 4387  df-on 4389  df-ilim 4390  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-recs 6334  df-frec 6420  df-map 6680  df-pnf 8029  df-mnf 8030  df-xr 8031  df-ltxr 8032  df-le 8033  df-sub 8165  df-neg 8166  df-reap 8567  df-inn 8955  df-2 9013  df-3 9014  df-4 9015  df-5 9016  df-6 9017  df-7 9018  df-8 9019  df-9 9020  df-n0 9212  df-z 9289  df-dec 9420  df-uz 9564  df-fz 10045  df-fzo 10179  df-seqfrec 10485  df-cj 10892  df-struct 12525  df-ndx 12526  df-slot 12527  df-base 12529  df-sets 12530  df-iress 12531  df-plusg 12613  df-mulr 12614  df-starv 12615  df-0g 12774  df-mgm 12843  df-sgrp 12888  df-mnd 12901  df-mhm 12934  df-grp 12971  df-minusg 12972  df-mulg 13085  df-subg 13134  df-ghm 13205  df-cmn 13250  df-mgp 13300  df-ur 13339  df-ring 13377  df-cring 13378  df-rhm 13527  df-subrg 13591  df-icnfld 13890  df-zring 13915

This theorem is referenced by:  mulgrhm2  13933