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Theorem opprqusdrng 33501
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 2735 . . . . . 6 (oppr𝑄) = (oppr𝑄)
2 eqid 2735 . . . . . 6 (1r𝑄) = (1r𝑄)
31, 2oppr1 20367 . . . . 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 33500 . . . . 5 (𝜑 → (1r‘(oppr𝑄)) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))
103, 9eqtrid 2787 . . . 4 (𝜑 → (1r𝑄) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼))))
11 eqid 2735 . . . . . 6 (0g𝑄) = (0g𝑄)
121, 11oppr0 20366 . . . . 5 (0g𝑄) = (0g‘(oppr𝑄))
1382idllidld 21282 . . . . . . 7 (𝜑𝐼 ∈ (LIdeal‘𝑅))
14 lidlnsg 21276 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
157, 13, 14syl2anc 584 . . . . . 6 (𝜑𝐼 ∈ (NrmSGrp‘𝑅))
164, 5, 6, 15opprqus0g 33498 . . . . 5 (𝜑 → (0g‘(oppr𝑄)) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼))))
1712, 16eqtrid 2787 . . . 4 (𝜑 → (0g𝑄) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼))))
1810, 17neeq12d 3000 . . 3 (𝜑 → ((1r𝑄) ≠ (0g𝑄) ↔ (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) ≠ (0g‘(𝑂 /s (𝑂 ~QG 𝐼)))))
19 eqid 2735 . . . . . . 7 (Base‘𝑄) = (Base‘𝑄)
201, 19opprbas 20358 . . . . . 6 (Base‘𝑄) = (Base‘(oppr𝑄))
21 eqid 2735 . . . . . . . . 9 (LIdeal‘𝑅) = (LIdeal‘𝑅)
224, 21lidlss 21240 . . . . . . . 8 (𝐼 ∈ (LIdeal‘𝑅) → 𝐼𝐵)
2313, 22syl 17 . . . . . . 7 (𝜑𝐼𝐵)
244, 5, 6, 7, 23opprqusbas 33496 . . . . . 6 (𝜑 → (Base‘(oppr𝑄)) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
2520, 24eqtrid 2787 . . . . 5 (𝜑 → (Base‘𝑄) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼))))
2617sneqd 4643 . . . . 5 (𝜑 → {(0g𝑄)} = {(0g‘(𝑂 /s (𝑂 ~QG 𝐼)))})
2725, 26difeq12d 4137 . . . 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 3975 . . . . . . . 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 33499 . . . . . . 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 2751 . . . . . 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 33499 . . . . . . 7 (((𝜑𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑦 ∈ (Base‘𝑄)) → (𝑦(.r‘(oppr𝑄))𝑥) = (𝑦(.r‘(𝑂 /s (𝑂 ~QG 𝐼)))𝑥))
3837, 35eqeq12d 2751 . . . . . 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 3331 . . . 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 3330 . . 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 2735 . . 3 (.r‘(oppr𝑄)) = (.r‘(oppr𝑄))
44 eqid 2735 . . . 4 (Unit‘𝑄) = (Unit‘𝑄)
4544, 1opprunit 20394 . . 3 (Unit‘𝑄) = (Unit‘(oppr𝑄))
46 eqid 2735 . . . . . 6 (2Ideal‘𝑅) = (2Ideal‘𝑅)
476, 46qusring 21303 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (2Ideal‘𝑅)) → 𝑄 ∈ Ring)
487, 8, 47syl2anc 584 . . . 4 (𝜑𝑄 ∈ Ring)
491opprring 20364 . . . 4 (𝑄 ∈ Ring → (oppr𝑄) ∈ Ring)
5048, 49syl 17 . . 3 (𝜑 → (oppr𝑄) ∈ Ring)
5120, 12, 3, 43, 45, 50isdrng4 33279 . 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 2735 . . 3 (Base‘(𝑂 /s (𝑂 ~QG 𝐼))) = (Base‘(𝑂 /s (𝑂 ~QG 𝐼)))
53 eqid 2735 . . 3 (0g‘(𝑂 /s (𝑂 ~QG 𝐼))) = (0g‘(𝑂 /s (𝑂 ~QG 𝐼)))
54 eqid 2735 . . 3 (1r‘(𝑂 /s (𝑂 ~QG 𝐼))) = (1r‘(𝑂 /s (𝑂 ~QG 𝐼)))
55 eqid 2735 . . 3 (.r‘(𝑂 /s (𝑂 ~QG 𝐼))) = (.r‘(𝑂 /s (𝑂 ~QG 𝐼)))
56 eqid 2735 . . 3 (Unit‘(𝑂 /s (𝑂 ~QG 𝐼))) = (Unit‘(𝑂 /s (𝑂 ~QG 𝐼)))
575opprring 20364 . . . . 5 (𝑅 ∈ Ring → 𝑂 ∈ Ring)
587, 57syl 17 . . . 4 (𝜑𝑂 ∈ Ring)
595, 7oppr2idl 33494 . . . . 5 (𝜑 → (2Ideal‘𝑅) = (2Ideal‘𝑂))
608, 59eleqtrd 2841 . . . 4 (𝜑𝐼 ∈ (2Ideal‘𝑂))
61 eqid 2735 . . . . 5 (𝑂 /s (𝑂 ~QG 𝐼)) = (𝑂 /s (𝑂 ~QG 𝐼))
62 eqid 2735 . . . . 5 (2Ideal‘𝑂) = (2Ideal‘𝑂)
6361, 62qusring 21303 . . . 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 33279 . 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 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  cdif 3960  wss 3963  {csn 4631  cfv 6563  (class class class)co 7431  Basecbs 17245  .rcmulr 17299  0gc0g 17486   /s cqus 17552  NrmSGrpcnsg 19152   ~QG cqg 19153  1rcur 20199  Ringcrg 20251  opprcoppr 20350  Unitcui 20372  DivRingcdr 20746  LIdealclidl 21234  2Idealc2idl 21277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-ec 8746  df-qs 8750  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-0g 17488  df-imas 17555  df-qus 17556  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-nsg 19155  df-eqg 19156  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-subrg 20587  df-drng 20748  df-lmod 20877  df-lss 20948  df-sra 21190  df-rgmod 21191  df-lidl 21236  df-2idl 21278
This theorem is referenced by:  qsdrng  33505
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