qusrhm - Metamath Proof Explorer

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Theorem qusrhm 21201

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 2729	. 2 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
3		eqid 2729	. 2 ⊢ (1_r‘𝑈) = (1_r‘𝑈)
4		eqid 2729	. 2 ⊢ (._r‘𝑅) = (._r‘𝑅)
5		eqid 2729	. 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 21200	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ Ring)
10		eqid 2729	. . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
11		eqid 2729	. . . . . . . . 9 ⊢ (opp_r‘𝑅) = (opp_r‘𝑅)
12		eqid 2729	. . . . . . . . 9 ⊢ (LIdeal‘(opp_r‘𝑅)) = (LIdeal‘(opp_r‘𝑅))
13	10, 11, 12, 8	2idlval 21176	. . . . . . . 8 ⊢ 𝐼 = ((LIdeal‘𝑅) ∩ (LIdeal‘(opp_r‘𝑅)))
14	13	elin2 4156	. . . . . . 7 ⊢ (𝑆 ∈ 𝐼 ↔ (𝑆 ∈ (LIdeal‘𝑅) ∧ 𝑆 ∈ (LIdeal‘(opp_r‘𝑅))))
15	14	simplbi 497	. . . . . 6 ⊢ (𝑆 ∈ 𝐼 → 𝑆 ∈ (LIdeal‘𝑅))
16	10	lidlsubg 21148	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ (LIdeal‘𝑅)) → 𝑆 ∈ (SubGrp‘𝑅))
17	15, 16	sylan2 593	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (SubGrp‘𝑅))
18		eqid 2729	. . . . . 6 ⊢ (𝑅 ~_QG 𝑆) = (𝑅 ~_QG 𝑆)
19	1, 18	eqger 19075	. . . . 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 6840	. . . . 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 17469	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝐹‘(1_r‘𝑅)) = [(1_r‘𝑅)](𝑅 ~_QG 𝑆))
25	7, 8, 2	qus1 21199	. . . 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 2764	. 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 21197	. . . . 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 20153	. . . . . . . 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 7545	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) → (𝑐(._r‘𝑅)𝑑) ∈ 𝑋)
35	28, 29, 20, 6, 30, 34, 4, 5	qusmulval 17477	. . . 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 17469	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑦) = [𝑦](𝑅 ~_QG 𝑆))
40	37, 38, 23	divsfval 17469	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑧) = [𝑧](𝑅 ~_QG 𝑆))
41	39, 40	oveq12d 7371	. . 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 17469	. . 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 2775	. 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘(𝑦(._r‘𝑅)𝑧)) = ((𝐹‘𝑦)(._r‘𝑈)(𝐹‘𝑧)))
44		ringabl 20184	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel)
45	44	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑅 ∈ Abel)
46		ablnsg 19744	. . . . 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 2831	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (NrmSGrp‘𝑅))
49	1, 7, 23	qusghm 19152	. . 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 20390	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝐹 ∈ (𝑅 RingHom 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3438 ↦ cmpt 5176 ‘cfv 6486 (class class class)co 7353 Er wer 8629 [cec 8630 Basecbs 17138 ._rcmulr 17180 /_s cqus 17427 SubGrpcsubg 19017 NrmSGrpcnsg 19018 ~_QG cqg 19019 GrpHom cghm 19109 Abelcabl 19678 1_rcur 20084 Ringcrg 20136 opp_rcoppr 20239 RingHom crh 20372 LIdealclidl 21131 2Idealc2idl 21174

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-ec 8634 df-qs 8638 df-map 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9351 df-inf 9352 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-fz 13429 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-0g 17363 df-imas 17430 df-qus 17431 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-mhm 18675 df-grp 18833 df-minusg 18834 df-sbg 18835 df-subg 19020 df-nsg 19021 df-eqg 19022 df-ghm 19110 df-cmn 19679 df-abl 19680 df-mgp 20044 df-rng 20056 df-ur 20085 df-ring 20138 df-oppr 20240 df-rhm 20375 df-subrg 20473 df-lmod 20783 df-lss 20853 df-sra 21095 df-rgmod 21096 df-lidl 21133 df-2idl 21175

This theorem is referenced by: znzrh2 21470

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