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Theorem opprqusdrng 33521
Description: The quotient of the opposite ring is a division ring iff the opposite of the quotient ring is. (Contributed by Thierry Arnoux, 13-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))
Assertion
Ref Expression
opprqusdrng (𝜑 → ((oppr𝑄) ∈ DivRing ↔ (𝑂 /s (𝑂 ~QG 𝐼)) ∈ DivRing))

Proof of Theorem opprqusdrng
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . . . 6 (oppr𝑄) = (oppr𝑄)
2 eqid 2737 . . . . . 6 (1r𝑄) = (1r𝑄)
31, 2oppr1 20350 . . . . 5 (1r𝑄) = (1r‘(oppr𝑄))
4 opprqus.b . . . . . 6 𝐵 = (Base‘𝑅)
5 opprqus.o . . . . . 6 𝑂 = (oppr𝑅)
6 opprqus.q . . . . . 6 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))
7 opprqus1r.r . . . . . 6 (𝜑𝑅 ∈ Ring)
8 opprqus1r.i . . . . . 6 (𝜑𝐼 ∈ (2Ideal‘𝑅))
94, 5, 6, 7, 8opprqus1r 33520 . . . . 5 (𝜑 → (1r‘(oppr𝑄)) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))
103, 9eqtrid 2789 . . . 4 (𝜑 → (1r𝑄) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))
11 eqid 2737 . . . . . 6 (0g𝑄) = (0g𝑄)
121, 11oppr0 20349 . . . . 5 (0g𝑄) = (0g‘(oppr𝑄))
1382idllidld 21264 . . . . . . 7 (𝜑𝐼 ∈ (LIdeal‘𝑅))
14 lidlnsg 21258 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
157, 13, 14syl2anc 584 . . . . . 6 (𝜑𝐼 ∈ (NrmSGrp‘𝑅))
164, 5, 6, 15opprqus0g 33518 . . . . 5 (𝜑 → (0g‘(oppr𝑄)) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼))))
1712, 16eqtrid 2789 . . . 4 (𝜑 → (0g𝑄) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼))))
1810, 17neeq12d 3002 . . 3 (𝜑 → ((1r𝑄) ≠ (0g𝑄) ↔ (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ≠ (0g‘(𝑂 /s (𝑂 ~QG 𝐼)))))
19 eqid 2737 . . . . . . 7 (Base‘𝑄) = (Base‘𝑄)
201, 19opprbas 20341 . . . . . 6 (Base‘𝑄) = (Base‘(oppr𝑄))
21 eqid 2737 . . . . . . . . 9 (LIdeal‘𝑅) = (LIdeal‘𝑅)
224, 21lidlss 21222 . . . . . . . 8 (𝐼 ∈ (LIdeal‘𝑅) → 𝐼𝐵)
2313, 22syl 17 . . . . . . 7 (𝜑𝐼𝐵)
244, 5, 6, 7, 23opprqusbas 33516 . . . . . 6 (𝜑 → (Base‘(oppr𝑄)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
2520, 24eqtrid 2789 . . . . 5 (𝜑 → (Base‘𝑄) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
2617sneqd 4638 . . . . 5 (𝜑 → {(0g𝑄)} = {(0g‘(𝑂 /s (𝑂 ~QG 𝐼)))})
2725, 26difeq12d 4127 . . . 4 (𝜑 → ((Base‘𝑄) ∖ {(0g𝑄)}) = ((Base‘(𝑂 /s (𝑂 ~QG 𝐼))) ∖ {(0g‘(𝑂 /s (𝑂 ~QG 𝐼)))}))
2825adantr 480 . . . . 5 ((𝜑𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) → (Base‘𝑄) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
297ad2antrr 726 . . . . . . . 8 (((𝜑𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑅 ∈ Ring)
308ad2antrr 726 . . . . . . . 8 (((𝜑𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝐼 ∈ (2Ideal‘𝑅))
31 simplr 769 . . . . . . . . 9 (((𝜑𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)}))
3231eldifad 3963 . . . . . . . 8 (((𝜑𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑥 ∈ (Base‘𝑄))
33 simpr 484 . . . . . . . 8 (((𝜑𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → 𝑦 ∈ (Base‘𝑄))
344, 5, 6, 29, 30, 19, 32, 33opprqusmulr 33519 . . . . . . 7 (((𝜑𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → (𝑥(.r‘(oppr𝑄))𝑦) = (𝑥(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑦))
3510ad2antrr 726 . . . . . . 7 (((𝜑𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → (1r𝑄) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))
3634, 35eqeq12d 2753 . . . . . 6 (((𝜑𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → ((𝑥(.r‘(oppr𝑄))𝑦) = (1r𝑄) ↔ (𝑥(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑦) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))))
374, 5, 6, 29, 30, 19, 33, 32opprqusmulr 33519 . . . . . . 7 (((𝜑𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → (𝑦(.r‘(oppr𝑄))𝑥) = (𝑦(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥))
3837, 35eqeq12d 2753 . . . . . 6 (((𝜑𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → ((𝑦(.r‘(oppr𝑄))𝑥) = (1r𝑄) ↔ (𝑦(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))))
3936, 38anbi12d 632 . . . . 5 (((𝜑𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → (((𝑥(.r‘(oppr𝑄))𝑦) = (1r𝑄) ∧ (𝑦(.r‘(oppr𝑄))𝑥) = (1r𝑄)) ↔ ((𝑥(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑦) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ (𝑦(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))))
4028, 39rexeqbidva 3333 . . . 4 ((𝜑𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) → (∃𝑦 ∈ (Base‘𝑄)((𝑥(.r‘(oppr𝑄))𝑦) = (1r𝑄) ∧ (𝑦(.r‘(oppr𝑄))𝑥) = (1r𝑄)) ↔ ∃𝑦 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))((𝑥(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑦) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ (𝑦(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))))
4127, 40raleqbidva 3332 . . 3 (𝜑 → (∀𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})∃𝑦 ∈ (Base‘𝑄)((𝑥(.r‘(oppr𝑄))𝑦) = (1r𝑄) ∧ (𝑦(.r‘(oppr𝑄))𝑥) = (1r𝑄)) ↔ ∀𝑥 ∈ ((Base‘(𝑂 /s (𝑂 ~QG 𝐼))) ∖ {(0g‘(𝑂 /s (𝑂 ~QG 𝐼)))})∃𝑦 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))((𝑥(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑦) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ (𝑦(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))))
4218, 41anbi12d 632 . 2 (𝜑 → (((1r𝑄) ≠ (0g𝑄) ∧ ∀𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})∃𝑦 ∈ (Base‘𝑄)((𝑥(.r‘(oppr𝑄))𝑦) = (1r𝑄) ∧ (𝑦(.r‘(oppr𝑄))𝑥) = (1r𝑄))) ↔ ((1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ≠ (0g‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ ∀𝑥 ∈ ((Base‘(𝑂 /s (𝑂 ~QG 𝐼))) ∖ {(0g‘(𝑂 /s (𝑂 ~QG 𝐼)))})∃𝑦 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))((𝑥(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑦) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ (𝑦(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))))))
43 eqid 2737 . . 3 (.r‘(oppr𝑄)) = (.r‘(oppr𝑄))
44 eqid 2737 . . . 4 (Unit‘𝑄) = (Unit‘𝑄)
4544, 1opprunit 20377 . . 3 (Unit‘𝑄) = (Unit‘(oppr𝑄))
46 eqid 2737 . . . . . 6 (2Ideal‘𝑅) = (2Ideal‘𝑅)
476, 46qusring 21285 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (2Ideal‘𝑅)) → 𝑄 ∈ Ring)
487, 8, 47syl2anc 584 . . . 4 (𝜑𝑄 ∈ Ring)
491opprring 20347 . . . 4 (𝑄 ∈ Ring → (oppr𝑄) ∈ Ring)
5048, 49syl 17 . . 3 (𝜑 → (oppr𝑄) ∈ Ring)
5120, 12, 3, 43, 45, 50isdrng4 33298 . 2 (𝜑 → ((oppr𝑄) ∈ DivRing ↔ ((1r𝑄) ≠ (0g𝑄) ∧ ∀𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})∃𝑦 ∈ (Base‘𝑄)((𝑥(.r‘(oppr𝑄))𝑦) = (1r𝑄) ∧ (𝑦(.r‘(oppr𝑄))𝑥) = (1r𝑄)))))
52 eqid 2737 . . 3 (Base‘(𝑂 /s (𝑂 ~QG 𝐼))) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))
53 eqid 2737 . . 3 (0g‘(𝑂 /s (𝑂 ~QG 𝐼))) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼)))
54 eqid 2737 . . 3 (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))
55 eqid 2737 . . 3 (.r‘(𝑂 /s (𝑂 ~QG 𝐼))) = (.r‘(𝑂 /s (𝑂 ~QG 𝐼)))
56 eqid 2737 . . 3 (Unit‘(𝑂 /s (𝑂 ~QG 𝐼))) = (Unit‘(𝑂 /s (𝑂 ~QG 𝐼)))
575opprring 20347 . . . . 5 (𝑅 ∈ Ring → 𝑂 ∈ Ring)
587, 57syl 17 . . . 4 (𝜑𝑂 ∈ Ring)
595, 7oppr2idl 33514 . . . . 5 (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂))
608, 59eleqtrd 2843 . . . 4 (𝜑𝐼 ∈ (2Ideal‘𝑂))
61 eqid 2737 . . . . 5 (𝑂 /s (𝑂 ~QG 𝐼)) = (𝑂 /s (𝑂 ~QG 𝐼))
62 eqid 2737 . . . . 5 (2Ideal‘𝑂) = (2Ideal‘𝑂)
6361, 62qusring 21285 . . . 4 ((𝑂 ∈ Ring ∧ 𝐼 ∈ (2Ideal‘𝑂)) → (𝑂 /s (𝑂 ~QG 𝐼)) ∈ Ring)
6458, 60, 63syl2anc 584 . . 3 (𝜑 → (𝑂 /s (𝑂 ~QG 𝐼)) ∈ Ring)
6552, 53, 54, 55, 56, 64isdrng4 33298 . 2 (𝜑 → ((𝑂 /s (𝑂 ~QG 𝐼)) ∈ DivRing ↔ ((1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ≠ (0g‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ ∀𝑥 ∈ ((Base‘(𝑂 /s (𝑂 ~QG 𝐼))) ∖ {(0g‘(𝑂 /s (𝑂 ~QG 𝐼)))})∃𝑦 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))((𝑥(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑦) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ∧ (𝑦(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))))))
6642, 51, 653bitr4d 311 1 (𝜑 → ((oppr𝑄) ∈ DivRing ↔ (𝑂 /s (𝑂 ~QG 𝐼)) ∈ DivRing))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  cdif 3948  wss 3951  {csn 4626  cfv 6561  (class class class)co 7431  Basecbs 17247  .rcmulr 17298  0gc0g 17484   /s cqus 17550  NrmSGrpcnsg 19139   ~QG cqg 19140  1rcur 20178  Ringcrg 20230  opprcoppr 20333  Unitcui 20355  DivRingcdr 20729  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-nsg 19142  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-dvdsr 20357  df-unit 20358  df-invr 20388  df-subrg 20570  df-drng 20731  df-lmod 20860  df-lss 20930  df-sra 21172  df-rgmod 21173  df-lidl 21218  df-2idl 21260
This theorem is referenced by:  qsdrng  33525
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