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

Theorem qsidomlem1 31149
 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 19377 . . 3 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
21ad2antrr 725 . 2 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝑅 ∈ Ring)
3 simplr 768 . 2 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ∈ (LIdeal‘𝑅))
4 qsidom.1 . . . . . . . . 9 𝑄 = (𝑅 /s (𝑅 ~QG 𝐼))
5 simpr 488 . . . . . . . . . . 11 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 𝐼 = (Base‘𝑅))
65oveq2d 7166 . . . . . . . . . 10 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (𝑅 ~QG 𝐼) = (𝑅 ~QG (Base‘𝑅)))
76oveq2d 7166 . . . . . . . . 9 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (𝑅 /s (𝑅 ~QG 𝐼)) = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
84, 7syl5eq 2805 . . . . . . . 8 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 𝑄 = (𝑅 /s (𝑅 ~QG (Base‘𝑅))))
98fveq2d 6662 . . . . . . 7 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (Base‘𝑄) = (Base‘(𝑅 /s (𝑅 ~QG (Base‘𝑅)))))
10 ringgrp 19370 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
111, 10syl 17 . . . . . . . . 9 (𝑅 ∈ CRing → 𝑅 ∈ Grp)
1211ad3antrrr 729 . . . . . . . 8 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 𝑅 ∈ Grp)
13 eqid 2758 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
14 eqid 2758 . . . . . . . . 9 (𝑅 /s (𝑅 ~QG (Base‘𝑅))) = (𝑅 /s (𝑅 ~QG (Base‘𝑅)))
1513, 14qustriv 31081 . . . . . . . 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 2793 . . . . . 6 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (Base‘𝑄) = {(Base‘𝑅)})
1817fveq2d 6662 . . . . 5 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = (♯‘{(Base‘𝑅)}))
19 fvex 6671 . . . . . 6 (Base‘𝑅) ∈ V
20 hashsng 13780 . . . . . 6 ((Base‘𝑅) ∈ V → (♯‘{(Base‘𝑅)}) = 1)
2119, 20ax-mp 5 . . . . 5 (♯‘{(Base‘𝑅)}) = 1
2218, 21eqtrdi 2809 . . . 4 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) = 1)
23 1red 10680 . . . . . 6 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 1 ∈ ℝ)
24 isidom 20145 . . . . . . . . . 10 (𝑄 ∈ IDomn ↔ (𝑄 ∈ CRing ∧ 𝑄 ∈ Domn))
2524simprbi 500 . . . . . . . . 9 (𝑄 ∈ IDomn → 𝑄 ∈ Domn)
26 domnnzr 20136 . . . . . . . . 9 (𝑄 ∈ Domn → 𝑄 ∈ NzRing)
2725, 26syl 17 . . . . . . . 8 (𝑄 ∈ IDomn → 𝑄 ∈ NzRing)
2827ad2antlr 726 . . . . . . 7 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → 𝑄 ∈ NzRing)
29 eqid 2758 . . . . . . . . 9 (Base‘𝑄) = (Base‘𝑄)
3029isnzr2hash 20105 . . . . . . . 8 (𝑄 ∈ NzRing ↔ (𝑄 ∈ Ring ∧ 1 < (♯‘(Base‘𝑄))))
3130simprbi 500 . . . . . . 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 10813 . . . . 5 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → (♯‘(Base‘𝑄)) ≠ 1)
3433neneqd 2956 . . . 4 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝐼 = (Base‘𝑅)) → ¬ (♯‘(Base‘𝑄)) = 1)
3522, 34pm2.65da 816 . . 3 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → ¬ 𝐼 = (Base‘𝑅))
3635neqned 2958 . 2 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ≠ (Base‘𝑅))
3725ad4antlr 732 . . . . . . 7 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → 𝑄 ∈ Domn)
38 ovex 7183 . . . . . . . . . 10 (𝑅 ~QG 𝐼) ∈ V
3938ecelqsi 8363 . . . . . . . . 9 (𝑥 ∈ (Base‘𝑅) → [𝑥](𝑅 ~QG 𝐼) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)))
4039ad3antlr 730 . . . . . . . 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 784 . . . . . . . . 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 2759 . . . . . . . . . 10 (𝑅 ∈ CRing → (Base‘𝑅) = (Base‘𝑅))
44 ovexd 7185 . . . . . . . . . 10 (𝑅 ∈ CRing → (𝑅 ~QG 𝐼) ∈ V)
45 id 22 . . . . . . . . . 10 (𝑅 ∈ CRing → 𝑅 ∈ CRing)
4642, 43, 44, 45qusbas 16876 . . . . . . . . 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 2854 . . . . . . 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 8363 . . . . . . . . 9 (𝑦 ∈ (Base‘𝑅) → [𝑦](𝑅 ~QG 𝐼) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝐼)))
5049ad2antlr 726 . . . . . . . 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 2854 . . . . . . 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 2758 . . . . . . . . . . . 12 (LIdeal‘𝑅) = (LIdeal‘𝑅)
5453lidlsubg 20056 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (SubGrp‘𝑅))
551, 54sylan 583 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (SubGrp‘𝑅))
5655ad4antr 731 . . . . . . . . 9 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → 𝐼 ∈ (SubGrp‘𝑅))
57 simpr 488 . . . . . . . . 9 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → (𝑥(.r𝑅)𝑦) ∈ 𝐼)
58 eqid 2758 . . . . . . . . . . 11 (𝑅 ~QG 𝐼) = (𝑅 ~QG 𝐼)
5958eqg0el 31078 . . . . . . . . . 10 ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([(𝑥(.r𝑅)𝑦)](𝑅 ~QG 𝐼) = 𝐼 ↔ (𝑥(.r𝑅)𝑦) ∈ 𝐼))
6059biimpar 481 . . . . . . . . 9 (((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → [(𝑥(.r𝑅)𝑦)](𝑅 ~QG 𝐼) = 𝐼)
6152, 56, 57, 60syl21anc 836 . . . . . . . 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 2759 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (Base‘𝑅) = (Base‘𝑅))
6413, 58eqger 18397 . . . . . . . . . . 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 486 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝑅 ∈ CRing)
6753crng2idl 20080 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → (LIdeal‘𝑅) = (2Ideal‘𝑅))
6867eleq2d 2837 . . . . . . . . . . . 12 (𝑅 ∈ CRing → (𝐼 ∈ (LIdeal‘𝑅) ↔ 𝐼 ∈ (2Ideal‘𝑅)))
6968biimpa 480 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (2Ideal‘𝑅))
70 eqid 2758 . . . . . . . . . . . 12 (2Ideal‘𝑅) = (2Ideal‘𝑅)
71 eqid 2758 . . . . . . . . . . . 12 (.r𝑅) = (.r𝑅)
7213, 58, 70, 712idlcpbl 20075 . . . . . . . . . . 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 684 . . . . . . . . . 10 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ((𝑔(𝑅 ~QG 𝐼)𝑒(𝑅 ~QG 𝐼)𝑓) → (𝑔(.r𝑅))(𝑅 ~QG 𝐼)(𝑒(.r𝑅)𝑓)))
741ad2antrr 725 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑅 ∈ Ring)
75 simprl 770 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑒 ∈ (Base‘𝑅))
76 simprr 772 . . . . . . . . . . 11 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ (𝑒 ∈ (Base‘𝑅) ∧ 𝑓 ∈ (Base‘𝑅))) → 𝑓 ∈ (Base‘𝑅))
7713, 71ringcl 19382 . . . . . . . . . . 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 2758 . . . . . . . . . 10 (.r𝑄) = (.r𝑄)
8062, 63, 65, 66, 73, 78, 71, 79qusmulval 16886 . . . . . . . . 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 31142 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
831, 82sylan 583 . . . . . . . . . . 11 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → 𝐼 ∈ (NrmSGrp‘𝑅))
84 eqid 2758 . . . . . . . . . . . 12 (0g𝑅) = (0g𝑅)
854, 84qus0 18405 . . . . . . . . . . 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 18399 . . . . . . . . . . 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 2795 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (0g𝑄) = 𝐼)
9089ad4antr 731 . . . . . . . 8 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → (0g𝑄) = 𝐼)
9161, 81, 903eqtr4d 2803 . . . . . . 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 2758 . . . . . . . . 9 (0g𝑄) = (0g𝑄)
9329, 79, 92domneq0 20138 . . . . . . . 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 480 . . . . . . 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 2769 . . . . . . . . 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 31078 . . . . . . . . . 10 ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([𝑥](𝑅 ~QG 𝐼) = 𝐼𝑥𝐼))
9997, 55, 98syl2anc 587 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ([𝑥](𝑅 ~QG 𝐼) = 𝐼𝑥𝐼))
10096, 99bitrd 282 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ([𝑥](𝑅 ~QG 𝐼) = (0g𝑄) ↔ 𝑥𝐼))
10189eqeq2d 2769 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ([𝑦](𝑅 ~QG 𝐼) = (0g𝑄) ↔ [𝑦](𝑅 ~QG 𝐼) = 𝐼))
10258eqg0el 31078 . . . . . . . . . 10 ((𝑅 ∈ Grp ∧ 𝐼 ∈ (SubGrp‘𝑅)) → ([𝑦](𝑅 ~QG 𝐼) = 𝐼𝑦𝐼))
10397, 55, 102syl2anc 587 . . . . . . . . 9 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ([𝑦](𝑅 ~QG 𝐼) = 𝐼𝑦𝐼))
104101, 103bitrd 282 . . . . . . . 8 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → ([𝑦](𝑅 ~QG 𝐼) = (0g𝑄) ↔ 𝑦𝐼))
105100, 104orbi12d 916 . . . . . . 7 ((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) → (([𝑥](𝑅 ~QG 𝐼) = (0g𝑄) ∨ [𝑦](𝑅 ~QG 𝐼) = (0g𝑄)) ↔ (𝑥𝐼𝑦𝐼)))
106105ad4antr 731 . . . . . 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 235 . . . . 5 ((((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥(.r𝑅)𝑦) ∈ 𝐼) → (𝑥𝐼𝑦𝐼))
108107ex 416 . . . 4 (((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼)))
109108anasss 470 . . 3 ((((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → ((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼)))
110109ralrimivva 3120 . 2 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) ∈ 𝐼 → (𝑥𝐼𝑦𝐼)))
11113, 71prmidl2 31137 . 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 837 1 (((𝑅 ∈ CRing ∧ 𝐼 ∈ (LIdeal‘𝑅)) ∧ 𝑄 ∈ IDomn) → 𝐼 ∈ (PrmIdeal‘𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  Vcvv 3409  {csn 4522   class class class wbr 5032  ‘cfv 6335  (class class class)co 7150   Er wer 8296  [cec 8297   / cqs 8298  1c1 10576   < clt 10713  ♯chash 13740  Basecbs 16541  .rcmulr 16624  0gc0g 16771   /s cqus 16836  Grpcgrp 18169  SubGrpcsubg 18340  NrmSGrpcnsg 18341   ~QG cqg 18342  Ringcrg 19365  CRingccrg 19366  LIdealclidl 20010  2Idealc2idl 20072  NzRingcnzr 20098  Domncdomn 20121  IDomncidom 20122  PrmIdealcprmidl 31131 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-tpos 7902  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-oadd 8116  df-er 8299  df-ec 8301  df-qs 8305  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-sup 8939  df-inf 8940  df-dju 9363  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-3 11738  df-4 11739  df-5 11740  df-6 11741  df-7 11742  df-8 11743  df-9 11744  df-n0 11935  df-xnn0 12007  df-z 12021  df-dec 12138  df-uz 12283  df-fz 12940  df-hash 13741  df-struct 16543  df-ndx 16544  df-slot 16545  df-base 16547  df-sets 16548  df-ress 16549  df-plusg 16636  df-mulr 16637  df-sca 16639  df-vsca 16640  df-ip 16641  df-tset 16642  df-ple 16643  df-ds 16645  df-0g 16773  df-imas 16839  df-qus 16840  df-mgm 17918  df-sgrp 17967  df-mnd 17978  df-grp 18172  df-minusg 18173  df-sbg 18174  df-subg 18343  df-nsg 18344  df-eqg 18345  df-cmn 18975  df-abl 18976  df-mgp 19308  df-ur 19320  df-ring 19367  df-cring 19368  df-oppr 19444  df-subrg 19601  df-lmod 19704  df-lss 19772  df-lsp 19812  df-sra 20012  df-rgmod 20013  df-lidl 20014  df-rsp 20015  df-2idl 20073  df-nzr 20099  df-domn 20125  df-idom 20126  df-prmidl 31132 This theorem is referenced by:  qsidom  31151
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