qusrhm - Metamath Proof Explorer

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Theorem qusrhm 21206

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 2730	. 2 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
3		eqid 2730	. 2 ⊢ (1_r‘𝑈) = (1_r‘𝑈)
4		eqid 2730	. 2 ⊢ (._r‘𝑅) = (._r‘𝑅)
5		eqid 2730	. 2 ⊢ (._r‘𝑈) = (._r‘𝑈)
6		simpl 482	. 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 21205	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ Ring)
10		eqid 2730	. . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
11		eqid 2730	. . . . . . . . 9 ⊢ (opp_r‘𝑅) = (opp_r‘𝑅)
12		eqid 2730	. . . . . . . . 9 ⊢ (LIdeal‘(opp_r‘𝑅)) = (LIdeal‘(opp_r‘𝑅))
13	10, 11, 12, 8	2idlval 21181	. . . . . . . 8 ⊢ 𝐼 = ((LIdeal‘𝑅) ∩ (LIdeal‘(opp_r‘𝑅)))
14	13	elin2 4151	. . . . . . 7 ⊢ (𝑆 ∈ 𝐼 ↔ (𝑆 ∈ (LIdeal‘𝑅) ∧ 𝑆 ∈ (LIdeal‘(opp_r‘𝑅))))
15	14	simplbi 497	. . . . . 6 ⊢ (𝑆 ∈ 𝐼 → 𝑆 ∈ (LIdeal‘𝑅))
16	10	lidlsubg 21153	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ (LIdeal‘𝑅)) → 𝑆 ∈ (SubGrp‘𝑅))
17	15, 16	sylan2 593	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (SubGrp‘𝑅))
18		eqid 2730	. . . . . 6 ⊢ (𝑅 ~_QG 𝑆) = (𝑅 ~_QG 𝑆)
19	1, 18	eqger 19083	. . . . 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 6831	. . . . 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 17443	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝐹‘(1_r‘𝑅)) = [(1_r‘𝑅)](𝑅 ~_QG 𝑆))
25	7, 8, 2	qus1 21204	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝑈 ∈ Ring ∧ [(1_r‘𝑅)](𝑅 ~_QG 𝑆) = (1_r‘𝑈)))
26	25	simprd 495	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → [(1_r‘𝑅)](𝑅 ~_QG 𝑆) = (1_r‘𝑈))
27	24, 26	eqtrd 2765	. 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 21202	. . . . 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 20161	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦(._r‘𝑅)𝑧) ∈ 𝑋)
32	31	3expb 1120	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(._r‘𝑅)𝑧) ∈ 𝑋)
33	32	adantlr 715	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(._r‘𝑅)𝑧) ∈ 𝑋)
34	33	caovclg 7533	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) → (𝑐(._r‘𝑅)𝑑) ∈ 𝑋)
35	28, 29, 20, 6, 30, 34, 4, 5	qusmulval 17451	. . . 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 1120	. . 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 480	. . . . 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 17443	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑦) = [𝑦](𝑅 ~_QG 𝑆))
40	37, 38, 23	divsfval 17443	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑧) = [𝑧](𝑅 ~_QG 𝑆))
41	39, 40	oveq12d 7359	. . 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 17443	. . 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 2776	. 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘(𝑦(._r‘𝑅)𝑧)) = ((𝐹‘𝑦)(._r‘𝑈)(𝐹‘𝑧)))
44		ringabl 20192	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel)
45	44	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑅 ∈ Abel)
46		ablnsg 19752	. . . . 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 2832	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (NrmSGrp‘𝑅))
49	1, 7, 23	qusghm 19160	. . 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 20397	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝐹 ∈ (𝑅 RingHom 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2110 Vcvv 3434 ↦ cmpt 5170 ‘cfv 6477 (class class class)co 7341 Er wer 8614 [cec 8615 Basecbs 17112 ._rcmulr 17154 /_s cqus 17401 SubGrpcsubg 19025 NrmSGrpcnsg 19026 ~_QG cqg 19027 GrpHom cghm 19117 Abelcabl 19686 1_rcur 20092 Ringcrg 20144 opp_rcoppr 20247 RingHom crh 20380 LIdealclidl 21136 2Idealc2idl 21179

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-tp 4579 df-op 4581 df-uni 4858 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-tpos 8151 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-er 8617 df-ec 8619 df-qs 8623 df-map 8747 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-sup 9321 df-inf 9322 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-nn 12118 df-2 12180 df-3 12181 df-4 12182 df-5 12183 df-6 12184 df-7 12185 df-8 12186 df-9 12187 df-n0 12374 df-z 12461 df-dec 12581 df-uz 12725 df-fz 13400 df-struct 17050 df-sets 17067 df-slot 17085 df-ndx 17097 df-base 17113 df-ress 17134 df-plusg 17166 df-mulr 17167 df-sca 17169 df-vsca 17170 df-ip 17171 df-tset 17172 df-ple 17173 df-ds 17175 df-0g 17337 df-imas 17404 df-qus 17405 df-mgm 18540 df-sgrp 18619 df-mnd 18635 df-mhm 18683 df-grp 18841 df-minusg 18842 df-sbg 18843 df-subg 19028 df-nsg 19029 df-eqg 19030 df-ghm 19118 df-cmn 19687 df-abl 19688 df-mgp 20052 df-rng 20064 df-ur 20093 df-ring 20146 df-oppr 20248 df-rhm 20383 df-subrg 20478 df-lmod 20788 df-lss 20858 df-sra 21100 df-rgmod 21101 df-lidl 21138 df-2idl 21180

This theorem is referenced by: znzrh2 21475

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