![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > qusmul | Structured version Visualization version GIF version |
Description: Value of the ring operation in a quotient ring. (Contributed by Thierry Arnoux, 1-Sep-2024.) |
Ref | Expression |
---|---|
qusmul.h | ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) |
qusmul.v | ⊢ 𝐵 = (Base‘𝑅) |
qusmul.p | ⊢ · = (.r‘𝑅) |
qusmul.a | ⊢ × = (.r‘𝑄) |
qusmul.r | ⊢ (𝜑 → 𝑅 ∈ CRing) |
qusmul.i | ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) |
qusmul.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
qusmul.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
qusmul | ⊢ (𝜑 → ([𝑋](𝑅 ~QG 𝐼) × [𝑌](𝑅 ~QG 𝐼)) = [(𝑋 · 𝑌)](𝑅 ~QG 𝐼)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qusmul.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
2 | qusmul.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
3 | qusmul.h | . . . 4 ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))) |
5 | qusmul.v | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) |
7 | qusmul.r | . . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing) | |
8 | 7 | crngringd 20060 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) |
9 | qusmul.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) | |
10 | eqid 2733 | . . . . . 6 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
11 | 10 | lidlsubg 20825 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (SubGrp‘𝑅)) |
12 | 8, 9, 11 | syl2anc 585 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ (SubGrp‘𝑅)) |
13 | eqid 2733 | . . . . 5 ⊢ (𝑅 ~QG 𝐼) = (𝑅 ~QG 𝐼) | |
14 | 5, 13 | eqger 19052 | . . . 4 ⊢ (𝐼 ∈ (SubGrp‘𝑅) → (𝑅 ~QG 𝐼) Er 𝐵) |
15 | 12, 14 | syl 17 | . . 3 ⊢ (𝜑 → (𝑅 ~QG 𝐼) Er 𝐵) |
16 | 10 | crng2idl 20864 | . . . . . 6 ⊢ (𝑅 ∈ CRing → (LIdeal‘𝑅) = (2Ideal‘𝑅)) |
17 | 7, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → (LIdeal‘𝑅) = (2Ideal‘𝑅)) |
18 | 9, 17 | eleqtrd 2836 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) |
19 | eqid 2733 | . . . . 5 ⊢ (2Ideal‘𝑅) = (2Ideal‘𝑅) | |
20 | qusmul.p | . . . . 5 ⊢ · = (.r‘𝑅) | |
21 | 5, 13, 19, 20 | 2idlcpbl 20858 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (2Ideal‘𝑅)) → ((𝑥(𝑅 ~QG 𝐼)𝑦 ∧ 𝑧(𝑅 ~QG 𝐼)𝑡) → (𝑥 · 𝑧)(𝑅 ~QG 𝐼)(𝑦 · 𝑡))) |
22 | 8, 18, 21 | syl2anc 585 | . . 3 ⊢ (𝜑 → ((𝑥(𝑅 ~QG 𝐼)𝑦 ∧ 𝑧(𝑅 ~QG 𝐼)𝑡) → (𝑥 · 𝑧)(𝑅 ~QG 𝐼)(𝑦 · 𝑡))) |
23 | 5, 20 | ringcl 20064 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵) → (𝑝 · 𝑞) ∈ 𝐵) |
24 | 23 | 3expb 1121 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → (𝑝 · 𝑞) ∈ 𝐵) |
25 | 8, 24 | sylan 581 | . . . 4 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → (𝑝 · 𝑞) ∈ 𝐵) |
26 | 25 | caovclg 7594 | . . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑡 ∈ 𝐵)) → (𝑦 · 𝑡) ∈ 𝐵) |
27 | qusmul.a | . . 3 ⊢ × = (.r‘𝑄) | |
28 | 4, 6, 15, 7, 22, 26, 20, 27 | qusmulval 17497 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ([𝑋](𝑅 ~QG 𝐼) × [𝑌](𝑅 ~QG 𝐼)) = [(𝑋 · 𝑌)](𝑅 ~QG 𝐼)) |
29 | 1, 2, 28 | mpd3an23 1464 | 1 ⊢ (𝜑 → ([𝑋](𝑅 ~QG 𝐼) × [𝑌](𝑅 ~QG 𝐼)) = [(𝑋 · 𝑌)](𝑅 ~QG 𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5147 ‘cfv 6540 (class class class)co 7404 Er wer 8696 [cec 8697 Basecbs 17140 .rcmulr 17194 /s cqus 17447 SubGrpcsubg 18994 ~QG cqg 18996 Ringcrg 20047 CRingccrg 20048 LIdealclidl 20771 2Idealc2idl 20843 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-tpos 8206 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-ec 8701 df-qs 8705 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-0g 17383 df-imas 17450 df-qus 17451 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-eqg 18999 df-cmn 19643 df-abl 19644 df-mgp 19980 df-ur 19997 df-ring 20049 df-cring 20050 df-oppr 20139 df-subrg 20349 df-lmod 20461 df-lss 20531 df-lsp 20571 df-sra 20773 df-rgmod 20774 df-lidl 20775 df-rsp 20776 df-2idl 20844 |
This theorem is referenced by: rhmqusker 32502 |
Copyright terms: Public domain | W3C validator |