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Theorem opprqusmulr 33498
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 2734 . . 3 (.r𝑄) = (.r𝑄)
3 eqid 2734 . . 3 (oppr𝑄) = (oppr𝑄)
4 eqid 2734 . . 3 (.r‘(oppr𝑄)) = (.r‘(oppr𝑄))
51, 2, 3, 4opprmul 20353 . 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 2734 . . . . . 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 21306 . . . . 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 7448 . . . . 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 2734 . . . . . . 7 (𝑂 /s (𝑂 ~QG 𝐼)) = (𝑂 /s (𝑂 ~QG 𝐼))
20 opprqus.o . . . . . . . 8 𝑂 = (oppr𝑅)
2120, 7opprbas 20357 . . . . . . 7 𝐵 = (Base‘𝑂)
22 eqid 2734 . . . . . . 7 (.r𝑂) = (.r𝑂)
23 eqid 2734 . . . . . . 7 (.r‘(𝑂 /s (𝑂 ~QG 𝐼))) = (.r‘(𝑂 /s (𝑂 ~QG 𝐼)))
2420opprring 20363 . . . . . . . . 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 33493 . . . . . . . . 9 (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂))
2811, 27eleqtrd 2840 . . . . . . . 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 21306 . . . . . 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 21281 . . . . . . . . . . . 12 (𝜑𝐼 ∈ (LIdeal‘𝑅))
32 eqid 2734 . . . . . . . . . . . . 13 (LIdeal‘𝑅) = (LIdeal‘𝑅)
337, 32lidlss 21239 . . . . . . . . . . . 12 (𝐼 ∈ (LIdeal‘𝑅) → 𝐼𝐵)
3431, 33syl 17 . . . . . . . . . . 11 (𝜑𝐼𝐵)
3520, 7oppreqg 33490 . . . . . . . . . . 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 8786 . . . . . . . 8 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → [𝑝](𝑅 ~QG 𝐼) = [𝑝](𝑂 ~QG 𝐼))
3917, 38eqtrd 2774 . . . . . . 7 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑋 = [𝑝](𝑂 ~QG 𝐼))
4037eceq2d 8786 . . . . . . . 8 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → [𝑞](𝑅 ~QG 𝐼) = [𝑞](𝑂 ~QG 𝐼))
4116, 40eqtrd 2774 . . . . . . 7 (((((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) ∧ 𝑞𝐵) ∧ 𝑌 = [𝑞](𝑅 ~QG 𝐼)) → 𝑌 = [𝑞](𝑂 ~QG 𝐼))
4239, 41oveq12d 7448 . . . . . 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 20353 . . . . . . . . 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 8783 . . . . . . 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 8786 . . . . . . 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 2776 . . . . . 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 2784 . . . . 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 2784 . . . 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 20357 . . . . . . . 8 𝐸 = (Base‘(oppr𝑄))
5250, 51eleqtrdi 2848 . . . . . . 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 7465 . . . . . . . . 9 (𝜑 → (𝑅 ~QG 𝐼) ∈ V)
5754, 55, 56, 9qusbas 17591 . . . . . . . 8 (𝜑 → (𝐵 / (𝑅 ~QG 𝐼)) = (Base‘𝑄))
581, 51eqtr3i 2764 . . . . . . . 8 (Base‘𝑄) = (Base‘(oppr𝑄))
5957, 58eqtr2di 2791 . . . . . . 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 2840 . . . . 5 (((𝜑𝑝𝐵) ∧ 𝑋 = [𝑝](𝑅 ~QG 𝐼)) → 𝑌 ∈ (𝐵 / (𝑅 ~QG 𝐼)))
62 elqsi 8808 . . . . 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 3159 . . 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 2848 . . . . 5 (𝜑𝑋 ∈ (Base‘(oppr𝑄)))
6766, 59eleqtrd 2840 . . . 4 (𝜑𝑋 ∈ (𝐵 / (𝑅 ~QG 𝐼)))
68 elqsi 8808 . . . 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 3159 . 2 (𝜑 → (𝑌(.r𝑄)𝑋) = (𝑋(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑌))
715, 70eqtrid 2786 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 1536  wcel 2105  wrex 3067  Vcvv 3477  wss 3962  cfv 6562  (class class class)co 7430  [cec 8741   / cqs 8742  Basecbs 17244  .rcmulr 17298   /s cqus 17551   ~QG cqg 19152  Ringcrg 20250  opprcoppr 20349  LIdealclidl 21233  2Idealc2idl 21276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-tpos 8249  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-ec 8745  df-qs 8749  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  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 17487  df-imas 17554  df-qus 17555  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-grp 18966  df-minusg 18967  df-sbg 18968  df-subg 19153  df-eqg 19155  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-oppr 20350  df-subrg 20586  df-lmod 20876  df-lss 20947  df-sra 21189  df-rgmod 21190  df-lidl 21235  df-2idl 21277
This theorem is referenced by:  opprqus1r  33499  opprqusdrng  33500  qsdrngi  33502
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