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Theorem qsidomlem1 33040
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 20140 . . 3 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
21ad2antrr 723 . 2 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝑅 ∈ Ring)
3 simplr 766 . 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 7417 . . . . . . . . . 10 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (𝑅 ~QG 𝐼) = (𝑅 ~QG (Base‘𝑅)))
76oveq2d 7417 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (𝑅 /s (𝑅 ~QG 𝐼)) = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
84, 7eqtrid 2776 . . . . . . . 8 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 𝑄 = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
98fveq2d 6885 . . . . . . 7 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (Base‘𝑄) = (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))))
10 ringgrp 20133 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
111, 10syl 17 . . . . . . . . 9 (𝑅 ∈ CRing → 𝑅 ∈ Grp)
1211ad3antrrr 727 . . . . . . . 8 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 𝑅 ∈ Grp)
13 eqid 2724 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
14 eqid 2724 . . . . . . . . 9 (𝑅 /s (𝑅 ~QG (Base‘𝑅))) = (𝑅 /s (𝑅 ~QG (Base‘𝑅)))
1513, 14qustriv 32946 . . . . . . . 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 2764 . . . . . 6 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (Base‘𝑄) = {(Base‘𝑅)})
1817fveq2d 6885 . . . . 5 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = (♯‘{(Base‘𝑅)}))
19 fvex 6894 . . . . . 6 (Base‘𝑅) ∈ V
20 hashsng 14326 . . . . . 6 ((Base‘𝑅) ∈ V → (♯‘{(Base‘𝑅)}) = 1)
2119, 20ax-mp 5 . . . . 5 (♯‘{(Base‘𝑅)}) = 1
2218, 21eqtrdi 2780 . . . 4 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = 1)
23 1red 11212 . . . . . 6 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 1 ∈ ℝ)
24 isidom 21206 . . . . . . . . . 10 (𝑄 ∈ IDomn ↔ (𝑄 ∈ CRing ∧ 𝑄 ∈ Domn))
2524simprbi 496 . . . . . . . . 9 (𝑄 ∈ IDomn → 𝑄 ∈ Domn)
26 domnnzr 21195 . . . . . . . . 9 (𝑄 ∈ Domn → 𝑄 ∈ NzRing)
2725, 26syl 17 . . . . . . . 8 (𝑄 ∈ IDomn → 𝑄 ∈ NzRing)
2827ad2antlr 724 . . . . . . 7 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 𝑄 ∈ NzRing)
29 eqid 2724 . . . . . . . . 9 (Base‘𝑄) = (Base‘𝑄)
3029isnzr2hash 20411 . . . . . . . 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 11346 . . . . 5 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) ≠ 1)
3433neneqd 2937 . . . 4 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → ¬ (♯‘(Base‘𝑄)) = 1)
3522, 34pm2.65da 814 . . 3 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → ¬ 𝐼 = (Base‘𝑅))
3635neqned 2939 . 2 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ≠ (Base‘𝑅))
3725ad4antlr 730 . . . . . . 7 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → 𝑄 ∈ Domn)
38 ovex 7434 . . . . . . . . . 10 (𝑅 ~QG 𝐼) ∈ V
3938ecelqsi 8763 . . . . . . . . 9 (𝑥 ∈ (Base‘𝑅) → [𝑥](𝑅 ~QG 𝐼) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)))
4039ad3antlr 728 . . . . . . . 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 782 . . . . . . . . 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 2725 . . . . . . . . . 10 (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘𝑅))
44 ovexd 7436 . . . . . . . . . 10 (𝑅 ∈ CRing → (𝑅 ~QG 𝐼) ∈ V)
45 id 22 . . . . . . . . . 10 (𝑅 ∈ CRing → 𝑅 ∈ CRing)
4642, 43, 44, 45qusbas 17490 . . . . . . . . 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 2827 . . . . . . 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 8763 . . . . . . . . 9 (𝑦 ∈ (Base‘𝑅) → [𝑦](𝑅 ~QG 𝐼) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)))
5049ad2antlr 724 . . . . . . . 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 2827 . . . . . . 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 2724 . . . . . . . . . . . 12 (LIdeal‘𝑅) = (LIdeal‘𝑅)
5453lidlsubg 21072 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (SubGrp‘𝑅))
551, 54sylan 579 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (SubGrp‘𝑅))
5655ad4antr 729 . . . . . . . . 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 2724 . . . . . . . . . . 11 (𝑅 ~QG 𝐼) = (𝑅 ~QG 𝐼)
5958eqg0el 32943 . . . . . . . . . 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 835 . . . . . . . 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 2725 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
6413, 58eqger 19095 . . . . . . . . . . 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 21126 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → (LIdeal‘𝑅) = (2Ideal‘𝑅))
6867eleq2d 2811 . . . . . . . . . . . 12 (𝑅 ∈ CRing → (𝐼 ∈ (LIdeal‘𝑅) ↔ 𝐼 ∈ (2Ideal‘𝑅)))
6968biimpa 476 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (2Ideal‘𝑅))
70 eqid 2724 . . . . . . . . . . . 12 (2Ideal‘𝑅) = (2Ideal‘𝑅)
71 eqid 2724 . . . . . . . . . . . 12 (.r𝑅) = (.r𝑅)
7213, 58, 70, 712idlcpbl 21119 . . . . . . . . . . 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 682 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ((𝑔(𝑅 ~QG 𝐼)𝑒(𝑅 ~QG 𝐼)𝑓) → (𝑔(.r𝑅))(𝑅 ~QG 𝐼)(𝑒(.r𝑅)𝑓)))
741ad2antrr 723 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring)
75 simprl 768 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑒 ∈ (Base‘𝑅))
76 simprr 770 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑓 ∈ (Base‘𝑅))
7713, 71ringcl 20145 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅)) → (𝑒(.r𝑅)𝑓) ∈ (Base‘𝑅))
7874, 75, 76, 77syl3anc 1368 . . . . . . . . . 10 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → (𝑒(.r𝑅)𝑓) ∈ (Base‘𝑅))
79 eqid 2724 . . . . . . . . . 10 (.r𝑄) = (.r𝑄)
8062, 63, 65, 66, 73, 78, 71, 79qusmulval 17500 . . . . . . . . 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 1364 . . . . . . . 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 33033 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
831, 82sylan 579 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
84 eqid 2724 . . . . . . . . . . . 12 (0g𝑅) = (0g𝑅)
854, 84qus0 19105 . . . . . . . . . . 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 19097 . . . . . . . . . . 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 2766 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (0g𝑄) = 𝐼)
9089ad4antr 729 . . . . . . . 8 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → (0g𝑄) = 𝐼)
9161, 81, 903eqtr4d 2774 . . . . . . 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 2724 . . . . . . . . 9 (0g𝑄) = (0g𝑄)
9329, 79, 92domneq0 21197 . . . . . . . 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 1370 . . . . . 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 2735 . . . . . . . . 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 32943 . . . . . . . . . 10 ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([𝑥](𝑅 ~QG 𝐼) = 𝐼𝑥𝐼))
9997, 55, 98syl2anc 583 . . . . . . . . 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 2735 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ([𝑦](𝑅 ~QG 𝐼) = (0g𝑄) ↔ [𝑦](𝑅 ~QG 𝐼) = 𝐼))
10258eqg0el 32943 . . . . . . . . . 10 ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([𝑦](𝑅 ~QG 𝐼) = 𝐼𝑦𝐼))
10397, 55, 102syl2anc 583 . . . . . . . . 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 915 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (([𝑥](𝑅 ~QG 𝐼) = (0g𝑄) ∨ [𝑦](𝑅 ~QG 𝐼) = (0g𝑄)) ↔ (𝑥𝐼𝑦𝐼)))
106105ad4antr 729 . . . . . 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 231 . . . . 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 3192 . 2 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼)))
11113, 71prmidl2 33028 . 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 836 1 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ∈ (PrmIdeal‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wo 844  w3a 1084   = wceq 1533  wcel 2098  wne 2932  wral 3053  Vcvv 3466  {csn 4620   class class class wbr 5138  cfv 6533  (class class class)co 7401   Er wer 8696  [cec 8697   / cqs 8698  1c1 11107   < clt 11245  chash 14287  Basecbs 17143  .rcmulr 17197  0gc0g 17384   /s cqus 17450  Grpcgrp 18853  SubGrpcsubg 19037  NrmSGrpcnsg 19038   ~QG cqg 19039  Ringcrg 20128  CRingccrg 20129  NzRingcnzr 20404  LIdealclidl 21055  2Idealc2idl 21096  Domncdomn 21180  IDomncidom 21181  PrmIdealcprmidl 33022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  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 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-tpos 8206  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-oadd 8465  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-dju 9892  df-card 9930  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-xnn0 12542  df-z 12556  df-dec 12675  df-uz 12820  df-fz 13482  df-hash 14288  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-mulr 17210  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-0g 17386  df-imas 17453  df-qus 17454  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-grp 18856  df-minusg 18857  df-sbg 18858  df-subg 19040  df-nsg 19041  df-eqg 19042  df-cmn 19692  df-abl 19693  df-mgp 20030  df-rng 20048  df-ur 20077  df-ring 20130  df-cring 20131  df-oppr 20226  df-nzr 20405  df-subrg 20461  df-lmod 20698  df-lss 20769  df-lsp 20809  df-sra 21011  df-rgmod 21012  df-lidl 21057  df-rsp 21058  df-2idl 21097  df-domn 21184  df-idom 21185  df-prmidl 33023
This theorem is referenced by:  qsidom  33042
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