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Theorem opprqusmulr 33519
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 2737 . . 3 (.r𝑄) = (.r𝑄)
3 eqid 2737 . . 3 (oppr𝑄) = (oppr𝑄)
4 eqid 2737 . . 3 (.r‘(oppr𝑄)) = (.r‘(oppr𝑄))
51, 2, 3, 4opprmul 20337 . 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 2737 . . . . . 6 (.r𝑅) = (.r𝑅)
9 opprqus1r.r . . . . . . 7 (𝜑𝑅 ∈ Ring)
109ad4antr 732 . . . . . 6 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑅 ∈ Ring)
11 opprqus1r.i . . . . . . 7 (𝜑𝐼 ∈ (2Ideal‘𝑅))
1211ad4antr 732 . . . . . 6 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝐼 ∈ (2Ideal‘𝑅))
13 simplr 769 . . . . . 6 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑞𝐵)
14 simp-4r 784 . . . . . 6 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑝𝐵)
156, 7, 8, 2, 10, 12, 13, 14qusmul2idl 21289 . . . . 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 484 . . . . . 6 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑌 = [𝑞](𝑅 ~QG 𝐼))
17 simpllr 776 . . . . . 6 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑋 = [𝑝](𝑅 ~QG 𝐼))
1816, 17oveq12d 7449 . . . . 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 2737 . . . . . . 7 (𝑂 /s (𝑂 ~QG 𝐼)) = (𝑂 /s (𝑂 ~QG 𝐼))
20 opprqus.o . . . . . . . 8 𝑂 = (oppr𝑅)
2120, 7opprbas 20341 . . . . . . 7 𝐵 = (Base‘𝑂)
22 eqid 2737 . . . . . . 7 (.r𝑂) = (.r𝑂)
23 eqid 2737 . . . . . . 7 (.r‘(𝑂 /s (𝑂 ~QG 𝐼))) = (.r‘(𝑂 /s (𝑂 ~QG 𝐼)))
2420opprring 20347 . . . . . . . . 9 (𝑅 ∈ Ring → 𝑂 ∈ Ring)
259, 24syl 17 . . . . . . . 8 (𝜑𝑂 ∈ Ring)
2625ad4antr 732 . . . . . . 7 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑂 ∈ Ring)
2720, 9oppr2idl 33514 . . . . . . . . 9 (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂))
2811, 27eleqtrd 2843 . . . . . . . 8 (𝜑𝐼 ∈ (2Ideal‘𝑂))
2928ad4antr 732 . . . . . . 7 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝐼 ∈ (2Ideal‘𝑂))
3019, 21, 22, 23, 26, 29, 14, 13qusmul2idl 21289 . . . . . 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 21264 . . . . . . . . . . . 12 (𝜑𝐼 ∈ (LIdeal‘𝑅))
32 eqid 2737 . . . . . . . . . . . . 13 (LIdeal‘𝑅) = (LIdeal‘𝑅)
337, 32lidlss 21222 . . . . . . . . . . . 12 (𝐼 ∈ (LIdeal‘𝑅) → 𝐼𝐵)
3431, 33syl 17 . . . . . . . . . . 11 (𝜑𝐼𝐵)
3520, 7oppreqg 33511 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐼𝐵) → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼))
369, 34, 35syl2anc 584 . . . . . . . . . 10 (𝜑 → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼))
3736ad4antr 732 . . . . . . . . 9 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → (𝑅 ~QG 𝐼) = (𝑂 ~QG 𝐼))
3837eceq2d 8788 . . . . . . . 8 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → [𝑝](𝑅 ~QG 𝐼) = [𝑝](𝑂 ~QG 𝐼))
3917, 38eqtrd 2777 . . . . . . 7 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑋 = [𝑝](𝑂 ~QG 𝐼))
4037eceq2d 8788 . . . . . . . 8 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → [𝑞](𝑅 ~QG 𝐼) = [𝑞](𝑂 ~QG 𝐼))
4116, 40eqtrd 2777 . . . . . . 7 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑌 = [𝑞](𝑂 ~QG 𝐼))
4239, 41oveq12d 7449 . . . . . 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 20337 . . . . . . . . 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 8785 . . . . . . 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 8788 . . . . . . 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 2779 . . . . . 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 2787 . . . . 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 2787 . . . 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 20341 . . . . . . . 8 𝐸 = (Base‘(oppr𝑄))
5250, 51eleqtrdi 2851 . . . . . . 7 (𝜑𝑌 ∈ (Base‘(oppr𝑄)))
5352ad2antrr 726 . . . . . 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 7466 . . . . . . . . 9 (𝜑 → (𝑅 ~QG 𝐼) ∈ V)
5754, 55, 56, 9qusbas 17590 . . . . . . . 8 (𝜑 → (𝐵 / (𝑅 ~QG 𝐼)) = (Base‘𝑄))
581, 51eqtr3i 2767 . . . . . . . 8 (Base‘𝑄) = (Base‘(oppr𝑄))
5957, 58eqtr2di 2794 . . . . . . 7 (𝜑 → (Base‘(oppr𝑄)) = (𝐵 / (𝑅 ~QG 𝐼)))
6059ad2antrr 726 . . . . . 6 (((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) → (Base‘(oppr𝑄)) = (𝐵 / (𝑅 ~QG 𝐼)))
6153, 60eleqtrd 2843 . . . . 5 (((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) → 𝑌 ∈ (𝐵 / (𝑅 ~QG 𝐼)))
62 elqsi 8810 . . . . 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 2851 . . . . 5 (𝜑𝑋 ∈ (Base‘(oppr𝑄)))
6766, 59eleqtrd 2843 . . . 4 (𝜑𝑋 ∈ (𝐵 / (𝑅 ~QG 𝐼)))
68 elqsi 8810 . . . 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 2789 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 395   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  wss 3951  cfv 6561  (class class class)co 7431  [cec 8743   / cqs 8744  Basecbs 17247  .rcmulr 17298   /s cqus 17550   ~QG cqg 19140  Ringcrg 20230  opprcoppr 20333  LIdealclidl 21216  2Idealc2idl 21259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-ec 8747  df-qs 8751  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-0g 17486  df-imas 17553  df-qus 17554  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-eqg 19143  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-oppr 20334  df-subrg 20570  df-lmod 20860  df-lss 20930  df-sra 21172  df-rgmod 21173  df-lidl 21218  df-2idl 21260
This theorem is referenced by:  opprqus1r  33520  opprqusdrng  33521  qsdrngi  33523
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