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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opprqus1r | Structured version Visualization version GIF version |
Description: The ring unity of the quotient of the opposite ring is the same as the ring unity of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 9-Mar-2025.) |
Ref | Expression |
---|---|
opprqus.b | ⊢ 𝐵 = (Base‘𝑅) |
opprqus.o | ⊢ 𝑂 = (oppr‘𝑅) |
opprqus.q | ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) |
opprqus1r.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
opprqus1r.i | ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) |
Ref | Expression |
---|---|
opprqus1r | ⊢ (𝜑 → (1r‘(oppr‘𝑄)) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2732 | . 2 ⊢ (Base‘(oppr‘𝑄)) = (Base‘(oppr‘𝑄)) | |
2 | fvexd 6894 | . 2 ⊢ (𝜑 → (oppr‘𝑄) ∈ V) | |
3 | ovexd 7429 | . 2 ⊢ (𝜑 → (𝑂 /s (𝑂 ~QG 𝐼)) ∈ V) | |
4 | opprqus.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
5 | opprqus.o | . . 3 ⊢ 𝑂 = (oppr‘𝑅) | |
6 | opprqus.q | . . 3 ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) | |
7 | opprqus1r.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
8 | opprqus1r.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (2Ideal‘𝑅)) | |
9 | 8 | 2idllidld 20807 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ (LIdeal‘𝑅)) |
10 | eqid 2732 | . . . . 5 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅) | |
11 | 4, 10 | lidlss 20783 | . . . 4 ⊢ (𝐼 ∈ (LIdeal‘𝑅) → 𝐼 ⊆ 𝐵) |
12 | 9, 11 | syl 17 | . . 3 ⊢ (𝜑 → 𝐼 ⊆ 𝐵) |
13 | 4, 5, 6, 7, 12 | opprqusbas 32512 | . 2 ⊢ (𝜑 → (Base‘(oppr‘𝑄)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))) |
14 | 7 | ad2antrr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) ∧ 𝑦 ∈ (Base‘(oppr‘𝑄))) → 𝑅 ∈ Ring) |
15 | 8 | ad2antrr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) ∧ 𝑦 ∈ (Base‘(oppr‘𝑄))) → 𝐼 ∈ (2Ideal‘𝑅)) |
16 | eqid 2732 | . . 3 ⊢ (Base‘𝑄) = (Base‘𝑄) | |
17 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → 𝑥 ∈ (Base‘(oppr‘𝑄))) | |
18 | eqid 2732 | . . . . . 6 ⊢ (oppr‘𝑄) = (oppr‘𝑄) | |
19 | 18, 16 | opprbas 20111 | . . . . 5 ⊢ (Base‘𝑄) = (Base‘(oppr‘𝑄)) |
20 | 17, 19 | eleqtrrdi 2844 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) → 𝑥 ∈ (Base‘𝑄)) |
21 | 20 | adantr 481 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) ∧ 𝑦 ∈ (Base‘(oppr‘𝑄))) → 𝑥 ∈ (Base‘𝑄)) |
22 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(oppr‘𝑄))) → 𝑦 ∈ (Base‘(oppr‘𝑄))) | |
23 | 22, 19 | eleqtrrdi 2844 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (Base‘(oppr‘𝑄))) → 𝑦 ∈ (Base‘𝑄)) |
24 | 23 | adantlr 713 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) ∧ 𝑦 ∈ (Base‘(oppr‘𝑄))) → 𝑦 ∈ (Base‘𝑄)) |
25 | 4, 5, 6, 14, 15, 16, 21, 24 | opprqusmulr 32515 | . 2 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘(oppr‘𝑄))) ∧ 𝑦 ∈ (Base‘(oppr‘𝑄))) → (𝑥(.r‘(oppr‘𝑄))𝑦) = (𝑥(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑦)) |
26 | 1, 2, 3, 13, 25 | urpropd 32309 | 1 ⊢ (𝜑 → (1r‘(oppr‘𝑄)) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3945 ‘cfv 6533 (class class class)co 7394 Basecbs 17128 /s cqus 17435 ~QG cqg 18976 1rcur 19965 Ringcrg 20016 opprcoppr 20103 LIdealclidl 20734 2Idealc2idl 20804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7840 df-1st 7959 df-2nd 7960 df-tpos 8195 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-1o 8450 df-er 8688 df-ec 8690 df-qs 8694 df-en 8925 df-dom 8926 df-sdom 8927 df-fin 8928 df-sup 9421 df-inf 9422 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-nn 12197 df-2 12259 df-3 12260 df-4 12261 df-5 12262 df-6 12263 df-7 12264 df-8 12265 df-9 12266 df-n0 12457 df-z 12543 df-dec 12662 df-uz 12807 df-fz 13469 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17129 df-ress 17158 df-plusg 17194 df-mulr 17195 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-0g 17371 df-imas 17438 df-qus 17439 df-mgm 18545 df-sgrp 18594 df-mnd 18605 df-grp 18799 df-minusg 18800 df-sbg 18801 df-subg 18977 df-eqg 18979 df-cmn 19616 df-abl 19617 df-mgp 19949 df-ur 19966 df-ring 20018 df-oppr 20104 df-subrg 20312 df-lmod 20424 df-lss 20494 df-sra 20736 df-rgmod 20737 df-lidl 20738 df-2idl 20805 |
This theorem is referenced by: opprqusdrng 32517 qsdrngi 32519 |
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