Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  qsidomlem2 Structured version   Visualization version   GIF version

Theorem qsidomlem2 32560
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 20061 . . . 4 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2 prmidlidl 32550 . . . 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 2732 . . . 4 (LIdeal‘𝑅) = (LIdeal‘𝑅)
64, 5quscrng 20870 . . 3 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝑄 ∈ CRing)
73, 6syldan 591 . 2 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ CRing)
85crng2idl 20869 . . . . . . . 8 (𝑅 ∈ CRing → (LIdeal‘𝑅) = (2Ideal‘𝑅))
98eleq2d 2819 . . . . . . 7 (𝑅 ∈ CRing → (𝐼 ∈ (LIdeal‘𝑅) ↔ 𝐼 ∈ (2Ideal‘𝑅)))
109biimpa 477 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (2Ideal‘𝑅))
113, 10syldan 591 . . . . 5 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝐼 ∈ (2Ideal‘𝑅))
12 eqid 2732 . . . . . 6 (2Ideal‘𝑅) = (2Ideal‘𝑅)
134, 12qusring 20865 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (2Ideal‘𝑅)) → 𝑄 ∈ Ring)
141, 11, 13syl2an2r 683 . . . 4 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ Ring)
15 eqid 2732 . . . . . . . . 9 (Base‘𝑄) = (Base‘𝑄)
16 eqid 2732 . . . . . . . . 9 (0g𝑄) = (0g𝑄)
1715, 16ring0cl 20077 . . . . . . . 8 (𝑄 ∈ Ring → (0g𝑄) ∈ (Base‘𝑄))
1814, 17syl 17 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → (0g𝑄) ∈ (Base‘𝑄))
1918snssd 4811 . . . . . 6 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → {(0g𝑄)} ⊆ (Base‘𝑄))
20 lidlnsg 32552 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
211, 20sylan 580 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
22 eqid 2732 . . . . . . . . . . . 12 (0g𝑅) = (0g𝑅)
234, 22qus0 19062 . . . . . . . . . . 11 (𝐼 ∈ (NrmSGrp‘𝑅) → [(0g𝑅)](𝑅 ~QG 𝐼) = (0g𝑄))
2421, 23syl 17 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → [(0g𝑅)](𝑅 ~QG 𝐼) = (0g𝑄))
255lidlsubg 20830 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (SubGrp‘𝑅))
261, 25sylan 580 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (SubGrp‘𝑅))
27 eqid 2732 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
28 eqid 2732 . . . . . . . . . . . 12 (𝑅 ~QG 𝐼) = (𝑅 ~QG 𝐼)
2927, 28, 22eqgid 19054 . . . . . . . . . . 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 2774 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (0g𝑄) = 𝐼)
323, 31syldan 591 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → (0g𝑄) = 𝐼)
3332sneqd 4639 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → {(0g𝑄)} = {𝐼})
34 eqid 2732 . . . . . . . . . . . 12 (.r𝑅) = (.r𝑅)
3527, 34isprmidlc 32554 . . . . . . . . . . 11 (𝑅 ∈ CRing → (𝐼 ∈ (PrmIdeal‘𝑅) ↔ (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼 ≠ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼)))))
3635biimpa 477 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → (𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼 ≠ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼))))
3736simp2d 1143 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝐼 ≠ (Base‘𝑅))
38 ringgrp 20054 . . . . . . . . . . . 12 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
391, 38syl 17 . . . . . . . . . . 11 (𝑅 ∈ CRing → 𝑅 ∈ Grp)
4039ad2antrr 724 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → 𝑅 ∈ Grp)
411ad2antrr 724 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → 𝑅 ∈ Ring)
423adantr 481 . . . . . . . . . . 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 485 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (Base‘𝑄) = {𝐼}) → (Base‘𝑄) = {𝐼})
4527, 4qustrivr 32465 . . . . . . . . . 10 ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅) ∧ (Base‘𝑄) = {𝐼}) → 𝐼 = (Base‘𝑅))
4640, 43, 44, 45syl3anc 1371 . . . . . . . . 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 4364 . . . . . 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 4345 . . . . 5 (((Base‘𝑄) ∖ {(0g𝑄)}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)}))
5351, 52sylib 217 . . . 4 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → ∃𝑥 𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)}))
5416, 15ringelnzr 20292 . . . . . 6 ((𝑄 ∈ Ring ∧ 𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) → 𝑄 ∈ NzRing)
5554ex 413 . . . . 5 (𝑄 ∈ Ring → (𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)}) → 𝑄 ∈ NzRing))
5655exlimdv 1936 . . . 4 (𝑄 ∈ Ring → (∃𝑥 𝑥 ∈ ((Base‘𝑄) ∖ {(0g𝑄)}) → 𝑄 ∈ NzRing))
5714, 53, 56sylc 65 . . 3 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → 𝑄 ∈ NzRing)
5836simp3d 1144 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼)))
5958ad7antr 736 . . . . . . . . . 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 782 . . . . . . . . . 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 767 . . . . . . . . . 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 789 . . . . . . . . . . . 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 736 . . . . . . . . . . . 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 2733 . . . . . . . . . . . . . 14 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
6827, 28eqger 19052 . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . 14 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝑅 ∈ CRing)
7127, 28, 12, 342idlcpbl 20863 . . . . . . . . . . . . . . 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 683 . . . . . . . . . . . . . 14 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ((𝑔(𝑅 ~QG 𝐼)𝑒(𝑅 ~QG 𝐼)𝑓) → (𝑔(.r𝑅))(𝑅 ~QG 𝐼)(𝑒(.r𝑅)𝑓)))
731ad2antrr 724 . . . . . . . . . . . . . . 15 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring)
74 simprl 769 . . . . . . . . . . . . . . 15 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑒 ∈ (Base‘𝑅))
75 simprr 771 . . . . . . . . . . . . . . 15 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑓 ∈ (Base‘𝑅))
7627, 34ringcl 20066 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Ring ∧ 𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅)) → (𝑒(.r𝑅)𝑓) ∈ (Base‘𝑅))
7773, 74, 75, 76syl3anc 1371 . . . . . . . . . . . . . 14 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → (𝑒(.r𝑅)𝑓) ∈ (Base‘𝑅))
78 eqid 2732 . . . . . . . . . . . . . 14 (.r𝑄) = (.r𝑄)
7966, 67, 69, 70, 72, 77, 34, 78qusmulval 17497 . . . . . . . . . . . . 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 1376 . . . . . . . . . . . 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 485 . . . . . . . . . . . . . 14 (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) → (𝑎(.r𝑄)𝑏) = (0g𝑄))
8281ad4antr 730 . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . 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 7423 . . . . . . . . . . . . 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 2780 . . . . . . . . . . . 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 2774 . . . . . . . . . . 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 32461 . . . . . . . . . . . 12 ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([(𝑥(.r𝑅)𝑦)](𝑅 ~QG 𝐼) = 𝐼 ↔ (𝑥(.r𝑅)𝑦) ∈ 𝐼))
9089biimpa 477 . . . . . . . . . . 11 (((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) ∧ [(𝑥(.r𝑅)𝑦)](𝑅 ~QG 𝐼) = 𝐼) → (𝑥(.r𝑅)𝑦) ∈ 𝐼)
9163, 65, 88, 90syl21anc 836 . . . . . . . . . 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 3274 . . . . . . . . . . . 12 (∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼)) → ((𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼))))
9392impl 456 . . . . . . . . . . 11 (((∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼)) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼)))
9493imp 407 . . . . . . . . . 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 838 . . . . . . . . 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 2743 . . . . . . . . . . 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 2734 . . . . . . . . . . 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 32461 . . . . . . . . . . . 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 305 . . . . . . . . . 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 2743 . . . . . . . . . . 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 2734 . . . . . . . . . . 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 32461 . . . . . . . . . . . 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 305 . . . . . . . . . 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 917 . . . . . . . . 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 231 . . . . . . . 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 767 . . . . . . . . . . 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 2733 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘𝑅))
111 ovexd 7440 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → (𝑅 ~QG 𝐼) ∈ V)
112 id 22 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → 𝑅 ∈ CRing)
113109, 110, 111, 112qusbas 17487 . . . . . . . . . . . 12 (𝑅 ∈ CRing → ((Base‘𝑅) / (𝑅 ~QG 𝐼)) = (Base‘𝑄))
114113ad4antr 730 . . . . . . . . . . 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 2836 . . . . . . . . . 10 (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) → 𝑏 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)))
116115ad2antrr 724 . . . . . . . . 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 8760 . . . . . . . . 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 774 . . . . . . . . 9 (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) → 𝑎 ∈ (Base‘𝑄))
121120, 114eleqtrrd 2836 . . . . . . . 8 (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) ∧ (𝑎(.r𝑄)𝑏) = (0g𝑄)) → 𝑎 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)))
122 elqsi 8760 . . . . . . . 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 413 . . . . 5 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ 𝑎 ∈ (Base‘𝑄)) ∧ 𝑏 ∈ (Base‘𝑄)) → ((𝑎(.r𝑄)𝑏) = (0g𝑄) → (𝑎 = (0g𝑄) ∨ 𝑏 = (0g𝑄))))
126125anasss 467 . . . 4 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) ∧ (𝑎 ∈ (Base‘𝑄) ∧ 𝑏 ∈ (Base‘𝑄))) → ((𝑎(.r𝑄)𝑏) = (0g𝑄) → (𝑎 = (0g𝑄) ∨ 𝑏 = (0g𝑄))))
127126ralrimivva 3200 . . 3 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (PrmIdeal‘𝑅)) → ∀𝑎 ∈ (Base‘𝑄)∀𝑏 ∈ (Base‘𝑄)((𝑎(.r𝑄)𝑏) = (0g𝑄) → (𝑎 = (0g𝑄) ∨ 𝑏 = (0g𝑄))))
12815, 78, 16isdomn 20902 . . 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 20914 . 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 205  wa 396  wo 845  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wne 2940  wral 3061  wrex 3070  Vcvv 3474  cdif 3944  wss 3947  c0 4321  {csn 4627   class class class wbr 5147  cfv 6540  (class class class)co 7405   Er wer 8696  [cec 8697   / cqs 8698  Basecbs 17140  .rcmulr 17194  0gc0g 17381   /s cqus 17447  Grpcgrp 18815  SubGrpcsubg 18994  NrmSGrpcnsg 18995   ~QG cqg 18996  Ringcrg 20049  CRingccrg 20050  NzRingcnzr 20283  LIdealclidl 20775  2Idealc2idl 20848  Domncdomn 20888  IDomncidom 20889  PrmIdealcprmidl 32541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-tpos 8207  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-ec 8701  df-qs 8705  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-inf 9434  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-fz 13481  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-sca 17209  df-vsca 17210  df-ip 17211  df-tset 17212  df-ple 17213  df-ds 17215  df-0g 17383  df-imas 17450  df-qus 17451  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-minusg 18819  df-sbg 18820  df-subg 18997  df-nsg 18998  df-eqg 18999  df-cmn 19644  df-abl 19645  df-mgp 19982  df-ur 19999  df-ring 20051  df-cring 20052  df-oppr 20142  df-nzr 20284  df-subrg 20353  df-lmod 20465  df-lss 20535  df-lsp 20575  df-sra 20777  df-rgmod 20778  df-lidl 20779  df-rsp 20780  df-2idl 20849  df-domn 20892  df-idom 20893  df-prmidl 32542
This theorem is referenced by:  qsidom  32561
  Copyright terms: Public domain W3C validator