Theorem mulgrhm 14097

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 14084	. 2 |- ZZ = ( Base ` ℤring )
2		zring1 14089	. 2 |- 1 = ( 1r ` ℤring )
3		mulgrhm.1	. 2 |- .1. = ( 1r ` R )
4		zringmulr 14087	. 2 |- x. = ( .r ` ℤring )
5		eqid 2193	. 2 |- ( .r ` R ) = ( .r ` R )
6		zringring 14081	. . 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 5925	. . . 4 |- ( n = 1 -> ( n .x. .1. ) = ( 1 .x. .1. ) )
11		1zzd 9344	. . . 4 |- ( R e. Ring -> 1 e. ZZ )
12		eqid 2193	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
13	12, 3	ringidcl 13516	. . . . . 6 |- ( R e. Ring -> .1. e. ( Base ` R ) )
14		mulgghm2.m	. . . . . . 7 |- .x. = (.g ` R )
15	12, 14	mulg1 13199	. . . . . 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 2270	. . . 4 |- ( R e. Ring -> ( 1 .x. .1. ) e. ( Base ` R ) )
18	9, 10, 11, 17	fvmptd3 5651	. . 3 |- ( R e. Ring -> ( F ` 1 ) = ( 1 .x. .1. ) )
19	18, 16	eqtrd 2226	. 2 |- ( R e. Ring -> ( F ` 1 ) = .1. )
20		ringgrp 13497	. . . . . . . 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 13214	. . . . . 6 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( y .x. .1. ) e. ( Base ` R ) )
25	12, 5, 3	ringlidm 13519	. . . . . 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 5934	. . . 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 13554	. . . . 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 13229	. . . . 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 2237	. . 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 5925	. . . 4 |- ( n = ( x x. y ) -> ( n .x. .1. ) = ( ( x x. y ) .x. .1. ) )
36		zmulcl 9370	. . . . 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 13214	. . . 4 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( x x. y ) .x. .1. ) e. ( Base ` R ) )
39	9, 35, 37, 38	fvmptd3 5651	. . 3 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` ( x x. y ) ) = ( ( x x. y ) .x. .1. ) )
40		oveq1 5925	. . . . 5 |- ( n = x -> ( n .x. .1. ) = ( x .x. .1. ) )
41	12, 14, 21, 29, 23	mulgcld 13214	. . . . 5 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x .x. .1. ) e. ( Base ` R ) )
42	9, 40, 29, 41	fvmptd3 5651	. . . 4 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` x ) = ( x .x. .1. ) )
43		oveq1 5925	. . . . 5 |- ( n = y -> ( n .x. .1. ) = ( y .x. .1. ) )
44	9, 43, 22, 24	fvmptd3 5651	. . . 4 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` y ) = ( y .x. .1. ) )
45	42, 44	oveq12d 5936	. . 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 2236	. 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 14096	. . 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 13661	1 |- ( R e. Ring -> F e. (ℤring RingHom R ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   |-> cmpt 4090   ` cfv 5254  (class class class)co 5918   1c1 7873   x. cmul 7877   ZZcz 9317   Basecbs 12618   .rcmulr 12696   Grpcgrp 13072  ._gcmg 13189   GrpHom cghm 13310   1rcur 13455   Ringcrg 13492   RingHom crh 13646  ℤ_ringczring 14078

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-addf 7994  ax-mulf 7995

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-tp 3626  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-map 6704  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-7 9046  df-8 9047  df-9 9048  df-n0 9241  df-z 9318  df-dec 9449  df-uz 9593  df-fz 10075  df-fzo 10209  df-seqfrec 10519  df-cj 10986  df-struct 12620  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-starv 12710  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-mhm 13031  df-grp 13075  df-minusg 13076  df-mulg 13190  df-subg 13240  df-ghm 13311  df-cmn 13356  df-mgp 13417  df-ur 13456  df-ring 13494  df-cring 13495  df-rhm 13648  df-subrg 13715  df-icnfld 14048  df-zring 14079

This theorem is referenced by:  mulgrhm2  14098