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Mirrors > Home > MPE Home > Th. List > qusrhm | Structured version Visualization version GIF version |
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.) |
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)) |
Ref | Expression |
---|---|
qusrhm | ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝐹 ∈ (𝑅 RingHom 𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qusrhm.x | . 2 ⊢ 𝑋 = (Base‘𝑅) | |
2 | eqid 2752 | . 2 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
3 | eqid 2752 | . 2 ⊢ (1r‘𝑈) = (1r‘𝑈) | |
4 | eqid 2752 | . 2 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | eqid 2752 | . 2 ⊢ (.r‘𝑈) = (.r‘𝑈) | |
6 | simpl 474 | . 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 19430 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ Ring) |
10 | eqid 2752 | . . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
11 | eqid 2752 | . . . . . . . . 9 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
12 | eqid 2752 | . . . . . . . . 9 ⊢ (LIdeal‘(oppr‘𝑅)) = (LIdeal‘(oppr‘𝑅)) | |
13 | 10, 11, 12, 8 | 2idlval 19427 | . . . . . . . 8 ⊢ 𝐼 = ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr‘𝑅))) |
14 | 13 | elin2 3936 | . . . . . . 7 ⊢ (𝑆 ∈ 𝐼 ↔ (𝑆 ∈ (LIdeal‘𝑅) ∧ 𝑆 ∈ (LIdeal‘(oppr‘𝑅)))) |
15 | 14 | simplbi 478 | . . . . . 6 ⊢ (𝑆 ∈ 𝐼 → 𝑆 ∈ (LIdeal‘𝑅)) |
16 | 10 | lidlsubg 19409 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ (LIdeal‘𝑅)) → 𝑆 ∈ (SubGrp‘𝑅)) |
17 | 15, 16 | sylan2 492 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (SubGrp‘𝑅)) |
18 | eqid 2752 | . . . . . 6 ⊢ (𝑅 ~QG 𝑆) = (𝑅 ~QG 𝑆) | |
19 | 1, 18 | eqger 17837 | . . . . 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 | fvex 6354 | . . . . . 6 ⊢ (Base‘𝑅) ∈ V | |
22 | 1, 21 | eqeltri 2827 | . . . . 5 ⊢ 𝑋 ∈ V |
23 | 22 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑋 ∈ V) |
24 | qusrhm.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ [𝑥](𝑅 ~QG 𝑆)) | |
25 | 20, 23, 24 | divsfval 16401 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝐹‘(1r‘𝑅)) = [(1r‘𝑅)](𝑅 ~QG 𝑆)) |
26 | 7, 8, 2 | qus1 19429 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝑈 ∈ Ring ∧ [(1r‘𝑅)](𝑅 ~QG 𝑆) = (1r‘𝑈))) |
27 | 26 | simprd 482 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → [(1r‘𝑅)](𝑅 ~QG 𝑆) = (1r‘𝑈)) |
28 | 25, 27 | eqtrd 2786 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝐹‘(1r‘𝑅)) = (1r‘𝑈)) |
29 | 7 | a1i 11 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))) |
30 | 1 | a1i 11 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑋 = (Base‘𝑅)) |
31 | 1, 18, 8, 4 | 2idlcpbl 19428 | . . . . 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))) |
32 | 1, 4 | ringcl 18753 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦(.r‘𝑅)𝑧) ∈ 𝑋) |
33 | 32 | 3expb 1113 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(.r‘𝑅)𝑧) ∈ 𝑋) |
34 | 33 | adantlr 753 | . . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(.r‘𝑅)𝑧) ∈ 𝑋) |
35 | 34 | caovclg 6983 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) → (𝑐(.r‘𝑅)𝑑) ∈ 𝑋) |
36 | 29, 30, 20, 6, 31, 35, 4, 5 | qusmulval 16409 | . . . 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)) |
37 | 36 | 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)) |
38 | 20 | adantr 472 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑅 ~QG 𝑆) Er 𝑋) |
39 | 22 | a1i 11 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑋 ∈ V) |
40 | 38, 39, 24 | divsfval 16401 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑦) = [𝑦](𝑅 ~QG 𝑆)) |
41 | 38, 39, 24 | divsfval 16401 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑧) = [𝑧](𝑅 ~QG 𝑆)) |
42 | 40, 41 | oveq12d 6823 | . . 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))) |
43 | 38, 39, 24 | divsfval 16401 | . . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘(𝑦(.r‘𝑅)𝑧)) = [(𝑦(.r‘𝑅)𝑧)](𝑅 ~QG 𝑆)) |
44 | 37, 42, 43 | 3eqtr4rd 2797 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘(𝑦(.r‘𝑅)𝑧)) = ((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧))) |
45 | ringabl 18772 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel) | |
46 | 45 | adantr 472 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑅 ∈ Abel) |
47 | ablnsg 18442 | . . . . 5 ⊢ (𝑅 ∈ Abel → (NrmSGrp‘𝑅) = (SubGrp‘𝑅)) | |
48 | 46, 47 | syl 17 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (NrmSGrp‘𝑅) = (SubGrp‘𝑅)) |
49 | 17, 48 | eleqtrrd 2834 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (NrmSGrp‘𝑅)) |
50 | 1, 7, 24 | qusghm 17890 | . . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝑅) → 𝐹 ∈ (𝑅 GrpHom 𝑈)) |
51 | 49, 50 | syl 17 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝐹 ∈ (𝑅 GrpHom 𝑈)) |
52 | 1, 2, 3, 4, 5, 6, 9, 28, 44, 51 | isrhm2d 18922 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝐹 ∈ (𝑅 RingHom 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1624 ∈ wcel 2131 Vcvv 3332 ↦ cmpt 4873 ‘cfv 6041 (class class class)co 6805 Er wer 7900 [cec 7901 Basecbs 16051 .rcmulr 16136 /s cqus 16359 SubGrpcsubg 17781 NrmSGrpcnsg 17782 ~QG cqg 17783 GrpHom cghm 17850 Abelcabl 18386 1rcur 18693 Ringcrg 18739 opprcoppr 18814 RingHom crh 18906 LIdealclidl 19364 2Idealc2idl 19425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-rep 4915 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-cnex 10176 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rmo 3050 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-int 4620 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-om 7223 df-1st 7325 df-2nd 7326 df-tpos 7513 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-1o 7721 df-oadd 7725 df-er 7903 df-ec 7905 df-qs 7909 df-map 8017 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-sup 8505 df-inf 8506 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-nn 11205 df-2 11263 df-3 11264 df-4 11265 df-5 11266 df-6 11267 df-7 11268 df-8 11269 df-9 11270 df-n0 11477 df-z 11562 df-dec 11678 df-uz 11872 df-fz 12512 df-struct 16053 df-ndx 16054 df-slot 16055 df-base 16057 df-sets 16058 df-ress 16059 df-plusg 16148 df-mulr 16149 df-sca 16151 df-vsca 16152 df-ip 16153 df-tset 16154 df-ple 16155 df-ds 16158 df-0g 16296 df-imas 16362 df-qus 16363 df-mgm 17435 df-sgrp 17477 df-mnd 17488 df-mhm 17528 df-grp 17618 df-minusg 17619 df-sbg 17620 df-subg 17784 df-nsg 17785 df-eqg 17786 df-ghm 17851 df-cmn 18387 df-abl 18388 df-mgp 18682 df-ur 18694 df-ring 18741 df-oppr 18815 df-rnghom 18909 df-subrg 18972 df-lmod 19059 df-lss 19127 df-sra 19366 df-rgmod 19367 df-lidl 19368 df-2idl 19426 |
This theorem is referenced by: znzrh2 20088 |
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