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Theorem opprqusmulr 32515
Description: The multiplication operation of the quotient of the opposite ring is the same as the multiplication operation of the opposite of the quotient ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
Hypotheses
Ref Expression
opprqus.b 𝐵 = (Base‘𝑅)
opprqus.o 𝑂 = (oppr𝑅)
opprqus.q 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))
opprqus1r.r (𝜑𝑅 ∈ Ring)
opprqus1r.i (𝜑𝐼 ∈ (2Ideal‘𝑅))
opprqusmulr.e 𝐸 = (Base‘𝑄)
opprqusmulr.x (𝜑𝑋𝐸)
opprqusmulr.y (𝜑𝑌𝐸)
Assertion
Ref Expression
opprqusmulr (𝜑 → (𝑋(.r‘(oppr𝑄))𝑌) = (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))

Proof of Theorem opprqusmulr
Dummy variables 𝑝 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opprqusmulr.e . . 3 𝐸 = (Base‘𝑄)
2 eqid 2732 . . 3 (.r𝑄) = (.r𝑄)
3 eqid 2732 . . 3 (oppr𝑄) = (oppr𝑄)
4 eqid 2732 . . 3 (.r‘(oppr𝑄)) = (.r‘(oppr𝑄))
51, 2, 3, 4opprmul 20107 . 2 (𝑋(.r‘(oppr𝑄))𝑌) = (𝑌(.r𝑄)𝑋)
6 opprqus.q . . . . . 6 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))
7 opprqus.b . . . . . 6 𝐵 = (Base‘𝑅)
8 eqid 2732 . . . . . 6 (.r𝑅) = (.r𝑅)
9 opprqus1r.r . . . . . . 7 (𝜑𝑅 ∈ Ring)
109ad4antr 730 . . . . . 6 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑅 ∈ Ring)
11 opprqus1r.i . . . . . . 7 (𝜑𝐼 ∈ (2Ideal‘𝑅))
1211ad4antr 730 . . . . . 6 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝐼 ∈ (2Ideal‘𝑅))
13 simplr 767 . . . . . 6 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑞𝐵)
14 simp-4r 782 . . . . . 6 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑝𝐵)
156, 7, 8, 2, 10, 12, 13, 14qusmul2 20813 . . . . 5 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → ([𝑞](𝑅 ~QG 𝐼)(.r𝑄)[𝑝](𝑅 ~QG 𝐼)) = [(𝑞(.r𝑅)𝑝)](𝑅 ~QG 𝐼))
16 simpr 485 . . . . . 6 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑌 = [𝑞](𝑅 ~QG 𝐼))
17 simpllr 774 . . . . . 6 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑋 = [𝑝](𝑅 ~QG 𝐼))
1816, 17oveq12d 7412 . . . . 5 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → (𝑌(.r𝑄)𝑋) = ([𝑞](𝑅 ~QG 𝐼)(.r𝑄)[𝑝](𝑅 ~QG 𝐼)))
19 eqid 2732 . . . . . . 7 (𝑂 /s (𝑂 ~QG 𝐼)) = (𝑂 /s (𝑂 ~QG 𝐼))
20 opprqus.o . . . . . . . 8 𝑂 = (oppr𝑅)
2120, 7opprbas 20111 . . . . . . 7 𝐵 = (Base‘𝑂)
22 eqid 2732 . . . . . . 7 (.r𝑂) = (.r𝑂)
23 eqid 2732 . . . . . . 7 (.r‘(𝑂 /s (𝑂 ~QG 𝐼))) = (.r‘(𝑂 /s (𝑂 ~QG 𝐼)))
2420opprring 20115 . . . . . . . . 9 (𝑅 ∈ Ring → 𝑂 ∈ Ring)
259, 24syl 17 . . . . . . . 8 (𝜑𝑂 ∈ Ring)
2625ad4antr 730 . . . . . . 7 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑂 ∈ Ring)
2720, 9oppr2idl 32510 . . . . . . . . 9 (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂))
2811, 27eleqtrd 2835 . . . . . . . 8 (𝜑𝐼 ∈ (2Ideal‘𝑂))
2928ad4antr 730 . . . . . . 7 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝐼 ∈ (2Ideal‘𝑂))
3019, 21, 22, 23, 26, 29, 14, 13qusmul2 20813 . . . . . 6 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → ([𝑝](𝑂 ~QG 𝐼)(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))[𝑞](𝑂 ~QG 𝐼)) = [(𝑝(.r𝑂)𝑞)](𝑂 ~QG 𝐼))
31112idllidld 20807 . . . . . . . . . . . 12 (𝜑𝐼 ∈ (LIdeal‘𝑅))
32 eqid 2732 . . . . . . . . . . . . 13 (LIdeal‘𝑅) = (LIdeal‘𝑅)
337, 32lidlss 20783 . . . . . . . . . . . 12 (𝐼 ∈ (LIdeal‘𝑅) → 𝐼𝐵)
3431, 33syl 17 . . . . . . . . . . 11 (𝜑𝐼𝐵)
3520, 7oppreqg 32507 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐼𝐵) → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼))
369, 34, 35syl2anc 584 . . . . . . . . . 10 (𝜑 → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼))
3736ad4antr 730 . . . . . . . . 9 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼))
3837eceq2d 8730 . . . . . . . 8 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → [𝑝](𝑅 ~QG 𝐼) = [𝑝](𝑂 ~QG 𝐼))
3917, 38eqtrd 2772 . . . . . . 7 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑋 = [𝑝](𝑂 ~QG 𝐼))
4037eceq2d 8730 . . . . . . . 8 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → [𝑞](𝑅 ~QG 𝐼) = [𝑞](𝑂 ~QG 𝐼))
4116, 40eqtrd 2772 . . . . . . 7 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑌 = [𝑞](𝑂 ~QG 𝐼))
4239, 41oveq12d 7412 . . . . . 6 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌) = ([𝑝](𝑂 ~QG 𝐼)(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))[𝑞](𝑂 ~QG 𝐼)))
437, 8, 20, 22opprmul 20107 . . . . . . . . 9 (𝑝(.r𝑂)𝑞) = (𝑞(.r𝑅)𝑝)
4443a1i 11 . . . . . . . 8 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → (𝑝(.r𝑂)𝑞) = (𝑞(.r𝑅)𝑝))
4544eceq1d 8727 . . . . . . 7 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → [(𝑝(.r𝑂)𝑞)](𝑅 ~QG 𝐼) = [(𝑞(.r𝑅)𝑝)](𝑅 ~QG 𝐼))
4637eceq2d 8730 . . . . . . 7 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → [(𝑝(.r𝑂)𝑞)](𝑅 ~QG 𝐼) = [(𝑝(.r𝑂)𝑞)](𝑂 ~QG 𝐼))
4745, 46eqtr3d 2774 . . . . . 6 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → [(𝑞(.r𝑅)𝑝)](𝑅 ~QG 𝐼) = [(𝑝(.r𝑂)𝑞)](𝑂 ~QG 𝐼))
4830, 42, 473eqtr4d 2782 . . . . 5 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌) = [(𝑞(.r𝑅)𝑝)](𝑅 ~QG 𝐼))
4915, 18, 483eqtr4d 2782 . . . 4 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → (𝑌(.r𝑄)𝑋) = (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))
50 opprqusmulr.y . . . . . . . 8 (𝜑𝑌𝐸)
513, 1opprbas 20111 . . . . . . . 8 𝐸 = (Base‘(oppr𝑄))
5250, 51eleqtrdi 2843 . . . . . . 7 (𝜑𝑌 ∈ (Base‘(oppr𝑄)))
5352ad2antrr 724 . . . . . 6 (((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) → 𝑌 ∈ (Base‘(oppr𝑄)))
546a1i 11 . . . . . . . . 9 (𝜑𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)))
557a1i 11 . . . . . . . . 9 (𝜑𝐵 = (Base‘𝑅))
56 ovexd 7429 . . . . . . . . 9 (𝜑 → (𝑅 ~QG 𝐼) ∈ V)
5754, 55, 56, 9qusbas 17475 . . . . . . . 8 (𝜑 → (𝐵 / (𝑅 ~QG 𝐼)) = (Base‘𝑄))
581, 51eqtr3i 2762 . . . . . . . 8 (Base‘𝑄) = (Base‘(oppr𝑄))
5957, 58eqtr2di 2789 . . . . . . 7 (𝜑 → (Base‘(oppr𝑄)) = (𝐵 / (𝑅 ~QG 𝐼)))
6059ad2antrr 724 . . . . . 6 (((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) → (Base‘(oppr𝑄)) = (𝐵 / (𝑅 ~QG 𝐼)))
6153, 60eleqtrd 2835 . . . . 5 (((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) → 𝑌 ∈ (𝐵 / (𝑅 ~QG 𝐼)))
62 elqsi 8749 . . . . 5 (𝑌 ∈ (𝐵 / (𝑅 ~QG 𝐼)) → ∃𝑞𝐵 𝑌 = [𝑞](𝑅 ~QG 𝐼))
6361, 62syl 17 . . . 4 (((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) → ∃𝑞𝐵 𝑌 = [𝑞](𝑅 ~QG 𝐼))
6449, 63r19.29a 3162 . . 3 (((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) → (𝑌(.r𝑄)𝑋) = (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))
65 opprqusmulr.x . . . . . 6 (𝜑𝑋𝐸)
6665, 51eleqtrdi 2843 . . . . 5 (𝜑𝑋 ∈ (Base‘(oppr𝑄)))
6766, 59eleqtrd 2835 . . . 4 (𝜑𝑋 ∈ (𝐵 / (𝑅 ~QG 𝐼)))
68 elqsi 8749 . . . 4 (𝑋 ∈ (𝐵 / (𝑅 ~QG 𝐼)) → ∃𝑝𝐵 𝑋 = [𝑝](𝑅 ~QG 𝐼))
6967, 68syl 17 . . 3 (𝜑 → ∃𝑝𝐵 𝑋 = [𝑝](𝑅 ~QG 𝐼))
7064, 69r19.29a 3162 . 2 (𝜑 → (𝑌(.r𝑄)𝑋) = (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))
715, 70eqtrid 2784 1 (𝜑 → (𝑋(.r‘(oppr𝑄))𝑌) = (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wrex 3070  Vcvv 3474  wss 3945  cfv 6533  (class class class)co 7394  [cec 8686   / cqs 8687  Basecbs 17128  .rcmulr 17182   /s cqus 17435   ~QG cqg 18976  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:  opprqus1r  32516  opprqusdrng  32517  qsdrngi  32519
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