Theorem mulgrhm 14456

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 14443	. 2 |- ZZ = ( Base ` ℤring )
2		zring1 14448	. 2 |- 1 = ( 1r ` ℤring )
3		mulgrhm.1	. 2 |- .1. = ( 1r ` R )
4		zringmulr 14446	. 2 |- x. = ( .r ` ℤring )
5		eqid 2206	. 2 |- ( .r ` R ) = ( .r ` R )
6		zringring 14440	. . 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 5969	. . . 4 |- ( n = 1 -> ( n .x. .1. ) = ( 1 .x. .1. ) )
11		1zzd 9429	. . . 4 |- ( R e. Ring -> 1 e. ZZ )
12		eqid 2206	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
13	12, 3	ringidcl 13867	. . . . . 6 |- ( R e. Ring -> .1. e. ( Base ` R ) )
14		mulgghm2.m	. . . . . . 7 |- .x. = (.g ` R )
15	12, 14	mulg1 13550	. . . . . 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 2283	. . . 4 |- ( R e. Ring -> ( 1 .x. .1. ) e. ( Base ` R ) )
18	9, 10, 11, 17	fvmptd3 5691	. . 3 |- ( R e. Ring -> ( F ` 1 ) = ( 1 .x. .1. ) )
19	18, 16	eqtrd 2239	. 2 |- ( R e. Ring -> ( F ` 1 ) = .1. )
20		ringgrp 13848	. . . . . . . 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 13565	. . . . . 6 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( y .x. .1. ) e. ( Base ` R ) )
25	12, 5, 3	ringlidm 13870	. . . . . 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 5978	. . . 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 13905	. . . . 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 1252	. . . 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 13580	. . . . 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 1252	. . . 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 2250	. . 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 5969	. . . 4 |- ( n = ( x x. y ) -> ( n .x. .1. ) = ( ( x x. y ) .x. .1. ) )
36		zmulcl 9456	. . . . 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 13565	. . . 4 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( x x. y ) .x. .1. ) e. ( Base ` R ) )
39	9, 35, 37, 38	fvmptd3 5691	. . 3 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` ( x x. y ) ) = ( ( x x. y ) .x. .1. ) )
40		oveq1 5969	. . . . 5 |- ( n = x -> ( n .x. .1. ) = ( x .x. .1. ) )
41	12, 14, 21, 29, 23	mulgcld 13565	. . . . 5 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x .x. .1. ) e. ( Base ` R ) )
42	9, 40, 29, 41	fvmptd3 5691	. . . 4 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` x ) = ( x .x. .1. ) )
43		oveq1 5969	. . . . 5 |- ( n = y -> ( n .x. .1. ) = ( y .x. .1. ) )
44	9, 43, 22, 24	fvmptd3 5691	. . . 4 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` y ) = ( y .x. .1. ) )
45	42, 44	oveq12d 5980	. . 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 2249	. 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 14455	. . 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 14012	1 |- ( R e. Ring -> F e. (ℤring RingHom R ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   |-> cmpt 4116   ` cfv 5285  (class class class)co 5962   1c1 7956   x. cmul 7960   ZZcz 9402   Basecbs 12917   .rcmulr 12995   Grpcgrp 13417  ._gcmg 13540   GrpHom cghm 13661   1rcur 13806   Ringcrg 13843   RingHom crh 13997  ℤ_ringczring 14437

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-mulrcl 8054  ax-addcom 8055  ax-mulcom 8056  ax-addass 8057  ax-mulass 8058  ax-distr 8059  ax-i2m1 8060  ax-0lt1 8061  ax-1rid 8062  ax-0id 8063  ax-rnegex 8064  ax-precex 8065  ax-cnre 8066  ax-pre-ltirr 8067  ax-pre-ltwlin 8068  ax-pre-lttrn 8069  ax-pre-apti 8070  ax-pre-ltadd 8071  ax-pre-mulgt0 8072  ax-addf 8077  ax-mulf 8078

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-tp 3646  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-iord 4426  df-on 4428  df-ilim 4429  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-recs 6409  df-frec 6495  df-map 6755  df-pnf 8139  df-mnf 8140  df-xr 8141  df-ltxr 8142  df-le 8143  df-sub 8275  df-neg 8276  df-reap 8678  df-inn 9067  df-2 9125  df-3 9126  df-4 9127  df-5 9128  df-6 9129  df-7 9130  df-8 9131  df-9 9132  df-n0 9326  df-z 9403  df-dec 9535  df-uz 9679  df-rp 9806  df-fz 10161  df-fzo 10295  df-seqfrec 10625  df-cj 11238  df-abs 11395  df-struct 12919  df-ndx 12920  df-slot 12921  df-base 12923  df-sets 12924  df-iress 12925  df-plusg 13007  df-mulr 13008  df-starv 13009  df-tset 13013  df-ple 13014  df-ds 13016  df-unif 13017  df-0g 13175  df-topgen 13177  df-mgm 13273  df-sgrp 13319  df-mnd 13334  df-mhm 13376  df-grp 13420  df-minusg 13421  df-mulg 13541  df-subg 13591  df-ghm 13662  df-cmn 13707  df-mgp 13768  df-ur 13807  df-ring 13845  df-cring 13846  df-rhm 13999  df-subrg 14066  df-bl 14393  df-mopn 14394  df-fg 14396  df-metu 14397  df-cnfld 14404  df-zring 14438

This theorem is referenced by:  mulgrhm2  14457