qusrhm - Metamath Proof Explorer

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Theorem qusrhm 21309

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 2740	. 2 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
3		eqid 2740	. 2 ⊢ (1_r‘𝑈) = (1_r‘𝑈)
4		eqid 2740	. 2 ⊢ (._r‘𝑅) = (._r‘𝑅)
5		eqid 2740	. 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 21308	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ Ring)
10		eqid 2740	. . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
11		eqid 2740	. . . . . . . . 9 ⊢ (opp_r‘𝑅) = (opp_r‘𝑅)
12		eqid 2740	. . . . . . . . 9 ⊢ (LIdeal‘(opp_r‘𝑅)) = (LIdeal‘(opp_r‘𝑅))
13	10, 11, 12, 8	2idlval 21284	. . . . . . . 8 ⊢ 𝐼 = ((LIdeal‘𝑅) ∩ (LIdeal‘(opp_r‘𝑅)))
14	13	elin2 4226	. . . . . . 7 ⊢ (𝑆 ∈ 𝐼 ↔ (𝑆 ∈ (LIdeal‘𝑅) ∧ 𝑆 ∈ (LIdeal‘(opp_r‘𝑅))))
15	14	simplbi 497	. . . . . 6 ⊢ (𝑆 ∈ 𝐼 → 𝑆 ∈ (LIdeal‘𝑅))
16	10	lidlsubg 21256	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ (LIdeal‘𝑅)) → 𝑆 ∈ (SubGrp‘𝑅))
17	15, 16	sylan2 592	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (SubGrp‘𝑅))
18		eqid 2740	. . . . . 6 ⊢ (𝑅 ~_QG 𝑆) = (𝑅 ~_QG 𝑆)
19	1, 18	eqger 19218	. . . . 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 6934	. . . . 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 17607	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝐹‘(1_r‘𝑅)) = [(1_r‘𝑅)](𝑅 ~_QG 𝑆))
25	7, 8, 2	qus1 21307	. . . 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 2780	. 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 21305	. . . . 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 20277	. . . . . . . 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 714	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(._r‘𝑅)𝑧) ∈ 𝑋)
34	33	caovclg 7642	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) → (𝑐(._r‘𝑅)𝑑) ∈ 𝑋)
35	28, 29, 20, 6, 30, 34, 4, 5	qusmulval 17615	. . . 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 17607	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑦) = [𝑦](𝑅 ~_QG 𝑆))
40	37, 38, 23	divsfval 17607	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑧) = [𝑧](𝑅 ~_QG 𝑆))
41	39, 40	oveq12d 7466	. . 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 17607	. . 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 2791	. 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘(𝑦(._r‘𝑅)𝑧)) = ((𝐹‘𝑦)(._r‘𝑈)(𝐹‘𝑧)))
44		ringabl 20304	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel)
45	44	adantr 480	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑅 ∈ Abel)
46		ablnsg 19889	. . . . 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 2847	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (NrmSGrp‘𝑅))
49	1, 7, 23	qusghm 19295	. . 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 20513	1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝐹 ∈ (𝑅 RingHom 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 Vcvv 3488 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 Er wer 8760 [cec 8761 Basecbs 17258 ._rcmulr 17312 /_s cqus 17565 SubGrpcsubg 19160 NrmSGrpcnsg 19161 ~_QG cqg 19162 GrpHom cghm 19252 Abelcabl 19823 1_rcur 20208 Ringcrg 20260 opp_rcoppr 20359 RingHom crh 20495 LIdealclidl 21239 2Idealc2idl 21282

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-ec 8765 df-qs 8769 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-fz 13568 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-0g 17501 df-imas 17568 df-qus 17569 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-grp 18976 df-minusg 18977 df-sbg 18978 df-subg 19163 df-nsg 19164 df-eqg 19165 df-ghm 19253 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-oppr 20360 df-rhm 20498 df-subrg 20597 df-lmod 20882 df-lss 20953 df-sra 21195 df-rgmod 21196 df-lidl 21241 df-2idl 21283

This theorem is referenced by: znzrh2 21587

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