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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > opprqusbas | Structured version Visualization version GIF version |
Description: The base of the quotient of the opposite ring is the same as the base 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)) |
opprqusbas.r | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
opprqusbas.i | ⊢ (𝜑 → 𝐼 ⊆ 𝐵) |
Ref | Expression |
---|---|
opprqusbas | ⊢ (𝜑 → (Base‘(oppr‘𝑄)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . . 3 ⊢ (oppr‘𝑄) = (oppr‘𝑄) | |
2 | eqid 2731 | . . 3 ⊢ (Base‘𝑄) = (Base‘𝑄) | |
3 | 1, 2 | opprbas 20233 | . 2 ⊢ (Base‘𝑄) = (Base‘(oppr‘𝑄)) |
4 | opprqusbas.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
5 | opprqusbas.i | . . . . 5 ⊢ (𝜑 → 𝐼 ⊆ 𝐵) | |
6 | opprqus.o | . . . . . 6 ⊢ 𝑂 = (oppr‘𝑅) | |
7 | opprqus.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
8 | 6, 7 | oppreqg 32872 | . . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ⊆ 𝐵) → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼)) |
9 | 4, 5, 8 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼)) |
10 | 9 | qseq2d 8764 | . . 3 ⊢ (𝜑 → (𝐵 / (𝑅 ~QG 𝐼)) = (𝐵 / (𝑂 ~QG 𝐼))) |
11 | opprqus.q | . . . . 5 ⊢ 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)) | |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))) |
13 | 7 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝑅)) |
14 | ovexd 7447 | . . . 4 ⊢ (𝜑 → (𝑅 ~QG 𝐼) ∈ V) | |
15 | 12, 13, 14, 4 | qusbas 17496 | . . 3 ⊢ (𝜑 → (𝐵 / (𝑅 ~QG 𝐼)) = (Base‘𝑄)) |
16 | eqidd 2732 | . . . 4 ⊢ (𝜑 → (𝑂 /s (𝑂 ~QG 𝐼)) = (𝑂 /s (𝑂 ~QG 𝐼))) | |
17 | 6, 7 | opprbas 20233 | . . . . 5 ⊢ 𝐵 = (Base‘𝑂) |
18 | 17 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝑂)) |
19 | ovexd 7447 | . . . 4 ⊢ (𝜑 → (𝑂 ~QG 𝐼) ∈ V) | |
20 | 6 | fvexi 6905 | . . . . 5 ⊢ 𝑂 ∈ V |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑂 ∈ V) |
22 | 16, 18, 19, 21 | qusbas 17496 | . . 3 ⊢ (𝜑 → (𝐵 / (𝑂 ~QG 𝐼)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))) |
23 | 10, 15, 22 | 3eqtr3d 2779 | . 2 ⊢ (𝜑 → (Base‘𝑄) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))) |
24 | 3, 23 | eqtr3id 2785 | 1 ⊢ (𝜑 → (Base‘(oppr‘𝑄)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 Vcvv 3473 ⊆ wss 3948 ‘cfv 6543 (class class class)co 7412 / cqs 8706 Basecbs 17149 /s cqus 17456 ~QG cqg 19039 opprcoppr 20225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8215 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-ec 8709 df-qs 8713 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-inf 9442 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-fz 13490 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-0g 17392 df-imas 17459 df-qus 17460 df-minusg 18860 df-eqg 19042 df-oppr 20226 |
This theorem is referenced by: opprqus0g 32879 opprqus1r 32881 opprqusdrng 32882 |
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