Theorem mulgrhm 14622

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 14609	. 2 |- ZZ = ( Base ` ℤring )
2		zring1 14614	. 2 |- 1 = ( 1r ` ℤring )
3		mulgrhm.1	. 2 |- .1. = ( 1r ` R )
4		zringmulr 14612	. 2 |- x. = ( .r ` ℤring )
5		eqid 2231	. 2 |- ( .r ` R ) = ( .r ` R )
6		zringring 14606	. . 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 6024	. . . 4 |- ( n = 1 -> ( n .x. .1. ) = ( 1 .x. .1. ) )
11		1zzd 9505	. . . 4 |- ( R e. Ring -> 1 e. ZZ )
12		eqid 2231	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
13	12, 3	ringidcl 14032	. . . . . 6 |- ( R e. Ring -> .1. e. ( Base ` R ) )
14		mulgghm2.m	. . . . . . 7 |- .x. = (.g ` R )
15	12, 14	mulg1 13715	. . . . . 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 2308	. . . 4 |- ( R e. Ring -> ( 1 .x. .1. ) e. ( Base ` R ) )
18	9, 10, 11, 17	fvmptd3 5740	. . 3 |- ( R e. Ring -> ( F ` 1 ) = ( 1 .x. .1. ) )
19	18, 16	eqtrd 2264	. 2 |- ( R e. Ring -> ( F ` 1 ) = .1. )
20		ringgrp 14013	. . . . . . . 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 533	. . . . . . 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 13730	. . . . . 6 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( y .x. .1. ) e. ( Base ` R ) )
25	12, 5, 3	ringlidm 14035	. . . . . 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 6033	. . . 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 531	. . . . 5 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> x e. ZZ )
30	12, 14, 5	mulgass2 14070	. . . . 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 1275	. . . 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 13745	. . . . 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 1275	. . . 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 2275	. . 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 6024	. . . 4 |- ( n = ( x x. y ) -> ( n .x. .1. ) = ( ( x x. y ) .x. .1. ) )
36		zmulcl 9532	. . . . 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 13730	. . . 4 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( x x. y ) .x. .1. ) e. ( Base ` R ) )
39	9, 35, 37, 38	fvmptd3 5740	. . 3 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` ( x x. y ) ) = ( ( x x. y ) .x. .1. ) )
40		oveq1 6024	. . . . 5 |- ( n = x -> ( n .x. .1. ) = ( x .x. .1. ) )
41	12, 14, 21, 29, 23	mulgcld 13730	. . . . 5 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x .x. .1. ) e. ( Base ` R ) )
42	9, 40, 29, 41	fvmptd3 5740	. . . 4 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` x ) = ( x .x. .1. ) )
43		oveq1 6024	. . . . 5 |- ( n = y -> ( n .x. .1. ) = ( y .x. .1. ) )
44	9, 43, 22, 24	fvmptd3 5740	. . . 4 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` y ) = ( y .x. .1. ) )
45	42, 44	oveq12d 6035	. . 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 2274	. 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 14621	. . 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 14178	1 |- ( R e. Ring -> F e. (ℤring RingHom R ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   |-> cmpt 4150   ` cfv 5326  (class class class)co 6017   1c1 8032   x. cmul 8036   ZZcz 9478   Basecbs 13081   .rcmulr 13160   Grpcgrp 13582  ._gcmg 13705   GrpHom cghm 13826   1rcur 13971   Ringcrg 14008   RingHom crh 14163  ℤ_ringczring 14603

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-addf 8153  ax-mulf 8154

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-tp 3677  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-map 6818  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-z 9479  df-dec 9611  df-uz 9755  df-rp 9888  df-fz 10243  df-fzo 10377  df-seqfrec 10709  df-cj 11402  df-abs 11559  df-struct 13083  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089  df-plusg 13172  df-mulr 13173  df-starv 13174  df-tset 13178  df-ple 13179  df-ds 13181  df-unif 13182  df-0g 13340  df-topgen 13342  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-mhm 13541  df-grp 13585  df-minusg 13586  df-mulg 13706  df-subg 13756  df-ghm 13827  df-cmn 13872  df-mgp 13933  df-ur 13972  df-ring 14010  df-cring 14011  df-rhm 14165  df-subrg 14232  df-bl 14559  df-mopn 14560  df-fg 14562  df-metu 14563  df-cnfld 14570  df-zring 14604

This theorem is referenced by:  mulgrhm2  14623