Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  qsidomlem1 Structured version   Visualization version   GIF version

Theorem qsidomlem1 33480
Description: If the quotient ring of a commutative ring relative to an ideal is an integral domain, that ideal must be prime. (Contributed by Thierry Arnoux, 16-Jan-2024.)
Hypothesis
Ref Expression
qsidom.1 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))
Assertion
Ref Expression
qsidomlem1 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ∈ (PrmIdeal‘𝑅))

Proof of Theorem qsidomlem1
Dummy variables 𝑦 𝑒 𝑓 𝑥 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 crngring 20242 . . 3 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
21ad2antrr 726 . 2 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝑅 ∈ Ring)
3 simplr 769 . 2 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ∈ (LIdeal‘𝑅))
4 qsidom.1 . . . . . . . . 9 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))
5 simpr 484 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 𝐼 = (Base‘𝑅))
65oveq2d 7447 . . . . . . . . . 10 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (𝑅 ~QG 𝐼) = (𝑅 ~QG (Base‘𝑅)))
76oveq2d 7447 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (𝑅 /s (𝑅 ~QG 𝐼)) = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
84, 7eqtrid 2789 . . . . . . . 8 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 𝑄 = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
98fveq2d 6910 . . . . . . 7 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (Base‘𝑄) = (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))))
10 ringgrp 20235 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
111, 10syl 17 . . . . . . . . 9 (𝑅 ∈ CRing → 𝑅 ∈ Grp)
1211ad3antrrr 730 . . . . . . . 8 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 𝑅 ∈ Grp)
13 eqid 2737 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
14 eqid 2737 . . . . . . . . 9 (𝑅 /s (𝑅 ~QG (Base‘𝑅))) = (𝑅 /s (𝑅 ~QG (Base‘𝑅)))
1513, 14qustriv 33392 . . . . . . . 8 (𝑅 ∈ Grp → (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))) = {(Base‘𝑅)})
1612, 15syl 17 . . . . . . 7 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))) = {(Base‘𝑅)})
179, 16eqtrd 2777 . . . . . 6 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (Base‘𝑄) = {(Base‘𝑅)})
1817fveq2d 6910 . . . . 5 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = (♯‘{(Base‘𝑅)}))
19 fvex 6919 . . . . . 6 (Base‘𝑅) ∈ V
20 hashsng 14408 . . . . . 6 ((Base‘𝑅) ∈ V → (♯‘{(Base‘𝑅)}) = 1)
2119, 20ax-mp 5 . . . . 5 (♯‘{(Base‘𝑅)}) = 1
2218, 21eqtrdi 2793 . . . 4 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = 1)
23 1red 11262 . . . . . 6 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 1 ∈ ℝ)
24 isidom 20725 . . . . . . . . . 10 (𝑄 ∈ IDomn ↔ (𝑄 ∈ CRing ∧ 𝑄 ∈ Domn))
2524simprbi 496 . . . . . . . . 9 (𝑄 ∈ IDomn → 𝑄 ∈ Domn)
26 domnnzr 20706 . . . . . . . . 9 (𝑄 ∈ Domn → 𝑄 ∈ NzRing)
2725, 26syl 17 . . . . . . . 8 (𝑄 ∈ IDomn → 𝑄 ∈ NzRing)
2827ad2antlr 727 . . . . . . 7 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 𝑄 ∈ NzRing)
29 eqid 2737 . . . . . . . . 9 (Base‘𝑄) = (Base‘𝑄)
3029isnzr2hash 20519 . . . . . . . 8 (𝑄 ∈ NzRing ↔ (𝑄 ∈ Ring ∧ 1 < (♯‘(Base‘𝑄))))
3130simprbi 496 . . . . . . 7 (𝑄 ∈ NzRing → 1 < (♯‘(Base‘𝑄)))
3228, 31syl 17 . . . . . 6 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 1 < (♯‘(Base‘𝑄)))
3323, 32gtned 11396 . . . . 5 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) ≠ 1)
3433neneqd 2945 . . . 4 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → ¬ (♯‘(Base‘𝑄)) = 1)
3522, 34pm2.65da 817 . . 3 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → ¬ 𝐼 = (Base‘𝑅))
3635neqned 2947 . 2 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ≠ (Base‘𝑅))
3725ad4antlr 733 . . . . . . 7 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → 𝑄 ∈ Domn)
38 ovex 7464 . . . . . . . . . 10 (𝑅 ~QG 𝐼) ∈ V
3938ecelqsi 8813 . . . . . . . . 9 (𝑥 ∈ (Base‘𝑅) → [𝑥](𝑅 ~QG 𝐼) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)))
4039ad3antlr 731 . . . . . . . 8 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → [𝑥](𝑅 ~QG 𝐼) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)))
41 simp-5l 785 . . . . . . . . 9 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → 𝑅 ∈ CRing)
424a1i 11 . . . . . . . . . 10 (𝑅 ∈ CRing → 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)))
43 eqidd 2738 . . . . . . . . . 10 (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘𝑅))
44 ovexd 7466 . . . . . . . . . 10 (𝑅 ∈ CRing → (𝑅 ~QG 𝐼) ∈ V)
45 id 22 . . . . . . . . . 10 (𝑅 ∈ CRing → 𝑅 ∈ CRing)
4642, 43, 44, 45qusbas 17590 . . . . . . . . 9 (𝑅 ∈ CRing → ((Base‘𝑅) / (𝑅 ~QG 𝐼)) = (Base‘𝑄))
4741, 46syl 17 . . . . . . . 8 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → ((Base‘𝑅) / (𝑅 ~QG 𝐼)) = (Base‘𝑄))
4840, 47eleqtrd 2843 . . . . . . 7 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → [𝑥](𝑅 ~QG 𝐼) ∈ (Base‘𝑄))
4938ecelqsi 8813 . . . . . . . . 9 (𝑦 ∈ (Base‘𝑅) → [𝑦](𝑅 ~QG 𝐼) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)))
5049ad2antlr 727 . . . . . . . 8 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → [𝑦](𝑅 ~QG 𝐼) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)))
5150, 47eleqtrd 2843 . . . . . . 7 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → [𝑦](𝑅 ~QG 𝐼) ∈ (Base‘𝑄))
5241, 1, 103syl 18 . . . . . . . . 9 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → 𝑅 ∈ Grp)
53 eqid 2737 . . . . . . . . . . . 12 (LIdeal‘𝑅) = (LIdeal‘𝑅)
5453lidlsubg 21233 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (SubGrp‘𝑅))
551, 54sylan 580 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (SubGrp‘𝑅))
5655ad4antr 732 . . . . . . . . 9 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → 𝐼 ∈ (SubGrp‘𝑅))
57 simpr 484 . . . . . . . . 9 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → (𝑥(.r𝑅)𝑦) ∈ 𝐼)
58 eqid 2737 . . . . . . . . . . 11 (𝑅 ~QG 𝐼) = (𝑅 ~QG 𝐼)
5958eqg0el 19201 . . . . . . . . . 10 ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([(𝑥(.r𝑅)𝑦)](𝑅 ~QG 𝐼) = 𝐼 ↔ (𝑥(.r𝑅)𝑦) ∈ 𝐼))
6059biimpar 477 . . . . . . . . 9 (((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → [(𝑥(.r𝑅)𝑦)](𝑅 ~QG 𝐼) = 𝐼)
6152, 56, 57, 60syl21anc 838 . . . . . . . 8 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → [(𝑥(.r𝑅)𝑦)](𝑅 ~QG 𝐼) = 𝐼)
624a1i 11 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼)))
63 eqidd 2738 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
6413, 58eqger 19196 . . . . . . . . . . 11 (𝐼 ∈ (SubGrp‘𝑅) → (𝑅 ~QG 𝐼) Er (Base‘𝑅))
6555, 64syl 17 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (𝑅 ~QG 𝐼) Er (Base‘𝑅))
66 simpl 482 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝑅 ∈ CRing)
6753crng2idl 21291 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → (LIdeal‘𝑅) = (2Ideal‘𝑅))
6867eleq2d 2827 . . . . . . . . . . . 12 (𝑅 ∈ CRing → (𝐼 ∈ (LIdeal‘𝑅) ↔ 𝐼 ∈ (2Ideal‘𝑅)))
6968biimpa 476 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (2Ideal‘𝑅))
70 eqid 2737 . . . . . . . . . . . 12 (2Ideal‘𝑅) = (2Ideal‘𝑅)
71 eqid 2737 . . . . . . . . . . . 12 (.r𝑅) = (.r𝑅)
7213, 58, 70, 712idlcpbl 21282 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (2Ideal‘𝑅)) → ((𝑔(𝑅 ~QG 𝐼)𝑒(𝑅 ~QG 𝐼)𝑓) → (𝑔(.r𝑅))(𝑅 ~QG 𝐼)(𝑒(.r𝑅)𝑓)))
731, 69, 72syl2an2r 685 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ((𝑔(𝑅 ~QG 𝐼)𝑒(𝑅 ~QG 𝐼)𝑓) → (𝑔(.r𝑅))(𝑅 ~QG 𝐼)(𝑒(.r𝑅)𝑓)))
741ad2antrr 726 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring)
75 simprl 771 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑒 ∈ (Base‘𝑅))
76 simprr 773 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑓 ∈ (Base‘𝑅))
7713, 71ringcl 20247 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅)) → (𝑒(.r𝑅)𝑓) ∈ (Base‘𝑅))
7874, 75, 76, 77syl3anc 1373 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → (𝑒(.r𝑅)𝑓) ∈ (Base‘𝑅))
79 eqid 2737 . . . . . . . . . 10 (.r𝑄) = (.r𝑄)
8062, 63, 65, 66, 73, 78, 71, 79qusmulval 17600 . . . . . . . . 9 (((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))
8180ad5ant134 1369 . . . . . . . 8 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → ([𝑥](𝑅 ~QG 𝐼)(.r𝑄)[𝑦](𝑅 ~QG 𝐼)) = [(𝑥(.r𝑅)𝑦)](𝑅 ~QG 𝐼))
82 lidlnsg 21258 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
831, 82sylan 580 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
84 eqid 2737 . . . . . . . . . . . 12 (0g𝑅) = (0g𝑅)
854, 84qus0 19207 . . . . . . . . . . 11 (𝐼 ∈ (NrmSGrp‘𝑅) → [(0g𝑅)](𝑅 ~QG 𝐼) = (0g𝑄))
8683, 85syl 17 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → [(0g𝑅)](𝑅 ~QG 𝐼) = (0g𝑄))
8713, 58, 84eqgid 19198 . . . . . . . . . . 11 (𝐼 ∈ (SubGrp‘𝑅) → [(0g𝑅)](𝑅 ~QG 𝐼) = 𝐼)
8855, 87syl 17 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → [(0g𝑅)](𝑅 ~QG 𝐼) = 𝐼)
8986, 88eqtr3d 2779 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (0g𝑄) = 𝐼)
9089ad4antr 732 . . . . . . . 8 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → (0g𝑄) = 𝐼)
9161, 81, 903eqtr4d 2787 . . . . . . 7 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → ([𝑥](𝑅 ~QG 𝐼)(.r𝑄)[𝑦](𝑅 ~QG 𝐼)) = (0g𝑄))
92 eqid 2737 . . . . . . . . 9 (0g𝑄) = (0g𝑄)
9329, 79, 92domneq0 20708 . . . . . . . 8 ((𝑄 ∈ Domn ∧ [𝑥](𝑅 ~QG 𝐼) ∈ (Base‘𝑄) ∧ [𝑦](𝑅 ~QG 𝐼) ∈ (Base‘𝑄)) → (([𝑥](𝑅 ~QG 𝐼)(.r𝑄)[𝑦](𝑅 ~QG 𝐼)) = (0g𝑄) ↔ ([𝑥](𝑅 ~QG 𝐼) = (0g𝑄) ∨ [𝑦](𝑅 ~QG 𝐼) = (0g𝑄))))
9493biimpa 476 . . . . . . 7 (((𝑄 ∈ Domn ∧ [𝑥](𝑅 ~QG 𝐼) ∈ (Base‘𝑄) ∧ [𝑦](𝑅 ~QG 𝐼) ∈ (Base‘𝑄)) ∧ ([𝑥](𝑅 ~QG 𝐼)(.r𝑄)[𝑦](𝑅 ~QG 𝐼)) = (0g𝑄)) → ([𝑥](𝑅 ~QG 𝐼) = (0g𝑄) ∨ [𝑦](𝑅 ~QG 𝐼) = (0g𝑄)))
9537, 48, 51, 91, 94syl31anc 1375 . . . . . 6 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → ([𝑥](𝑅 ~QG 𝐼) = (0g𝑄) ∨ [𝑦](𝑅 ~QG 𝐼) = (0g𝑄)))
9689eqeq2d 2748 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ([𝑥](𝑅 ~QG 𝐼) = (0g𝑄) ↔ [𝑥](𝑅 ~QG 𝐼) = 𝐼))
9766, 1, 103syl 18 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝑅 ∈ Grp)
9858eqg0el 19201 . . . . . . . . . 10 ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([𝑥](𝑅 ~QG 𝐼) = 𝐼𝑥𝐼))
9997, 55, 98syl2anc 584 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ([𝑥](𝑅 ~QG 𝐼) = 𝐼𝑥𝐼))
10096, 99bitrd 279 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ([𝑥](𝑅 ~QG 𝐼) = (0g𝑄) ↔ 𝑥𝐼))
10189eqeq2d 2748 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ([𝑦](𝑅 ~QG 𝐼) = (0g𝑄) ↔ [𝑦](𝑅 ~QG 𝐼) = 𝐼))
10258eqg0el 19201 . . . . . . . . . 10 ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([𝑦](𝑅 ~QG 𝐼) = 𝐼𝑦𝐼))
10397, 55, 102syl2anc 584 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ([𝑦](𝑅 ~QG 𝐼) = 𝐼𝑦𝐼))
104101, 103bitrd 279 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ([𝑦](𝑅 ~QG 𝐼) = (0g𝑄) ↔ 𝑦𝐼))
105100, 104orbi12d 919 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (([𝑥](𝑅 ~QG 𝐼) = (0g𝑄) ∨ [𝑦](𝑅 ~QG 𝐼) = (0g𝑄)) ↔ (𝑥𝐼𝑦𝐼)))
106105ad4antr 732 . . . . . 6 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → (([𝑥](𝑅 ~QG 𝐼) = (0g𝑄) ∨ [𝑦](𝑅 ~QG 𝐼) = (0g𝑄)) ↔ (𝑥𝐼𝑦𝐼)))
10795, 106mpbid 232 . . . . 5 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → (𝑥𝐼𝑦𝐼))
108107ex 412 . . . 4 (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼)))
109108anasss 466 . . 3 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼)))
110109ralrimivva 3202 . 2 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼)))
11113, 71prmidl2 33469 . 2 (((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝐼 ≠ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼)))) → 𝐼 ∈ (PrmIdeal‘𝑅))
1122, 3, 36, 110, 111syl22anc 839 1 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ∈ (PrmIdeal‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  Vcvv 3480  {csn 4626   class class class wbr 5143  cfv 6561  (class class class)co 7431   Er wer 8742  [cec 8743   / cqs 8744  1c1 11156   < clt 11295  chash 14369  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-oadd 8510  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-dju 9941  df-card 9979  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-xnn0 12600  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-hash 14370  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
  Copyright terms: Public domain W3C validator