Theorem mulgrhm 14944

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 14931	. 2 |- ZZ = ( Base ` ℤring )
2		zring1 14936	. 2 |- 1 = ( 1r ` ℤring )
3		mulgrhm.1	. 2 |- .1. = ( 1r ` R )
4		zringmulr 14934	. 2 |- x. = ( .r ` ℤring )
5		eqid 2238	. 2 |- ( .r ` R ) = ( .r ` R )
6		zringring 14928	. . 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 6092	. . . 4 |- ( n = 1 -> ( n .x. .1. ) = ( 1 .x. .1. ) )
11		1zzd 9671	. . . 4 |- ( R e. Ring -> 1 e. ZZ )
12		eqid 2238	. . . . . . 7 |- ( Base ` R ) = ( Base ` R )
13	12, 3	ringidcl 14325	. . . . . 6 |- ( R e. Ring -> .1. e. ( Base ` R ) )
14		mulgghm2.m	. . . . . . 7 |- .x. = (.g ` R )
15	12, 14	mulg1 13932	. . . . . 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 2315	. . . 4 |- ( R e. Ring -> ( 1 .x. .1. ) e. ( Base ` R ) )
18	9, 10, 11, 17	fvmptd3 5799	. . 3 |- ( R e. Ring -> ( F ` 1 ) = ( 1 .x. .1. ) )
19	18, 16	eqtrd 2271	. 2 |- ( R e. Ring -> ( F ` 1 ) = .1. )
20		ringgrp 14305	. . . . . . . 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 537	. . . . . . 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 13947	. . . . . 6 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( y .x. .1. ) e. ( Base ` R ) )
25	12, 5, 3	ringlidm 14328	. . . . . 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 6101	. . . 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 535	. . . . 5 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> x e. ZZ )
30	12, 14, 5	mulgass2 14363	. . . . 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 1280	. . . 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 13962	. . . . 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 1280	. . . 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 2282	. . 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 6092	. . . 4 |- ( n = ( x x. y ) -> ( n .x. .1. ) = ( ( x x. y ) .x. .1. ) )
36		zmulcl 9698	. . . . 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 13947	. . . 4 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( x x. y ) .x. .1. ) e. ( Base ` R ) )
39	9, 35, 37, 38	fvmptd3 5799	. . 3 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` ( x x. y ) ) = ( ( x x. y ) .x. .1. ) )
40		oveq1 6092	. . . . 5 |- ( n = x -> ( n .x. .1. ) = ( x .x. .1. ) )
41	12, 14, 21, 29, 23	mulgcld 13947	. . . . 5 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x .x. .1. ) e. ( Base ` R ) )
42	9, 40, 29, 41	fvmptd3 5799	. . . 4 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` x ) = ( x .x. .1. ) )
43		oveq1 6092	. . . . 5 |- ( n = y -> ( n .x. .1. ) = ( y .x. .1. ) )
44	9, 43, 22, 24	fvmptd3 5799	. . . 4 |- ( ( R e. Ring /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( F ` y ) = ( y .x. .1. ) )
45	42, 44	oveq12d 6103	. . 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 2281	. 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 14943	. . 3 |- ( ( R e. Grp /\ .1. e. ( Base ` R ) ) -> F e. (ℤring GrpHom R ) )
48	20, 13, 47	syl2anc 415	. 2 |- ( R e. Ring -> F e. (ℤring GrpHom R ) )
49	1, 2, 3, 4, 5, 7, 8, 19, 46, 48	isrhm2d 14472	1 |- ( R e. Ring -> F e. (ℤring RingHom R ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   |-> cmpt 4192   ` cfv 5377  (class class class)co 6085   1c1 8180   x. cmul 8184   ZZcz 9644   Basecbs 13352   .rcmulr 13432   Grpcgrp 13805  ._gcmg 13922   GrpHom cghm 14043   1rcur 14262   Ringcrg 14300   RingHom crh 14457  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-map 6924  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-fzo 10550  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-mhm 13766  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-rhm 14459  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:  mulgrhm2  14945