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

Theorem qsdrngi 32519
Description: A quotient by a maximal left and maximal right ideal is a division ring. (Contributed by Thierry Arnoux, 9-Mar-2025.)
Hypotheses
Ref Expression
qsdrng.0 𝑂 = (oppr𝑅)
qsdrng.q 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))
qsdrng.r (𝜑𝑅 ∈ NzRing)
qsdrngi.1 (𝜑𝑀 ∈ (MaxIdeal‘𝑅))
qsdrngi.2 (𝜑𝑀 ∈ (MaxIdeal‘𝑂))
Assertion
Ref Expression
qsdrngi (𝜑𝑄 ∈ DivRing)

Proof of Theorem qsdrngi
Dummy variables 𝑟 𝑢 𝑣 𝑥 𝑠 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 qsdrng.q . . . 4 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))
2 eqid 2732 . . . 4 (Base‘𝑅) = (Base‘𝑅)
3 qsdrng.r . . . . 5 (𝜑𝑅 ∈ NzRing)
4 nzrring 20247 . . . . 5 (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
53, 4syl 17 . . . 4 (𝜑𝑅 ∈ Ring)
6 qsdrngi.1 . . . . . . 7 (𝜑𝑀 ∈ (MaxIdeal‘𝑅))
72mxidlidl 32494 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ∈ (LIdeal‘𝑅))
85, 6, 7syl2anc 584 . . . . . 6 (𝜑𝑀 ∈ (LIdeal‘𝑅))
9 qsdrng.0 . . . . . . . . 9 𝑂 = (oppr𝑅)
109opprring 20115 . . . . . . . 8 (𝑅 ∈ Ring → 𝑂 ∈ Ring)
115, 10syl 17 . . . . . . 7 (𝜑𝑂 ∈ Ring)
12 qsdrngi.2 . . . . . . 7 (𝜑𝑀 ∈ (MaxIdeal‘𝑂))
13 eqid 2732 . . . . . . . 8 (Base‘𝑂) = (Base‘𝑂)
1413mxidlidl 32494 . . . . . . 7 ((𝑂 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑂)) → 𝑀 ∈ (LIdeal‘𝑂))
1511, 12, 14syl2anc 584 . . . . . 6 (𝜑𝑀 ∈ (LIdeal‘𝑂))
168, 15elind 4191 . . . . 5 (𝜑𝑀 ∈ ((LIdeal‘𝑅) ∩ (LIdeal‘𝑂)))
17 eqid 2732 . . . . . 6 (LIdeal‘𝑅) = (LIdeal‘𝑅)
18 eqid 2732 . . . . . 6 (LIdeal‘𝑂) = (LIdeal‘𝑂)
19 eqid 2732 . . . . . 6 (2Ideal‘𝑅) = (2Ideal‘𝑅)
2017, 9, 18, 192idlval 20806 . . . . 5 (2Ideal‘𝑅) = ((LIdeal‘𝑅) ∩ (LIdeal‘𝑂))
2116, 20eleqtrrdi 2844 . . . 4 (𝜑𝑀 ∈ (2Ideal‘𝑅))
222mxidlnr 32495 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdeal‘𝑅)) → 𝑀 ≠ (Base‘𝑅))
235, 6, 22syl2anc 584 . . . 4 (𝜑𝑀 ≠ (Base‘𝑅))
241, 2, 5, 3, 21, 23qsnzr 32489 . . 3 (𝜑𝑄 ∈ NzRing)
25 eqid 2732 . . . 4 (1r𝑄) = (1r𝑄)
26 eqid 2732 . . . 4 (0g𝑄) = (0g𝑄)
2725, 26nzrnz 20246 . . 3 (𝑄 ∈ NzRing → (1r𝑄) ≠ (0g𝑄))
2824, 27syl 17 . 2 (𝜑 → (1r𝑄) ≠ (0g𝑄))
29 eqid 2732 . . . . . . . . . . . . . 14 (Base‘𝑄) = (Base‘𝑄)
30 eqid 2732 . . . . . . . . . . . . . 14 (.r𝑄) = (.r𝑄)
31 eqid 2732 . . . . . . . . . . . . . 14 (Unit‘𝑄) = (Unit‘𝑄)
321, 19qusring 20811 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (2Ideal‘𝑅)) → 𝑄 ∈ Ring)
335, 21, 32syl2anc 584 . . . . . . . . . . . . . . . 16 (𝜑𝑄 ∈ Ring)
3433ad10antr 742 . . . . . . . . . . . . . . 15 (((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) → 𝑄 ∈ Ring)
3534adantr 481 . . . . . . . . . . . . . 14 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → 𝑄 ∈ Ring)
36 eldifi 4123 . . . . . . . . . . . . . . . 16 (𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)}) → 𝑢 ∈ (Base‘𝑄))
3736adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) → 𝑢 ∈ (Base‘𝑄))
3837ad10antr 742 . . . . . . . . . . . . . 14 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → 𝑢 ∈ (Base‘𝑄))
39 ovex 7427 . . . . . . . . . . . . . . . . 17 (𝑅 ~QG 𝑀) ∈ V
4039ecelqsi 8752 . . . . . . . . . . . . . . . 16 (𝑟 ∈ (Base‘𝑅) → [𝑟](𝑅 ~QG 𝑀) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
4140ad4antlr 731 . . . . . . . . . . . . . . 15 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → [𝑟](𝑅 ~QG 𝑀) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
421a1i 11 . . . . . . . . . . . . . . . . . 18 (𝜑𝑄 = (𝑅 /s (𝑅 ~QG 𝑀)))
43 eqidd 2733 . . . . . . . . . . . . . . . . . 18 (𝜑 → (Base‘𝑅) = (Base‘𝑅))
44 ovexd 7429 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑅 ~QG 𝑀) ∈ V)
4542, 43, 44, 3qusbas 17475 . . . . . . . . . . . . . . . . 17 (𝜑 → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = (Base‘𝑄))
4645adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = (Base‘𝑄))
4746ad10antr 742 . . . . . . . . . . . . . . 15 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = (Base‘𝑄))
4841, 47eleqtrd 2835 . . . . . . . . . . . . . 14 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → [𝑟](𝑅 ~QG 𝑀) ∈ (Base‘𝑄))
4939ecelqsi 8752 . . . . . . . . . . . . . . . 16 (𝑠 ∈ (Base‘𝑅) → [𝑠](𝑅 ~QG 𝑀) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
5049ad2antlr 725 . . . . . . . . . . . . . . 15 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → [𝑠](𝑅 ~QG 𝑀) ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
5150, 47eleqtrd 2835 . . . . . . . . . . . . . 14 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → [𝑠](𝑅 ~QG 𝑀) ∈ (Base‘𝑄))
52 simpllr 774 . . . . . . . . . . . . . . . 16 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → 𝑣 = [𝑟](𝑅 ~QG 𝑀))
53 simp-9r 792 . . . . . . . . . . . . . . . . 17 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → 𝑢 = [𝑥](𝑅 ~QG 𝑀))
5453eqcomd 2738 . . . . . . . . . . . . . . . 16 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → [𝑥](𝑅 ~QG 𝑀) = 𝑢)
5552, 54oveq12d 7412 . . . . . . . . . . . . . . 15 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = ([𝑟](𝑅 ~QG 𝑀)(.r𝑄)𝑢))
56 simp-7r 788 . . . . . . . . . . . . . . 15 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄))
5755, 56eqtr3d 2774 . . . . . . . . . . . . . 14 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → ([𝑟](𝑅 ~QG 𝑀)(.r𝑄)𝑢) = (1r𝑄))
58 eqid 2732 . . . . . . . . . . . . . . . 16 (oppr𝑄) = (oppr𝑄)
59 eqid 2732 . . . . . . . . . . . . . . . 16 (.r‘(oppr𝑄)) = (.r‘(oppr𝑄))
6029, 30, 58, 59opprmul 20107 . . . . . . . . . . . . . . 15 ([𝑠](𝑅 ~QG 𝑀)(.r‘(oppr𝑄))𝑢) = (𝑢(.r𝑄)[𝑠](𝑅 ~QG 𝑀))
61 simp-5r 784 . . . . . . . . . . . . . . . 16 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀))))
625ad3antrrr 728 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑅 ∈ Ring)
6362ad8antr 738 . . . . . . . . . . . . . . . . . 18 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → 𝑅 ∈ Ring)
6421ad3antrrr 728 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑀 ∈ (2Ideal‘𝑅))
6564ad8antr 738 . . . . . . . . . . . . . . . . . 18 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → 𝑀 ∈ (2Ideal‘𝑅))
662, 9, 1, 63, 65, 29, 51, 38opprqusmulr 32515 . . . . . . . . . . . . . . . . 17 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → ([𝑠](𝑅 ~QG 𝑀)(.r‘(oppr𝑄))𝑢) = ([𝑠](𝑅 ~QG 𝑀)(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))𝑢))
67 simpr 485 . . . . . . . . . . . . . . . . . 18 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → 𝑤 = [𝑠](𝑅 ~QG 𝑀))
682, 17lidlss 20783 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑀 ∈ (LIdeal‘𝑅) → 𝑀 ⊆ (Base‘𝑅))
698, 68syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑀 ⊆ (Base‘𝑅))
709, 2oppreqg 32507 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ Ring ∧ 𝑀 ⊆ (Base‘𝑅)) → (𝑅 ~QG 𝑀) = (𝑂 ~QG 𝑀))
715, 69, 70syl2anc 584 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑅 ~QG 𝑀) = (𝑂 ~QG 𝑀))
7271ad10antr 742 . . . . . . . . . . . . . . . . . . . . 21 (((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) → (𝑅 ~QG 𝑀) = (𝑂 ~QG 𝑀))
7372adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → (𝑅 ~QG 𝑀) = (𝑂 ~QG 𝑀))
7473eceq2d 8730 . . . . . . . . . . . . . . . . . . 19 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → [𝑥](𝑅 ~QG 𝑀) = [𝑥](𝑂 ~QG 𝑀))
7553, 74eqtr2d 2773 . . . . . . . . . . . . . . . . . 18 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → [𝑥](𝑂 ~QG 𝑀) = 𝑢)
7667, 75oveq12d 7412 . . . . . . . . . . . . . . . . 17 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = ([𝑠](𝑅 ~QG 𝑀)(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))𝑢))
7766, 76eqtr4d 2775 . . . . . . . . . . . . . . . 16 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → ([𝑠](𝑅 ~QG 𝑀)(.r‘(oppr𝑄))𝑢) = (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)))
7858, 25oppr1 20118 . . . . . . . . . . . . . . . . . . 19 (1r𝑄) = (1r‘(oppr𝑄))
792, 9, 1, 5, 21opprqus1r 32516 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (1r‘(oppr𝑄)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀))))
8078, 79eqtrid 2784 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1r𝑄) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀))))
8180ad10antr 742 . . . . . . . . . . . . . . . . 17 (((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) → (1r𝑄) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀))))
8281adantr 481 . . . . . . . . . . . . . . . 16 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → (1r𝑄) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀))))
8361, 77, 823eqtr4d 2782 . . . . . . . . . . . . . . 15 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → ([𝑠](𝑅 ~QG 𝑀)(.r‘(oppr𝑄))𝑢) = (1r𝑄))
8460, 83eqtr3id 2786 . . . . . . . . . . . . . 14 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → (𝑢(.r𝑄)[𝑠](𝑅 ~QG 𝑀)) = (1r𝑄))
8529, 26, 25, 30, 31, 35, 38, 48, 51, 57, 84ringinveu 32320 . . . . . . . . . . . . 13 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → [𝑠](𝑅 ~QG 𝑀) = [𝑟](𝑅 ~QG 𝑀))
8685, 67, 523eqtr4rd 2783 . . . . . . . . . . . 12 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → 𝑣 = 𝑤)
8786oveq2d 7410 . . . . . . . . . . 11 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → (𝑢(.r𝑄)𝑣) = (𝑢(.r𝑄)𝑤))
8867oveq2d 7410 . . . . . . . . . . 11 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → (𝑢(.r𝑄)𝑤) = (𝑢(.r𝑄)[𝑠](𝑅 ~QG 𝑀)))
8987, 88, 843eqtrd 2776 . . . . . . . . . 10 ((((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Base‘𝑅)) ∧ 𝑤 = [𝑠](𝑅 ~QG 𝑀)) → (𝑢(.r𝑄)𝑣) = (1r𝑄))
90 simp-4r 782 . . . . . . . . . . . 12 ((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀))))
9171qseq2d 8745 . . . . . . . . . . . . . 14 (𝜑 → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = ((Base‘𝑅) / (𝑂 ~QG 𝑀)))
9291ad9antr 740 . . . . . . . . . . . . 13 ((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = ((Base‘𝑅) / (𝑂 ~QG 𝑀)))
93 eqidd 2733 . . . . . . . . . . . . . 14 ((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → (𝑂 /s (𝑂 ~QG 𝑀)) = (𝑂 /s (𝑂 ~QG 𝑀)))
949, 2opprbas 20111 . . . . . . . . . . . . . . 15 (Base‘𝑅) = (Base‘𝑂)
9594a1i 11 . . . . . . . . . . . . . 14 ((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → (Base‘𝑅) = (Base‘𝑂))
96 ovexd 7429 . . . . . . . . . . . . . 14 ((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → (𝑂 ~QG 𝑀) ∈ V)
979fvexi 6893 . . . . . . . . . . . . . . 15 𝑂 ∈ V
9897a1i 11 . . . . . . . . . . . . . 14 ((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → 𝑂 ∈ V)
9993, 95, 96, 98qusbas 17475 . . . . . . . . . . . . 13 ((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → ((Base‘𝑅) / (𝑂 ~QG 𝑀)) = (Base‘(𝑂 /s (𝑂 ~QG 𝑀))))
10092, 99eqtr2d 2773 . . . . . . . . . . . 12 ((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → (Base‘(𝑂 /s (𝑂 ~QG 𝑀))) = ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
10190, 100eleqtrd 2835 . . . . . . . . . . 11 ((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → 𝑤 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
102 elqsi 8749 . . . . . . . . . . 11 (𝑤 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)) → ∃𝑠 ∈ (Base‘𝑅)𝑤 = [𝑠](𝑅 ~QG 𝑀))
103101, 102syl 17 . . . . . . . . . 10 ((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → ∃𝑠 ∈ (Base‘𝑅)𝑤 = [𝑠](𝑅 ~QG 𝑀))
10489, 103r19.29a 3162 . . . . . . . . 9 ((((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ 𝑟 ∈ (Base‘𝑅)) ∧ 𝑣 = [𝑟](𝑅 ~QG 𝑀)) → (𝑢(.r𝑄)𝑣) = (1r𝑄))
105 simp-4r 782 . . . . . . . . . . 11 ((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) → 𝑣 ∈ (Base‘𝑄))
10646ad6antr 734 . . . . . . . . . . 11 ((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) → ((Base‘𝑅) / (𝑅 ~QG 𝑀)) = (Base‘𝑄))
107105, 106eleqtrrd 2836 . . . . . . . . . 10 ((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) → 𝑣 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
108 elqsi 8749 . . . . . . . . . 10 (𝑣 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)) → ∃𝑟 ∈ (Base‘𝑅)𝑣 = [𝑟](𝑅 ~QG 𝑀))
109107, 108syl 17 . . . . . . . . 9 ((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) → ∃𝑟 ∈ (Base‘𝑅)𝑣 = [𝑟](𝑅 ~QG 𝑀))
110104, 109r19.29a 3162 . . . . . . . 8 ((((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) ∧ 𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀)))) → (𝑢(.r𝑄)𝑣) = (1r𝑄))
111 eqid 2732 . . . . . . . . . 10 (oppr𝑂) = (oppr𝑂)
112 eqid 2732 . . . . . . . . . 10 (𝑂 /s (𝑂 ~QG 𝑀)) = (𝑂 /s (𝑂 ~QG 𝑀))
1133ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑅 ∈ NzRing)
1149opprnzr 20251 . . . . . . . . . . 11 (𝑅 ∈ NzRing → 𝑂 ∈ NzRing)
115113, 114syl 17 . . . . . . . . . 10 ((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑂 ∈ NzRing)
11612ad3antrrr 728 . . . . . . . . . 10 ((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑀 ∈ (MaxIdeal‘𝑂))
1176ad3antrrr 728 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑀 ∈ (MaxIdeal‘𝑅))
1189, 62, 117opprmxidlabs 32511 . . . . . . . . . 10 ((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑀 ∈ (MaxIdeal‘(oppr𝑂)))
119 simplr 767 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑥 ∈ (Base‘𝑅))
12094a1i 11 . . . . . . . . . . 11 ((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → (Base‘𝑅) = (Base‘𝑂))
121119, 120eleqtrd 2835 . . . . . . . . . 10 ((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → 𝑥 ∈ (Base‘𝑂))
122 simplr 767 . . . . . . . . . . . 12 (((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥𝑀) → 𝑢 = [𝑥](𝑅 ~QG 𝑀))
1235ringgrpd 20025 . . . . . . . . . . . . . . 15 (𝜑𝑅 ∈ Grp)
124123ad4antr 730 . . . . . . . . . . . . . 14 (((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥𝑀) → 𝑅 ∈ Grp)
125 lidlnsg 32479 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdeal‘𝑅)) → 𝑀 ∈ (NrmSGrp‘𝑅))
1265, 8, 125syl2anc 584 . . . . . . . . . . . . . . . 16 (𝜑𝑀 ∈ (NrmSGrp‘𝑅))
127 nsgsubg 19012 . . . . . . . . . . . . . . . 16 (𝑀 ∈ (NrmSGrp‘𝑅) → 𝑀 ∈ (SubGrp‘𝑅))
128126, 127syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑀 ∈ (SubGrp‘𝑅))
129128ad4antr 730 . . . . . . . . . . . . . 14 (((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥𝑀) → 𝑀 ∈ (SubGrp‘𝑅))
130 simpr 485 . . . . . . . . . . . . . 14 (((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥𝑀) → 𝑥𝑀)
131 eqid 2732 . . . . . . . . . . . . . . . 16 (𝑅 ~QG 𝑀) = (𝑅 ~QG 𝑀)
132131eqg0el 32399 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Grp ∧ 𝑀 ∈ (SubGrp‘𝑅)) → ([𝑥](𝑅 ~QG 𝑀) = 𝑀𝑥𝑀))
133132biimpar 478 . . . . . . . . . . . . . 14 (((𝑅 ∈ Grp ∧ 𝑀 ∈ (SubGrp‘𝑅)) ∧ 𝑥𝑀) → [𝑥](𝑅 ~QG 𝑀) = 𝑀)
134124, 129, 130, 133syl21anc 836 . . . . . . . . . . . . 13 (((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥𝑀) → [𝑥](𝑅 ~QG 𝑀) = 𝑀)
135 eqid 2732 . . . . . . . . . . . . . . 15 (0g𝑅) = (0g𝑅)
1362, 131, 135eqgid 19034 . . . . . . . . . . . . . 14 (𝑀 ∈ (SubGrp‘𝑅) → [(0g𝑅)](𝑅 ~QG 𝑀) = 𝑀)
137129, 136syl 17 . . . . . . . . . . . . 13 (((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥𝑀) → [(0g𝑅)](𝑅 ~QG 𝑀) = 𝑀)
138134, 137eqtr4d 2775 . . . . . . . . . . . 12 (((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥𝑀) → [𝑥](𝑅 ~QG 𝑀) = [(0g𝑅)](𝑅 ~QG 𝑀))
1391, 135qus0 19040 . . . . . . . . . . . . . 14 (𝑀 ∈ (NrmSGrp‘𝑅) → [(0g𝑅)](𝑅 ~QG 𝑀) = (0g𝑄))
140126, 139syl 17 . . . . . . . . . . . . 13 (𝜑 → [(0g𝑅)](𝑅 ~QG 𝑀) = (0g𝑄))
141140ad4antr 730 . . . . . . . . . . . 12 (((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥𝑀) → [(0g𝑅)](𝑅 ~QG 𝑀) = (0g𝑄))
142122, 138, 1413eqtrd 2776 . . . . . . . . . . 11 (((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥𝑀) → 𝑢 = (0g𝑄))
143 eldifsnneq 4788 . . . . . . . . . . . 12 (𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)}) → ¬ 𝑢 = (0g𝑄))
144143ad4antlr 731 . . . . . . . . . . 11 (((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑥𝑀) → ¬ 𝑢 = (0g𝑄))
145142, 144pm2.65da 815 . . . . . . . . . 10 ((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → ¬ 𝑥𝑀)
146111, 112, 115, 116, 118, 121, 145qsdrngilem 32518 . . . . . . . . 9 ((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → ∃𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))(𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀))))
147146ad2antrr 724 . . . . . . . 8 ((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) → ∃𝑤 ∈ (Base‘(𝑂 /s (𝑂 ~QG 𝑀)))(𝑤(.r‘(𝑂 /s (𝑂 ~QG 𝑀)))[𝑥](𝑂 ~QG 𝑀)) = (1r‘(𝑂 /s (𝑂 ~QG 𝑀))))
148110, 147r19.29a 3162 . . . . . . 7 ((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) → (𝑢(.r𝑄)𝑣) = (1r𝑄))
149 simpllr 774 . . . . . . . . 9 ((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) → 𝑢 = [𝑥](𝑅 ~QG 𝑀))
150149oveq2d 7410 . . . . . . . 8 ((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) → (𝑣(.r𝑄)𝑢) = (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)))
151 simpr 485 . . . . . . . 8 ((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) → (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄))
152150, 151eqtrd 2772 . . . . . . 7 ((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) → (𝑣(.r𝑄)𝑢) = (1r𝑄))
153148, 152jca 512 . . . . . 6 ((((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Base‘𝑄)) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄)) → ((𝑢(.r𝑄)𝑣) = (1r𝑄) ∧ (𝑣(.r𝑄)𝑢) = (1r𝑄)))
154153anasss 467 . . . . 5 (((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) ∧ (𝑣 ∈ (Base‘𝑄) ∧ (𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄))) → ((𝑢(.r𝑄)𝑣) = (1r𝑄) ∧ (𝑣(.r𝑄)𝑢) = (1r𝑄)))
1559, 1, 113, 117, 116, 119, 145qsdrngilem 32518 . . . . 5 ((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → ∃𝑣 ∈ (Base‘𝑄)(𝑣(.r𝑄)[𝑥](𝑅 ~QG 𝑀)) = (1r𝑄))
156154, 155reximddv 3171 . . . 4 ((((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) ∧ 𝑥 ∈ (Base‘𝑅)) ∧ 𝑢 = [𝑥](𝑅 ~QG 𝑀)) → ∃𝑣 ∈ (Base‘𝑄)((𝑢(.r𝑄)𝑣) = (1r𝑄) ∧ (𝑣(.r𝑄)𝑢) = (1r𝑄)))
15737, 46eleqtrrd 2836 . . . . 5 ((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) → 𝑢 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)))
158 elqsi 8749 . . . . 5 (𝑢 ∈ ((Base‘𝑅) / (𝑅 ~QG 𝑀)) → ∃𝑥 ∈ (Base‘𝑅)𝑢 = [𝑥](𝑅 ~QG 𝑀))
159157, 158syl 17 . . . 4 ((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) → ∃𝑥 ∈ (Base‘𝑅)𝑢 = [𝑥](𝑅 ~QG 𝑀))
160156, 159r19.29a 3162 . . 3 ((𝜑𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})) → ∃𝑣 ∈ (Base‘𝑄)((𝑢(.r𝑄)𝑣) = (1r𝑄) ∧ (𝑣(.r𝑄)𝑢) = (1r𝑄)))
161160ralrimiva 3146 . 2 (𝜑 → ∀𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})∃𝑣 ∈ (Base‘𝑄)((𝑢(.r𝑄)𝑣) = (1r𝑄) ∧ (𝑣(.r𝑄)𝑢) = (1r𝑄)))
16229, 26, 25, 30, 31, 33isdrng4 32321 . 2 (𝜑 → (𝑄 ∈ DivRing ↔ ((1r𝑄) ≠ (0g𝑄) ∧ ∀𝑢 ∈ ((Base‘𝑄) ∖ {(0g𝑄)})∃𝑣 ∈ (Base‘𝑄)((𝑢(.r𝑄)𝑣) = (1r𝑄) ∧ (𝑣(.r𝑄)𝑢) = (1r𝑄)))))
16328, 161, 162mpbir2and 711 1 (𝜑𝑄 ∈ DivRing)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2940  wral 3061  wrex 3070  Vcvv 3474  cdif 3942  cin 3944  wss 3945  {csn 4623  cfv 6533  (class class class)co 7394  [cec 8686   / cqs 8687  Basecbs 17128  .rcmulr 17182  0gc0g 17369   /s cqus 17435  Grpcgrp 18796  SubGrpcsubg 18974  NrmSGrpcnsg 18975   ~QG cqg 18976  1rcur 19965  Ringcrg 20016  opprcoppr 20103  Unitcui 20123  NzRingcnzr 20243  DivRingcdr 20267  LIdealclidl 20734  2Idealc2idl 20804  MaxIdealcmxidl 32490
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 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709  ax-cnex 11150  ax-resscn 11151  ax-1cn 11152  ax-icn 11153  ax-addcl 11154  ax-addrcl 11155  ax-mulcl 11156  ax-mulrcl 11157  ax-mulcom 11158  ax-addass 11159  ax-mulass 11160  ax-distr 11161  ax-i2m1 11162  ax-1ne0 11163  ax-1rid 11164  ax-rnegex 11165  ax-rrecex 11166  ax-cnre 11167  ax-pre-lttri 11168  ax-pre-lttrn 11169  ax-pre-ltadd 11170  ax-pre-mulgt0 11171
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 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-se 5626  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  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-isom 6542  df-riota 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-of 7654  df-om 7840  df-1st 7959  df-2nd 7960  df-supp 8131  df-tpos 8195  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-rdg 8394  df-1o 8450  df-2o 8451  df-er 8688  df-ec 8690  df-qs 8694  df-map 8807  df-ixp 8877  df-en 8925  df-dom 8926  df-sdom 8927  df-fin 8928  df-fsupp 9347  df-sup 9421  df-inf 9422  df-oi 9489  df-card 9918  df-pnf 11234  df-mnf 11235  df-xr 11236  df-ltxr 11237  df-le 11238  df-sub 11430  df-neg 11431  df-nn 12197  df-2 12259  df-3 12260  df-4 12261  df-5 12262  df-6 12263  df-7 12264  df-8 12265  df-9 12266  df-n0 12457  df-z 12543  df-dec 12662  df-uz 12807  df-fz 13469  df-fzo 13612  df-seq 13951  df-hash 14275  df-struct 17064  df-sets 17081  df-slot 17099  df-ndx 17111  df-base 17129  df-ress 17158  df-plusg 17194  df-mulr 17195  df-sca 17197  df-vsca 17198  df-ip 17199  df-tset 17200  df-ple 17201  df-ds 17203  df-hom 17205  df-cco 17206  df-0g 17371  df-gsum 17372  df-prds 17377  df-pws 17379  df-imas 17438  df-qus 17439  df-mre 17514  df-mrc 17515  df-acs 17517  df-mgm 18545  df-sgrp 18594  df-mnd 18605  df-mhm 18649  df-submnd 18650  df-grp 18799  df-minusg 18800  df-sbg 18801  df-mulg 18925  df-subg 18977  df-nsg 18978  df-eqg 18979  df-ghm 19058  df-cntz 19149  df-cmn 19616  df-abl 19617  df-mgp 19949  df-ur 19966  df-ring 20018  df-oppr 20104  df-dvdsr 20125  df-unit 20126  df-invr 20156  df-nzr 20244  df-drng 20269  df-subrg 20312  df-lmod 20424  df-lss 20494  df-lsp 20534  df-lmhm 20584  df-lbs 20637  df-sra 20736  df-rgmod 20737  df-lidl 20738  df-rsp 20739  df-2idl 20805  df-dsmm 21222  df-frlm 21237  df-uvc 21273  df-mxidl 32491
This theorem is referenced by:  qsdrng  32521
  Copyright terms: Public domain W3C validator