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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 2826 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | 3 | a1i 11 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (Base‘𝑅) = (Base‘𝑅)) |
5 | eqid 2826 | . 2 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
6 | eqid 2826 | . 2 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
7 | qus1.o | . 2 ⊢ 1 = (1r‘𝑅) | |
8 | eqid 2826 | . . . . . . 7 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
9 | eqid 2826 | . . . . . . 7 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
10 | eqid 2826 | . . . . . . 7 ⊢ (LIdeal‘(oppr‘𝑅)) = (LIdeal‘(oppr‘𝑅)) | |
11 | qusring.i | . . . . . . 7 ⊢ 𝐼 = (2Ideal‘𝑅) | |
12 | 8, 9, 10, 11 | 2idlval 19595 | . . . . . 6 ⊢ 𝐼 = ((LIdeal‘𝑅) ∩ (LIdeal‘(oppr‘𝑅))) |
13 | 12 | elin2 4029 | . . . . 5 ⊢ (𝑆 ∈ 𝐼 ↔ (𝑆 ∈ (LIdeal‘𝑅) ∧ 𝑆 ∈ (LIdeal‘(oppr‘𝑅)))) |
14 | 13 | simplbi 493 | . . . 4 ⊢ (𝑆 ∈ 𝐼 → 𝑆 ∈ (LIdeal‘𝑅)) |
15 | 8 | lidlsubg 19577 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ (LIdeal‘𝑅)) → 𝑆 ∈ (SubGrp‘𝑅)) |
16 | 14, 15 | sylan2 588 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (SubGrp‘𝑅)) |
17 | eqid 2826 | . . . 4 ⊢ (𝑅 ~QG 𝑆) = (𝑅 ~QG 𝑆) | |
18 | 3, 17 | eqger 17996 | . . 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 18935 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Abel) | |
21 | 20 | adantr 474 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑅 ∈ Abel) |
22 | ablnsg 18604 | . . . . 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 2910 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑆 ∈ (NrmSGrp‘𝑅)) |
25 | 3, 17, 5 | eqgcpbl 18000 | . . 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 19596 | . 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 476 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → 𝑅 ∈ Ring) | |
29 | 2, 4, 5, 6, 7, 19, 26, 27, 28 | qusring2 18975 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ∈ 𝐼) → (𝑈 ∈ Ring ∧ [ 1 ](𝑅 ~QG 𝑆) = (1r‘𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 class class class wbr 4874 ‘cfv 6124 (class class class)co 6906 Er wer 8007 [cec 8008 Basecbs 16223 +gcplusg 16306 .rcmulr 16307 /s cqus 16519 SubGrpcsubg 17940 NrmSGrpcnsg 17941 ~QG cqg 17942 Abelcabl 18548 1rcur 18856 Ringcrg 18902 opprcoppr 18977 LIdealclidl 19532 2Idealc2idl 19593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-rep 4995 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329 ax-pre-mulgt0 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rmo 3126 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-pss 3815 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4660 df-int 4699 df-iun 4743 df-br 4875 df-opab 4937 df-mpt 4954 df-tr 4977 df-id 5251 df-eprel 5256 df-po 5264 df-so 5265 df-fr 5302 df-we 5304 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-pred 5921 df-ord 5967 df-on 5968 df-lim 5969 df-suc 5970 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-om 7328 df-1st 7429 df-2nd 7430 df-tpos 7618 df-wrecs 7673 df-recs 7735 df-rdg 7773 df-1o 7827 df-oadd 7831 df-er 8010 df-ec 8012 df-qs 8016 df-en 8224 df-dom 8225 df-sdom 8226 df-fin 8227 df-sup 8618 df-inf 8619 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-le 10398 df-sub 10588 df-neg 10589 df-nn 11352 df-2 11415 df-3 11416 df-4 11417 df-5 11418 df-6 11419 df-7 11420 df-8 11421 df-9 11422 df-n0 11620 df-z 11706 df-dec 11823 df-uz 11970 df-fz 12621 df-struct 16225 df-ndx 16226 df-slot 16227 df-base 16229 df-sets 16230 df-ress 16231 df-plusg 16319 df-mulr 16320 df-sca 16322 df-vsca 16323 df-ip 16324 df-tset 16325 df-ple 16326 df-ds 16328 df-0g 16456 df-imas 16522 df-qus 16523 df-mgm 17596 df-sgrp 17638 df-mnd 17649 df-grp 17780 df-minusg 17781 df-sbg 17782 df-subg 17943 df-nsg 17944 df-eqg 17945 df-cmn 18549 df-abl 18550 df-mgp 18845 df-ur 18857 df-ring 18904 df-oppr 18978 df-subrg 19135 df-lmod 19222 df-lss 19290 df-sra 19534 df-rgmod 19535 df-lidl 19536 df-2idl 19594 |
This theorem is referenced by: qusring 19598 qusrhm 19599 |
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