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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 2738 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | 3 | a1i 11 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (Base‘𝑅) = (Base‘𝑅)) |
5 | eqid 2738 | . 2 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
6 | eqid 2738 | . 2 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
7 | qus1.o | . 2 ⊢ 1 = (1r‘𝑅) | |
8 | eqid 2738 | . . . . . . 7 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
9 | eqid 2738 | . . . . . . 7 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
10 | eqid 2738 | . . . . . . 7 ⊢ (LIdeal‘(oppr‘𝑅)) = (LIdeal‘(oppr‘𝑅)) | |
11 | qusring.i | . . . . . . 7 ⊢ 𝐼 = (2Ideal‘𝑅) | |
12 | 8, 9, 10, 11 | 2idlval 20504 | . . . . . 6 ⊢ 𝐼 = ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr‘𝑅))) |
13 | 12 | elin2 4131 | . . . . 5 ⊢ (𝑆 ∈ 𝐼 ↔ (𝑆 ∈ (LIdeal‘𝑅) ∧ 𝑆 ∈ (LIdeal‘(oppr‘𝑅)))) |
14 | 13 | simplbi 498 | . . . 4 ⊢ (𝑆 ∈ 𝐼 → 𝑆 ∈ (LIdeal‘𝑅)) |
15 | 8 | lidlsubg 20486 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ (LIdeal‘𝑅)) → 𝑆 ∈ (SubGrp‘𝑅)) |
16 | 14, 15 | sylan2 593 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (SubGrp‘𝑅)) |
17 | eqid 2738 | . . . 4 ⊢ (𝑅 ~QG 𝑆) = (𝑅 ~QG 𝑆) | |
18 | 3, 17 | eqger 18806 | . . 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 19819 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel) | |
21 | 20 | adantr 481 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑅 ∈ Abel) |
22 | ablnsg 19448 | . . . . 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 2842 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (NrmSGrp‘𝑅)) |
25 | 3, 17, 5 | eqgcpbl 18810 | . . 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 20505 | . 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 483 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑅 ∈ Ring) | |
29 | 2, 4, 5, 6, 7, 19, 26, 27, 28 | qusring2 19859 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝑈 ∈ Ring ∧ [ 1 ](𝑅 ~QG 𝑆) = (1r‘𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 Er wer 8495 [cec 8496 Basecbs 16912 +gcplusg 16962 .rcmulr 16963 /s cqus 17216 SubGrpcsubg 18749 NrmSGrpcnsg 18750 ~QG cqg 18751 Abelcabl 19387 1rcur 19737 Ringcrg 19783 opprcoppr 19861 LIdealclidl 20432 2Idealc2idl 20502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-ec 8500 df-qs 8504 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-0g 17152 df-imas 17219 df-qus 17220 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-nsg 18753 df-eqg 18754 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-oppr 19862 df-subrg 20022 df-lmod 20125 df-lss 20194 df-sra 20434 df-rgmod 20435 df-lidl 20436 df-2idl 20503 |
This theorem is referenced by: qusring 20507 qusrhm 20508 |
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