Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > qus1 | Structured version Visualization version GIF version |
Description: The multiplicative identity of the quotient ring. (Contributed by Mario Carneiro, 14-Jun-2015.) |
Ref | Expression |
---|---|
qusring.u | ⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆)) |
qusring.i | ⊢ 𝐼 = (2Ideal‘𝑅) |
qus1.o | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
qus1 | ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝑈 ∈ Ring ∧ [ 1 ](𝑅 ~QG 𝑆) = (1r‘𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qusring.u | . . 3 ⊢ 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆)) | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))) |
3 | eqid 2739 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | 3 | a1i 11 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (Base‘𝑅) = (Base‘𝑅)) |
5 | eqid 2739 | . 2 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
6 | eqid 2739 | . 2 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
7 | qus1.o | . 2 ⊢ 1 = (1r‘𝑅) | |
8 | eqid 2739 | . . . . . . 7 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
9 | eqid 2739 | . . . . . . 7 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
10 | eqid 2739 | . . . . . . 7 ⊢ (LIdeal‘(oppr‘𝑅)) = (LIdeal‘(oppr‘𝑅)) | |
11 | qusring.i | . . . . . . 7 ⊢ 𝐼 = (2Ideal‘𝑅) | |
12 | 8, 9, 10, 11 | 2idlval 20138 | . . . . . 6 ⊢ 𝐼 = ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr‘𝑅))) |
13 | 12 | elin2 4097 | . . . . 5 ⊢ (𝑆 ∈ 𝐼 ↔ (𝑆 ∈ (LIdeal‘𝑅) ∧ 𝑆 ∈ (LIdeal‘(oppr‘𝑅)))) |
14 | 13 | simplbi 501 | . . . 4 ⊢ (𝑆 ∈ 𝐼 → 𝑆 ∈ (LIdeal‘𝑅)) |
15 | 8 | lidlsubg 20120 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ (LIdeal‘𝑅)) → 𝑆 ∈ (SubGrp‘𝑅)) |
16 | 14, 15 | sylan2 596 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (SubGrp‘𝑅)) |
17 | eqid 2739 | . . . 4 ⊢ (𝑅 ~QG 𝑆) = (𝑅 ~QG 𝑆) | |
18 | 3, 17 | eqger 18461 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝑅) → (𝑅 ~QG 𝑆) Er (Base‘𝑅)) |
19 | 16, 18 | syl 17 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝑅 ~QG 𝑆) Er (Base‘𝑅)) |
20 | ringabl 19465 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel) | |
21 | 20 | adantr 484 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑅 ∈ Abel) |
22 | ablnsg 19099 | . . . . 5 ⊢ (𝑅 ∈ Abel → (NrmSGrp‘𝑅) = (SubGrp‘𝑅)) | |
23 | 21, 22 | syl 17 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (NrmSGrp‘𝑅) = (SubGrp‘𝑅)) |
24 | 16, 23 | eleqtrrd 2837 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (NrmSGrp‘𝑅)) |
25 | 3, 17, 5 | eqgcpbl 18465 | . . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝑅) → ((𝑎(𝑅 ~QG 𝑆)𝑐 ∧ 𝑏(𝑅 ~QG 𝑆)𝑑) → (𝑎(+g‘𝑅)𝑏)(𝑅 ~QG 𝑆)(𝑐(+g‘𝑅)𝑑))) |
26 | 24, 25 | syl 17 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → ((𝑎(𝑅 ~QG 𝑆)𝑐 ∧ 𝑏(𝑅 ~QG 𝑆)𝑑) → (𝑎(+g‘𝑅)𝑏)(𝑅 ~QG 𝑆)(𝑐(+g‘𝑅)𝑑))) |
27 | 3, 17, 11, 6 | 2idlcpbl 20139 | . 2 ⊢ ((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))) |
28 | simpl 486 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑅 ∈ Ring) | |
29 | 2, 4, 5, 6, 7, 19, 26, 27, 28 | qusring2 19505 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝑈 ∈ Ring ∧ [ 1 ](𝑅 ~QG 𝑆) = (1r‘𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 class class class wbr 5040 ‘cfv 6350 (class class class)co 7183 Er wer 8330 [cec 8331 Basecbs 16599 +gcplusg 16681 .rcmulr 16682 /s cqus 16894 SubGrpcsubg 18404 NrmSGrpcnsg 18405 ~QG cqg 18406 Abelcabl 19038 1rcur 19383 Ringcrg 19429 opprcoppr 19507 LIdealclidl 20074 2Idealc2idl 20136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-1st 7727 df-2nd 7728 df-tpos 7934 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-1o 8144 df-er 8333 df-ec 8335 df-qs 8339 df-en 8569 df-dom 8570 df-sdom 8571 df-fin 8572 df-sup 8992 df-inf 8993 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-nn 11730 df-2 11792 df-3 11793 df-4 11794 df-5 11795 df-6 11796 df-7 11797 df-8 11798 df-9 11799 df-n0 11990 df-z 12076 df-dec 12193 df-uz 12338 df-fz 12995 df-struct 16601 df-ndx 16602 df-slot 16603 df-base 16605 df-sets 16606 df-ress 16607 df-plusg 16694 df-mulr 16695 df-sca 16697 df-vsca 16698 df-ip 16699 df-tset 16700 df-ple 16701 df-ds 16703 df-0g 16831 df-imas 16897 df-qus 16898 df-mgm 17981 df-sgrp 18030 df-mnd 18041 df-grp 18235 df-minusg 18236 df-sbg 18237 df-subg 18407 df-nsg 18408 df-eqg 18409 df-cmn 19039 df-abl 19040 df-mgp 19372 df-ur 19384 df-ring 19431 df-oppr 19508 df-subrg 19665 df-lmod 19768 df-lss 19836 df-sra 20076 df-rgmod 20077 df-lidl 20078 df-2idl 20137 |
This theorem is referenced by: qusring 20141 qusrhm 20142 |
Copyright terms: Public domain | W3C validator |