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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 2818 | . 2 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
3 | eqid 2818 | . 2 ⊢ (1r‘𝑈) = (1r‘𝑈) | |
4 | eqid 2818 | . 2 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | eqid 2818 | . 2 ⊢ (.r‘𝑈) = (.r‘𝑈) | |
6 | simpl 483 | . 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 19937 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ Ring) |
10 | eqid 2818 | . . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
11 | eqid 2818 | . . . . . . . . 9 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
12 | eqid 2818 | . . . . . . . . 9 ⊢ (LIdeal‘(oppr‘𝑅)) = (LIdeal‘(oppr‘𝑅)) | |
13 | 10, 11, 12, 8 | 2idlval 19934 | . . . . . . . 8 ⊢ 𝐼 = ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr‘𝑅))) |
14 | 13 | elin2 4171 | . . . . . . 7 ⊢ (𝑆 ∈ 𝐼 ↔ (𝑆 ∈ (LIdeal‘𝑅) ∧ 𝑆 ∈ (LIdeal‘(oppr‘𝑅)))) |
15 | 14 | simplbi 498 | . . . . . 6 ⊢ (𝑆 ∈ 𝐼 → 𝑆 ∈ (LIdeal‘𝑅)) |
16 | 10 | lidlsubg 19916 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ (LIdeal‘𝑅)) → 𝑆 ∈ (SubGrp‘𝑅)) |
17 | 15, 16 | sylan2 592 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (SubGrp‘𝑅)) |
18 | eqid 2818 | . . . . . 6 ⊢ (𝑅 ~QG 𝑆) = (𝑅 ~QG 𝑆) | |
19 | 1, 18 | eqger 18268 | . . . . 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 6677 | . . . . 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 16808 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝐹‘(1r‘𝑅)) = [(1r‘𝑅)](𝑅 ~QG 𝑆)) |
25 | 7, 8, 2 | qus1 19936 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝑈 ∈ Ring ∧ [(1r‘𝑅)](𝑅 ~QG 𝑆) = (1r‘𝑈))) |
26 | 25 | simprd 496 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → [(1r‘𝑅)](𝑅 ~QG 𝑆) = (1r‘𝑈)) |
27 | 24, 26 | eqtrd 2853 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝐹‘(1r‘𝑅)) = (1r‘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 19935 | . . . . 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 19240 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦(.r‘𝑅)𝑧) ∈ 𝑋) |
32 | 31 | 3expb 1112 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(.r‘𝑅)𝑧) ∈ 𝑋) |
33 | 32 | adantlr 711 | . . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(.r‘𝑅)𝑧) ∈ 𝑋) |
34 | 33 | caovclg 7329 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) → (𝑐(.r‘𝑅)𝑑) ∈ 𝑋) |
35 | 28, 29, 20, 6, 30, 34, 4, 5 | qusmulval 16816 | . . . 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 1112 | . . 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 481 | . . . . 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 16808 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑦) = [𝑦](𝑅 ~QG 𝑆)) |
40 | 37, 38, 23 | divsfval 16808 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑧) = [𝑧](𝑅 ~QG 𝑆)) |
41 | 39, 40 | oveq12d 7163 | . . 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 16808 | . . 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 2864 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘(𝑦(.r‘𝑅)𝑧)) = ((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧))) |
44 | ringabl 19259 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel) | |
45 | 44 | adantr 481 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑅 ∈ Abel) |
46 | ablnsg 18896 | . . . . 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 2913 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (NrmSGrp‘𝑅)) |
49 | 1, 7, 23 | qusghm 18333 | . . 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 19409 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝐹 ∈ (𝑅 RingHom 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 Er wer 8275 [cec 8276 Basecbs 16471 .rcmulr 16554 /s cqus 16766 SubGrpcsubg 18211 NrmSGrpcnsg 18212 ~QG cqg 18213 GrpHom cghm 18293 Abelcabl 18836 1rcur 19180 Ringcrg 19226 opprcoppr 19301 RingHom crh 19393 LIdealclidl 19871 2Idealc2idl 19932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-ec 8280 df-qs 8284 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-0g 16703 df-imas 16769 df-qus 16770 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-nsg 18215 df-eqg 18216 df-ghm 18294 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-rnghom 19396 df-subrg 19462 df-lmod 19565 df-lss 19633 df-sra 19873 df-rgmod 19874 df-lidl 19875 df-2idl 19933 |
This theorem is referenced by: znzrh2 20620 |
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