qusrhm - Metamath Proof Explorer

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Theorem qusrhm 21377

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 2765	. 2 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
3		eqid 2765	. 2 ⊢ (1_r‘𝑈) = (1_r‘𝑈)
4		eqid 2765	. 2 ⊢ (._r‘𝑅) = (._r‘𝑅)
5		eqid 2765	. 2 ⊢ (._r‘𝑈) = (._r‘𝑈)
6		simpl 487	. 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 21376	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ Ring)
10		eqid 2765	. . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
11		eqid 2765	. . . . . . . . 9 ⊢ (opp_r‘𝑅) = (opp_r‘𝑅)
12		eqid 2765	. . . . . . . . 9 ⊢ (LIdeal‘(opp_r‘𝑅)) = (LIdeal‘(opp_r‘𝑅))
13	10, 11, 12, 8	2idlval 21352	. . . . . . . 8 ⊢ 𝐼 = ((LIdeal‘𝑅) ∩ (LIdeal‘(opp_r‘𝑅)))
14	13	elin2 4158	. . . . . . 7 ⊢ (𝑆 ∈ 𝐼 ↔ (𝑆 ∈ (LIdeal‘𝑅) ∧ 𝑆 ∈ (LIdeal‘(opp_r‘𝑅))))
15	14	simplbi 501	. . . . . 6 ⊢ (𝑆 ∈ 𝐼 → 𝑆 ∈ (LIdeal‘𝑅))
16	10	lidlsubg 21317	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ (LIdeal‘𝑅)) → 𝑆 ∈ (SubGrp‘𝑅))
17	15, 16	sylan2 604	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (SubGrp‘𝑅))
18		eqid 2765	. . . . . 6 ⊢ (𝑅 ~_QG 𝑆) = (𝑅 ~_QG 𝑆)
19	1, 18	eqger 19237	. . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝑅) → (𝑅 ~_QG 𝑆) Er 𝑋)
20	17, 19	syl 18	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝑅 ~_QG 𝑆) Er 𝑋)
21	1	fvexi 6885	. . . . 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 17591	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝐹‘(1_r‘𝑅)) = [(1_r‘𝑅)](𝑅 ~_QG 𝑆))
25	7, 8, 2	qus1 21375	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝑈 ∈ Ring ∧ [(1_r‘𝑅)](𝑅 ~_QG 𝑆) = (1_r‘𝑈)))
26	25	simprd 500	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → [(1_r‘𝑅)](𝑅 ~_QG 𝑆) = (1_r‘𝑈))
27	24, 26	eqtrd 2800	. 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 21373	. . . . 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 20323	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦(._r‘𝑅)𝑧) ∈ 𝑋)
32	31	3expb 1136	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(._r‘𝑅)𝑧) ∈ 𝑋)
33	32	adantlr 727	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(._r‘𝑅)𝑧) ∈ 𝑋)
34	33	caovclg 7592	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) → (𝑐(._r‘𝑅)𝑑) ∈ 𝑋)
35	28, 29, 20, 6, 30, 34, 4, 5	qusmulval 17599	. . . 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 1136	. . 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 485	. . . . 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 17591	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑦) = [𝑦](𝑅 ~_QG 𝑆))
40	37, 38, 23	divsfval 17591	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑧) = [𝑧](𝑅 ~_QG 𝑆))
41	39, 40	oveq12d 7418	. . 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 17591	. . 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 2811	. 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘(𝑦(._r‘𝑅)𝑧)) = ((𝐹‘𝑦)(._r‘𝑈)(𝐹‘𝑧)))
44		ringabl 20355	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel)
45	44	adantr 485	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑅 ∈ Abel)
46		ablnsg 19908	. . . . 5 ⊢ (𝑅 ∈ Abel → (NrmSGrp‘𝑅) = (SubGrp‘𝑅))
47	45, 46	syl 18	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (NrmSGrp‘𝑅) = (SubGrp‘𝑅))
48	17, 47	eleqtrrd 2868	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (NrmSGrp‘𝑅))
49	1, 7, 23	qusghm 19316	. . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝑅) → 𝐹 ∈ (𝑅 GrpHom 𝑈))
50	48, 49	syl 18	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝐹 ∈ (𝑅 GrpHom 𝑈))
51	1, 2, 3, 4, 5, 6, 9, 27, 43, 50	isrhm2d 20560	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝐹 ∈ (𝑅 RingHom 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1563 ∈ wcel 2145 Vcvv 3457 ↦ cmpt 5186 ‘cfv 6525 (class class class)co 7400 Er wer 8679 [cec 8680 Basecbs 17259 ._rcmulr 17301 /_s cqus 17549 SubGrpcsubg 19177 NrmSGrpcnsg 19178 ~_QG cqg 19179 GrpHom cghm 19274 Abelcabl 19842 1_rcur 20254 Ringcrg 20306 opp_rcoppr 20409 RingHom crh 20542 LIdealclidl 21299 2Idealc2idl 21350

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-tpos 8210 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-er 8682 df-ec 8684 df-qs 8688 df-map 8814 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-sup 9390 df-inf 9391 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-uz 12854 df-fz 13527 df-struct 17197 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-mulr 17314 df-sca 17316 df-vsca 17317 df-ip 17318 df-tset 17319 df-ple 17320 df-ds 17322 df-0g 17484 df-imas 17552 df-qus 17553 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-mhm 18831 df-grp 18993 df-minusg 18994 df-sbg 18995 df-subg 19180 df-nsg 19181 df-eqg 19182 df-ghm 19275 df-cmn 19843 df-abl 19844 df-mgp 20208 df-rng 20222 df-ur 20255 df-ring 20308 df-oppr 20410 df-rhm 20545 df-subrg 20646 df-lmod 20952 df-lss 21022 df-sra 21263 df-rgmod 21264 df-lidl 21301 df-2idl 21351

This theorem is referenced by: znzrh2 21655

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