qusrhm - Metamath Proof Explorer

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Theorem qusrhm 19699

Description: If 𝑆 is a two-sided ideal in 𝑅, then the "natural map" from elements to their cosets is a ring homomorphism from 𝑅 to 𝑅 / 𝑆. (Contributed by Mario Carneiro, 15-Jun-2015.)

**Hypotheses**
Ref	Expression
qusring.u	⊢ 𝑈 = (𝑅 /_s (𝑅 ~_QG 𝑆))
qusring.i	⊢ 𝐼 = (2Ideal‘𝑅)
qusrhm.x	⊢ 𝑋 = (Base‘𝑅)
qusrhm.f	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ [𝑥](𝑅 ~_QG 𝑆))

**Assertion**
Ref	Expression
qusrhm	⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝐹 ∈ (𝑅 RingHom 𝑈))

Distinct variable groups: 𝑥,𝐼 𝑥,𝑅 𝑥,𝑆 𝑥,𝑈 𝑥,𝑋 Allowed substitution hint: 𝐹(𝑥)

Proof of Theorem qusrhm Dummy variables 𝑦 𝑧 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qusrhm.x	. 2 ⊢ 𝑋 = (Base‘𝑅)
2		eqid 2795	. 2 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
3		eqid 2795	. 2 ⊢ (1_r‘𝑈) = (1_r‘𝑈)
4		eqid 2795	. 2 ⊢ (._r‘𝑅) = (._r‘𝑅)
5		eqid 2795	. 2 ⊢ (._r‘𝑈) = (._r‘𝑈)
6		simpl 483	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑅 ∈ Ring)
7		qusring.u	. . 3 ⊢ 𝑈 = (𝑅 /_s (𝑅 ~_QG 𝑆))
8		qusring.i	. . 3 ⊢ 𝐼 = (2Ideal‘𝑅)
9	7, 8	qusring 19698	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ Ring)
10		eqid 2795	. . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
11		eqid 2795	. . . . . . . . 9 ⊢ (opp_r‘𝑅) = (opp_r‘𝑅)
12		eqid 2795	. . . . . . . . 9 ⊢ (LIdeal‘(opp_r‘𝑅)) = (LIdeal‘(opp_r‘𝑅))
13	10, 11, 12, 8	2idlval 19695	. . . . . . . 8 ⊢ 𝐼 = ((LIdeal‘𝑅) ∩ (LIdeal‘(opp_r‘𝑅)))
14	13	elin2 4095	. . . . . . 7 ⊢ (𝑆 ∈ 𝐼 ↔ (𝑆 ∈ (LIdeal‘𝑅) ∧ 𝑆 ∈ (LIdeal‘(opp_r‘𝑅))))
15	14	simplbi 498	. . . . . 6 ⊢ (𝑆 ∈ 𝐼 → 𝑆 ∈ (LIdeal‘𝑅))
16	10	lidlsubg 19677	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ (LIdeal‘𝑅)) → 𝑆 ∈ (SubGrp‘𝑅))
17	15, 16	sylan2 592	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (SubGrp‘𝑅))
18		eqid 2795	. . . . . 6 ⊢ (𝑅 ~_QG 𝑆) = (𝑅 ~_QG 𝑆)
19	1, 18	eqger 18083	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝑅) → (𝑅 ~_QG 𝑆) Er 𝑋)
20	17, 19	syl 17	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝑅 ~_QG 𝑆) Er 𝑋)
21	1	fvexi 6552	. . . . 5 ⊢ 𝑋 ∈ V
22	21	a1i 11	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑋 ∈ V)
23		qusrhm.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ [𝑥](𝑅 ~_QG 𝑆))
24	20, 22, 23	divsfval 16649	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝐹‘(1_r‘𝑅)) = [(1_r‘𝑅)](𝑅 ~_QG 𝑆))
25	7, 8, 2	qus1 19697	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝑈 ∈ Ring ∧ [(1_r‘𝑅)](𝑅 ~_QG 𝑆) = (1_r‘𝑈)))
26	25	simprd 496	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → [(1_r‘𝑅)](𝑅 ~_QG 𝑆) = (1_r‘𝑈))
27	24, 26	eqtrd 2831	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝐹‘(1_r‘𝑅)) = (1_r‘𝑈))
28	7	a1i 11	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑈 = (𝑅 /_s (𝑅 ~_QG 𝑆)))
29	1	a1i 11	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑋 = (Base‘𝑅))
30	1, 18, 8, 4	2idlcpbl 19696	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → ((𝑎(𝑅 ~_QG 𝑆)𝑐 ∧ 𝑏(𝑅 ~_QG 𝑆)𝑑) → (𝑎(._r‘𝑅)𝑏)(𝑅 ~_QG 𝑆)(𝑐(._r‘𝑅)𝑑)))
31	1, 4	ringcl 19001	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦(._r‘𝑅)𝑧) ∈ 𝑋)
32	31	3expb 1113	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(._r‘𝑅)𝑧) ∈ 𝑋)
33	32	adantlr 711	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(._r‘𝑅)𝑧) ∈ 𝑋)
34	33	caovclg 7196	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) → (𝑐(._r‘𝑅)𝑑) ∈ 𝑋)
35	28, 29, 20, 6, 30, 34, 4, 5	qusmulval 16657	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → ([𝑦](𝑅 ~_QG 𝑆)(._r‘𝑈)[𝑧](𝑅 ~_QG 𝑆)) = [(𝑦(._r‘𝑅)𝑧)](𝑅 ~_QG 𝑆))
36	35	3expb 1113	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ([𝑦](𝑅 ~_QG 𝑆)(._r‘𝑈)[𝑧](𝑅 ~_QG 𝑆)) = [(𝑦(._r‘𝑅)𝑧)](𝑅 ~_QG 𝑆))
37	20	adantr 481	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑅 ~_QG 𝑆) Er 𝑋)
38	21	a1i 11	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑋 ∈ V)
39	37, 38, 23	divsfval 16649	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑦) = [𝑦](𝑅 ~_QG 𝑆))
40	37, 38, 23	divsfval 16649	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑧) = [𝑧](𝑅 ~_QG 𝑆))
41	39, 40	oveq12d 7034	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐹‘𝑦)(._r‘𝑈)(𝐹‘𝑧)) = ([𝑦](𝑅 ~_QG 𝑆)(._r‘𝑈)[𝑧](𝑅 ~_QG 𝑆)))
42	37, 38, 23	divsfval 16649	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘(𝑦(._r‘𝑅)𝑧)) = [(𝑦(._r‘𝑅)𝑧)](𝑅 ~_QG 𝑆))
43	36, 41, 42	3eqtr4rd 2842	. 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘(𝑦(._r‘𝑅)𝑧)) = ((𝐹‘𝑦)(._r‘𝑈)(𝐹‘𝑧)))
44		ringabl 19020	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel)
45	44	adantr 481	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑅 ∈ Abel)
46		ablnsg 18690	. . . . 5 ⊢ (𝑅 ∈ Abel → (NrmSGrp‘𝑅) = (SubGrp‘𝑅))
47	45, 46	syl 17	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (NrmSGrp‘𝑅) = (SubGrp‘𝑅))
48	17, 47	eleqtrrd 2886	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (NrmSGrp‘𝑅))
49	1, 7, 23	qusghm 18136	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝑅) → 𝐹 ∈ (𝑅 GrpHom 𝑈))
50	48, 49	syl 17	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝐹 ∈ (𝑅 GrpHom 𝑈))
51	1, 2, 3, 4, 5, 6, 9, 27, 43, 50	isrhm2d 19170	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝐹 ∈ (𝑅 RingHom 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 Vcvv 3437 ↦ cmpt 5041 ‘cfv 6225 (class class class)co 7016 Er wer 8136 [cec 8137 Basecbs 16312 ._rcmulr 16395 /_s cqus 16607 SubGrpcsubg 18027 NrmSGrpcnsg 18028 ~_QG cqg 18029 GrpHom cghm 18096 Abelcabl 18634 1_rcur 18941 Ringcrg 18987 opp_rcoppr 19062 RingHom crh 19154 LIdealclidl 19632 2Idealc2idl 19693

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-tpos 7743 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-er 8139 df-ec 8141 df-qs 8145 df-map 8258 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-sup 8752 df-inf 8753 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-z 11830 df-dec 11948 df-uz 12094 df-fz 12743 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-sca 16410 df-vsca 16411 df-ip 16412 df-tset 16413 df-ple 16414 df-ds 16416 df-0g 16544 df-imas 16610 df-qus 16611 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-mhm 17774 df-grp 17864 df-minusg 17865 df-sbg 17866 df-subg 18030 df-nsg 18031 df-eqg 18032 df-ghm 18097 df-cmn 18635 df-abl 18636 df-mgp 18930 df-ur 18942 df-ring 18989 df-oppr 19063 df-rnghom 19157 df-subrg 19223 df-lmod 19326 df-lss 19394 df-sra 19634 df-rgmod 19635 df-lidl 19636 df-2idl 19694

This theorem is referenced by: znzrh2 20374

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