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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  qsidomlem2 Structured version   Visualization version   GIF version

Theorem qsidomlem2 33481
Description: A quotient by a prime ideal is an integral domain. (Contributed by Thierry Arnoux, 16-Jan-2024.)
Hypothesis
Ref Expression
qsidom.1 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))
Assertion
Ref Expression
qsidomlem2 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ IDomn)

Proof of Theorem qsidomlem2
Dummy variables 𝑎 𝑦 𝑏 𝑒 𝑓 𝑥 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 crngring 20242 . . . 4 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2 prmidlidl 33472 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝐼 ∈ (LIdeal‘𝑅))
31, 2sylan 580 . . 3 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝐼 ∈ (LIdeal‘𝑅))
4 qsidom.1 . . . 4 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))
5 eqid 2737 . . . 4 (LIdeal‘𝑅) = (LIdeal‘𝑅)
64, 5quscrng 21293 . . 3 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝑄 ∈ CRing)
73, 6syldan 591 . 2 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ CRing)
85crng2idl 21291 . . . . . . . 8 (𝑅 ∈ CRing → (LIdeal‘𝑅) = (2Ideal‘𝑅))
98eleq2d 2827 . . . . . . 7 (𝑅 ∈ CRing → (𝐼 ∈ (LIdeal‘𝑅) ↔ 𝐼 ∈ (2Ideal‘𝑅)))
109biimpa 476 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (2Ideal‘𝑅))
113, 10syldan 591 . . . . 5 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝐼 ∈ (2Ideal‘𝑅))
12 eqid 2737 . . . . . 6 (2Ideal‘𝑅) = (2Ideal‘𝑅)
134, 12qusring 21285 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (2Ideal‘𝑅)) → 𝑄 ∈ Ring)
141, 11, 13syl2an2r 685 . . . 4 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ Ring)
15 eqid 2737 . . . . . . . . 9 (Base‘𝑄) = (Base‘𝑄)
16 eqid 2737 . . . . . . . . 9 (0g𝑄) = (0g𝑄)
1715, 16ring0cl 20264 . . . . . . . 8 (𝑄 ∈ Ring → (0g𝑄) ∈ (Base‘𝑄))
1814, 17syl 17 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → (0g𝑄) ∈ (Base‘𝑄))
1918snssd 4809 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → {(0g𝑄)} ⊆ (Base‘𝑄))
20 lidlnsg 21258 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
211, 20sylan 580 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
22 eqid 2737 . . . . . . . . . . . 12 (0g𝑅) = (0g𝑅)
234, 22qus0 19207 . . . . . . . . . . 11 (𝐼 ∈ (NrmSGrp‘𝑅) → [(0g𝑅)](𝑅 ~QG 𝐼) = (0g𝑄))
2421, 23syl 17 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → [(0g𝑅)](𝑅 ~QG 𝐼) = (0g𝑄))
255lidlsubg 21233 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (SubGrp‘𝑅))
261, 25sylan 580 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (SubGrp‘𝑅))
27 eqid 2737 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
28 eqid 2737 . . . . . . . . . . . 12 (𝑅 ~QG 𝐼) = (𝑅 ~QG 𝐼)
2927, 28, 22eqgid 19198 . . . . . . . . . . 11 (𝐼 ∈ (SubGrp‘𝑅) → [(0g𝑅)](𝑅 ~QG 𝐼) = 𝐼)
3026, 29syl 17 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → [(0g𝑅)](𝑅 ~QG 𝐼) = 𝐼)
3124, 30eqtr3d 2779 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (0g𝑄) = 𝐼)
323, 31syldan 591 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → (0g𝑄) = 𝐼)
3332sneqd 4638 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → {(0g𝑄)} = {𝐼})
34 eqid 2737 . . . . . . . . . . . 12 (.r𝑅) = (.r𝑅)
3527, 34isprmidlc 33475 . . . . . . . . . . 11 (𝑅 ∈ CRing → (𝐼 ∈ (PrmIdeal‘𝑅) ↔ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼 ≠ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼)))))
3635biimpa 476 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼 ≠ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼))))
3736simp2d 1144 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝐼 ≠ (Base‘𝑅))
38 ringgrp 20235 . . . . . . . . . . . 12 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
391, 38syl 17 . . . . . . . . . . 11 (𝑅 ∈ CRing → 𝑅 ∈ Grp)
4039ad2antrr 726 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → 𝑅 ∈ Grp)
411ad2antrr 726 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → 𝑅 ∈ Ring)
423adantr 480 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → 𝐼 ∈ (LIdeal‘𝑅))
4341, 42, 25syl2anc 584 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → 𝐼 ∈ (SubGrp‘𝑅))
44 simpr 484 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → (Base‘𝑄) = {𝐼})
4527, 4qustrivr 33393 . . . . . . . . . 10 ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅) ∧ (Base‘𝑄) = {𝐼}) → 𝐼 = (Base‘𝑅))
4640, 43, 44, 45syl3anc 1373 . . . . . . . . 9 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → 𝐼 = (Base‘𝑅))
4737, 46mteqand 3033 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → (Base‘𝑄) ≠ {𝐼})
4847necomd 2996 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → {𝐼} ≠ (Base‘𝑄))
4933, 48eqnetrd 3008 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → {(0g𝑄)} ≠ (Base‘𝑄))
50 pssdifn0 4368 . . . . . 6 (({(0g𝑄)} ⊆ (Base‘𝑄) ∧ {(0g𝑄)} ≠ (Base‘𝑄)) → ((Base‘𝑄) ∖ {(0g𝑄)}) ≠ ∅)
5119, 49, 50syl2anc 584 . . . . 5 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → ((Base‘𝑄) ∖ {(0g𝑄)}) ≠ ∅)
52 n0 4353 . . . . 5 (((Base‘𝑄) ∖ {(0g𝑄)}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)}))
5351, 52sylib 218 . . . 4 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → ∃𝑥 𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)}))
5416, 15ringelnzr 20523 . . . . . 6 ((𝑄 ∈ Ring ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) → 𝑄 ∈ NzRing)
5554ex 412 . . . . 5 (𝑄 ∈ Ring → (𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)}) → 𝑄 ∈ NzRing))
5655exlimdv 1933 . . . 4 (𝑄 ∈ Ring → (∃𝑥 𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)}) → 𝑄 ∈ NzRing))
5714, 53, 56sylc 65 . . 3 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ NzRing)
5836simp3d 1145 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼)))
5958ad7antr 738 . . . . . . . . . 10 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼)))
60 simp-4r 784 . . . . . . . . . 10 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → 𝑥 ∈ (Base‘𝑅))
61 simplr 769 . . . . . . . . . 10 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → 𝑦 ∈ (Base‘𝑅))
62 simp-8l 791 . . . . . . . . . . . 12 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → 𝑅 ∈ CRing)
6362, 39syl 17 . . . . . . . . . . 11 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → 𝑅 ∈ Grp)
643ad7antr 738 . . . . . . . . . . . 12 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → 𝐼 ∈ (LIdeal‘𝑅))
6562, 64, 26syl2anc 584 . . . . . . . . . . 11 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → 𝐼 ∈ (SubGrp‘𝑅))
664a1i 11 . . . . . . . . . . . . . 14 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)))
67 eqidd 2738 . . . . . . . . . . . . . 14 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
6827, 28eqger 19196 . . . . . . . . . . . . . . 15 (𝐼 ∈ (SubGrp‘𝑅) → (𝑅 ~QG 𝐼) Er (Base‘𝑅))
6926, 68syl 17 . . . . . . . . . . . . . 14 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (𝑅 ~QG 𝐼) Er (Base‘𝑅))
70 simpl 482 . . . . . . . . . . . . . 14 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝑅 ∈ CRing)
7127, 28, 12, 342idlcpbl 21282 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (2Ideal‘𝑅)) → ((𝑔(𝑅 ~QG 𝐼)𝑒(𝑅 ~QG 𝐼)𝑓) → (𝑔(.r𝑅))(𝑅 ~QG 𝐼)(𝑒(.r𝑅)𝑓)))
721, 10, 71syl2an2r 685 . . . . . . . . . . . . . 14 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ((𝑔(𝑅 ~QG 𝐼)𝑒(𝑅 ~QG 𝐼)𝑓) → (𝑔(.r𝑅))(𝑅 ~QG 𝐼)(𝑒(.r𝑅)𝑓)))
731ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring)
74 simprl 771 . . . . . . . . . . . . . . 15 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑒 ∈ (Base‘𝑅))
75 simprr 773 . . . . . . . . . . . . . . 15 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑓 ∈ (Base‘𝑅))
7627, 34ringcl 20247 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅)) → (𝑒(.r𝑅)𝑓) ∈ (Base‘𝑅))
7773, 74, 75, 76syl3anc 1373 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → (𝑒(.r𝑅)𝑓) ∈ (Base‘𝑅))
78 eqid 2737 . . . . . . . . . . . . . 14 (.r𝑄) = (.r𝑄)
7966, 67, 69, 70, 72, 77, 34, 78qusmulval 17600 . . . . . . . . . . . . 13 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ([𝑥](𝑅 ~QG 𝐼)(.r𝑄)[𝑦](𝑅 ~QG 𝐼)) = [(𝑥(.r𝑅)𝑦)](𝑅 ~QG 𝐼))
8062, 64, 60, 61, 79syl211anc 1378 . . . . . . . . . . . 12 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → ([𝑥](𝑅 ~QG 𝐼)(.r𝑄)[𝑦](𝑅 ~QG 𝐼)) = [(𝑥(.r𝑅)𝑦)](𝑅 ~QG 𝐼))
81 simpr 484 . . . . . . . . . . . . . 14 (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) → (𝑎(.r𝑄)𝑏) = (0g𝑄))
8281ad4antr 732 . . . . . . . . . . . . 13 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (𝑎(.r𝑄)𝑏) = (0g𝑄))
83 simpllr 776 . . . . . . . . . . . . . 14 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → 𝑎 = [𝑥](𝑅 ~QG 𝐼))
84 simpr 484 . . . . . . . . . . . . . 14 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → 𝑏 = [𝑦](𝑅 ~QG 𝐼))
8583, 84oveq12d 7449 . . . . . . . . . . . . 13 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (𝑎(.r𝑄)𝑏) = ([𝑥](𝑅 ~QG 𝐼)(.r𝑄)[𝑦](𝑅 ~QG 𝐼)))
8662, 64, 31syl2anc 584 . . . . . . . . . . . . 13 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (0g𝑄) = 𝐼)
8782, 85, 863eqtr3d 2785 . . . . . . . . . . . 12 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → ([𝑥](𝑅 ~QG 𝐼)(.r𝑄)[𝑦](𝑅 ~QG 𝐼)) = 𝐼)
8880, 87eqtr3d 2779 . . . . . . . . . . 11 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → [(𝑥(.r𝑅)𝑦)](𝑅 ~QG 𝐼) = 𝐼)
8928eqg0el 19201 . . . . . . . . . . . 12 ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([(𝑥(.r𝑅)𝑦)](𝑅 ~QG 𝐼) = 𝐼 ↔ (𝑥(.r𝑅)𝑦) ∈ 𝐼))
9089biimpa 476 . . . . . . . . . . 11 (((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) ∧ [(𝑥(.r𝑅)𝑦)](𝑅 ~QG 𝐼) = 𝐼) → (𝑥(.r𝑅)𝑦) ∈ 𝐼)
9163, 65, 88, 90syl21anc 838 . . . . . . . . . 10 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (𝑥(.r𝑅)𝑦) ∈ 𝐼)
92 rsp2 3277 . . . . . . . . . . . 12 (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼)) → ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼))))
9392impl 455 . . . . . . . . . . 11 (((∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼)))
9493imp 406 . . . . . . . . . 10 ((((∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → (𝑥𝐼𝑦𝐼))
9559, 60, 61, 91, 94syl1111anc 841 . . . . . . . . 9 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (𝑥𝐼𝑦𝐼))
9686eqeq2d 2748 . . . . . . . . . . 11 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (𝑎 = (0g𝑄) ↔ 𝑎 = 𝐼))
9783eqeq1d 2739 . . . . . . . . . . 11 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (𝑎 = 𝐼 ↔ [𝑥](𝑅 ~QG 𝐼) = 𝐼))
9828eqg0el 19201 . . . . . . . . . . . 12 ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([𝑥](𝑅 ~QG 𝐼) = 𝐼𝑥𝐼))
9963, 65, 98syl2anc 584 . . . . . . . . . . 11 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → ([𝑥](𝑅 ~QG 𝐼) = 𝐼𝑥𝐼))
10096, 97, 993bitrrd 306 . . . . . . . . . 10 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (𝑥𝐼𝑎 = (0g𝑄)))
10186eqeq2d 2748 . . . . . . . . . . 11 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (𝑏 = (0g𝑄) ↔ 𝑏 = 𝐼))
10284eqeq1d 2739 . . . . . . . . . . 11 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (𝑏 = 𝐼 ↔ [𝑦](𝑅 ~QG 𝐼) = 𝐼))
10328eqg0el 19201 . . . . . . . . . . . 12 ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([𝑦](𝑅 ~QG 𝐼) = 𝐼𝑦𝐼))
10463, 65, 103syl2anc 584 . . . . . . . . . . 11 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → ([𝑦](𝑅 ~QG 𝐼) = 𝐼𝑦𝐼))
105101, 102, 1043bitrrd 306 . . . . . . . . . 10 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (𝑦𝐼𝑏 = (0g𝑄)))
106100, 105orbi12d 919 . . . . . . . . 9 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → ((𝑥𝐼𝑦𝐼) ↔ (𝑎 = (0g𝑄) ∨ 𝑏 = (0g𝑄))))
10795, 106mpbid 232 . . . . . . . 8 (((((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑏 = [𝑦](𝑅 ~QG 𝐼)) → (𝑎 = (0g𝑄) ∨ 𝑏 = (0g𝑄)))
108 simplr 769 . . . . . . . . . . 11 (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) → 𝑏 ∈ (Base‘𝑄))
1094a1i 11 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)))
110 eqidd 2738 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘𝑅))
111 ovexd 7466 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → (𝑅 ~QG 𝐼) ∈ V)
112 id 22 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → 𝑅 ∈ CRing)
113109, 110, 111, 112qusbas 17590 . . . . . . . . . . . 12 (𝑅 ∈ CRing → ((Base‘𝑅) / (𝑅 ~QG 𝐼)) = (Base‘𝑄))
114113ad4antr 732 . . . . . . . . . . 11 (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) → ((Base‘𝑅) / (𝑅 ~QG 𝐼)) = (Base‘𝑄))
115108, 114eleqtrrd 2844 . . . . . . . . . 10 (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) → 𝑏 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)))
116115ad2antrr 726 . . . . . . . . 9 (((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) → 𝑏 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)))
117 elqsi 8810 . . . . . . . . 9 (𝑏 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)) → ∃𝑦 ∈ (Base‘𝑅)𝑏 = [𝑦](𝑅 ~QG 𝐼))
118116, 117syl 17 . . . . . . . 8 (((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) → ∃𝑦 ∈ (Base‘𝑅)𝑏 = [𝑦](𝑅 ~QG 𝐼))
119107, 118r19.29a 3162 . . . . . . 7 (((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑎 = [𝑥](𝑅 ~QG 𝐼)) → (𝑎 = (0g𝑄) ∨ 𝑏 = (0g𝑄)))
120 simpllr 776 . . . . . . . . 9 (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) → 𝑎 ∈ (Base‘𝑄))
121120, 114eleqtrrd 2844 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) → 𝑎 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)))
122 elqsi 8810 . . . . . . . 8 (𝑎 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)) → ∃𝑥 ∈ (Base‘𝑅)𝑎 = [𝑥](𝑅 ~QG 𝐼))
123121, 122syl 17 . . . . . . 7 (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) → ∃𝑥 ∈ (Base‘𝑅)𝑎 = [𝑥](𝑅 ~QG 𝐼))
124119, 123r19.29a 3162 . . . . . 6 (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) → (𝑎 = (0g𝑄) ∨ 𝑏 = (0g𝑄)))
125124ex 412 . . . . 5 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) → ((𝑎(.r𝑄)𝑏) = (0g𝑄) → (𝑎 = (0g𝑄) ∨ 𝑏 = (0g𝑄))))
126125anasss 466 . . . 4 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (𝑎 ∈ (Base‘𝑄) ∧ 𝑏 ∈ (Base‘𝑄))) → ((𝑎(.r𝑄)𝑏) = (0g𝑄) → (𝑎 = (0g𝑄) ∨ 𝑏 = (0g𝑄))))
127126ralrimivva 3202 . . 3 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → ∀𝑎 ∈ (Base‘𝑄)∀𝑏 ∈ (Base‘𝑄)((𝑎(.r𝑄)𝑏) = (0g𝑄) → (𝑎 = (0g𝑄) ∨ 𝑏 = (0g𝑄))))
12815, 78, 16isdomn 20705 . . 3 (𝑄 ∈ Domn ↔ (𝑄 ∈ NzRing ∧ ∀𝑎 ∈ (Base‘𝑄)∀𝑏 ∈ (Base‘𝑄)((𝑎(.r𝑄)𝑏) = (0g𝑄) → (𝑎 = (0g𝑄) ∨ 𝑏 = (0g𝑄)))))
12957, 127, 128sylanbrc 583 . 2 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ Domn)
130 isidom 20725 . 2 (𝑄 ∈ IDomn ↔ (𝑄 ∈ CRing ∧ 𝑄 ∈ Domn))
1317, 129, 130sylanbrc 583 1 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ IDomn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  wrex 3070  Vcvv 3480  cdif 3948  wss 3951  c0 4333  {csn 4626   class class class wbr 5143  cfv 6561  (class class class)co 7431   Er wer 8742  [cec 8743   / cqs 8744  Basecbs 17247  .rcmulr 17298  0gc0g 17484   /s cqus 17550  Grpcgrp 18951  SubGrpcsubg 19138  NrmSGrpcnsg 19139   ~QG cqg 19140  Ringcrg 20230  CRingccrg 20231  NzRingcnzr 20512  Domncdomn 20692  IDomncidom 20693  LIdealclidl 21216  2Idealc2idl 21259  PrmIdealcprmidl 33463
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-int 4947  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-cring 20233  df-oppr 20334  df-nzr 20513  df-subrg 20570  df-domn 20695  df-idom 20696  df-lmod 20860  df-lss 20930  df-lsp 20970  df-sra 21172  df-rgmod 21173  df-lidl 21218  df-rsp 21219  df-2idl 21260  df-prmidl 33464
This theorem is referenced by:  qsidom  33482
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