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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 2823 | . 2 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
3 | eqid 2823 | . 2 ⊢ (1r‘𝑈) = (1r‘𝑈) | |
4 | eqid 2823 | . 2 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | eqid 2823 | . 2 ⊢ (.r‘𝑈) = (.r‘𝑈) | |
6 | simpl 485 | . 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 20011 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑈 ∈ Ring) |
10 | eqid 2823 | . . . . . . . . 9 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
11 | eqid 2823 | . . . . . . . . 9 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
12 | eqid 2823 | . . . . . . . . 9 ⊢ (LIdeal‘(oppr‘𝑅)) = (LIdeal‘(oppr‘𝑅)) | |
13 | 10, 11, 12, 8 | 2idlval 20008 | . . . . . . . 8 ⊢ 𝐼 = ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr‘𝑅))) |
14 | 13 | elin2 4176 | . . . . . . 7 ⊢ (𝑆 ∈ 𝐼 ↔ (𝑆 ∈ (LIdeal‘𝑅) ∧ 𝑆 ∈ (LIdeal‘(oppr‘𝑅)))) |
15 | 14 | simplbi 500 | . . . . . 6 ⊢ (𝑆 ∈ 𝐼 → 𝑆 ∈ (LIdeal‘𝑅)) |
16 | 10 | lidlsubg 19990 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ (LIdeal‘𝑅)) → 𝑆 ∈ (SubGrp‘𝑅)) |
17 | 15, 16 | sylan2 594 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (SubGrp‘𝑅)) |
18 | eqid 2823 | . . . . . 6 ⊢ (𝑅 ~QG 𝑆) = (𝑅 ~QG 𝑆) | |
19 | 1, 18 | eqger 18332 | . . . . 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 6686 | . . . . 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 16822 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝐹‘(1r‘𝑅)) = [(1r‘𝑅)](𝑅 ~QG 𝑆)) |
25 | 7, 8, 2 | qus1 20010 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝑈 ∈ Ring ∧ [(1r‘𝑅)](𝑅 ~QG 𝑆) = (1r‘𝑈))) |
26 | 25 | simprd 498 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → [(1r‘𝑅)](𝑅 ~QG 𝑆) = (1r‘𝑈)) |
27 | 24, 26 | eqtrd 2858 | . 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 20009 | . . . . 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 19313 | . . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦(.r‘𝑅)𝑧) ∈ 𝑋) |
32 | 31 | 3expb 1116 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(.r‘𝑅)𝑧) ∈ 𝑋) |
33 | 32 | adantlr 713 | . . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(.r‘𝑅)𝑧) ∈ 𝑋) |
34 | 33 | caovclg 7342 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑐 ∈ 𝑋 ∧ 𝑑 ∈ 𝑋)) → (𝑐(.r‘𝑅)𝑑) ∈ 𝑋) |
35 | 28, 29, 20, 6, 30, 34, 4, 5 | qusmulval 16830 | . . . 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 1116 | . . 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 483 | . . . . 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 16822 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑦) = [𝑦](𝑅 ~QG 𝑆)) |
40 | 37, 38, 23 | divsfval 16822 | . . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑧) = [𝑧](𝑅 ~QG 𝑆)) |
41 | 39, 40 | oveq12d 7176 | . . 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 16822 | . . 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 2869 | . 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘(𝑦(.r‘𝑅)𝑧)) = ((𝐹‘𝑦)(.r‘𝑈)(𝐹‘𝑧))) |
44 | ringabl 19332 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel) | |
45 | 44 | adantr 483 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑅 ∈ Abel) |
46 | ablnsg 18969 | . . . . 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 2918 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (NrmSGrp‘𝑅)) |
49 | 1, 7, 23 | qusghm 18397 | . . 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 19482 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝐹 ∈ (𝑅 RingHom 𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 Er wer 8288 [cec 8289 Basecbs 16485 .rcmulr 16568 /s cqus 16780 SubGrpcsubg 18275 NrmSGrpcnsg 18276 ~QG cqg 18277 GrpHom cghm 18357 Abelcabl 18909 1rcur 19253 Ringcrg 19299 opprcoppr 19374 RingHom crh 19466 LIdealclidl 19944 2Idealc2idl 20006 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-ec 8293 df-qs 8297 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-0g 16717 df-imas 16783 df-qus 16784 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-nsg 18279 df-eqg 18280 df-ghm 18358 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-rnghom 19469 df-subrg 19535 df-lmod 19638 df-lss 19706 df-sra 19946 df-rgmod 19947 df-lidl 19948 df-2idl 20007 |
This theorem is referenced by: znzrh2 20694 |
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