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Theorem qsdrngi 32884
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 = (opprβ€˜π‘…)
qsdrng.q 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))
qsdrng.r (πœ‘ β†’ 𝑅 ∈ NzRing)
qsdrngi.1 (πœ‘ β†’ 𝑀 ∈ (MaxIdealβ€˜π‘…))
qsdrngi.2 (πœ‘ β†’ 𝑀 ∈ (MaxIdealβ€˜π‘‚))
Assertion
Ref Expression
qsdrngi (πœ‘ β†’ 𝑄 ∈ DivRing)

Proof of Theorem qsdrngi
Dummy variables π‘Ÿ 𝑒 𝑣 π‘₯ 𝑠 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 qsdrng.q . . . 4 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀))
2 eqid 2731 . . . 4 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
3 qsdrng.r . . . . 5 (πœ‘ β†’ 𝑅 ∈ NzRing)
4 nzrring 20408 . . . . 5 (𝑅 ∈ NzRing β†’ 𝑅 ∈ Ring)
53, 4syl 17 . . . 4 (πœ‘ β†’ 𝑅 ∈ Ring)
6 qsdrngi.1 . . . . . . 7 (πœ‘ β†’ 𝑀 ∈ (MaxIdealβ€˜π‘…))
72mxidlidl 32854 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) β†’ 𝑀 ∈ (LIdealβ€˜π‘…))
85, 6, 7syl2anc 583 . . . . . 6 (πœ‘ β†’ 𝑀 ∈ (LIdealβ€˜π‘…))
9 qsdrng.0 . . . . . . . . 9 𝑂 = (opprβ€˜π‘…)
109opprring 20239 . . . . . . . 8 (𝑅 ∈ Ring β†’ 𝑂 ∈ Ring)
115, 10syl 17 . . . . . . 7 (πœ‘ β†’ 𝑂 ∈ Ring)
12 qsdrngi.2 . . . . . . 7 (πœ‘ β†’ 𝑀 ∈ (MaxIdealβ€˜π‘‚))
13 eqid 2731 . . . . . . . 8 (Baseβ€˜π‘‚) = (Baseβ€˜π‘‚)
1413mxidlidl 32854 . . . . . . 7 ((𝑂 ∈ Ring ∧ 𝑀 ∈ (MaxIdealβ€˜π‘‚)) β†’ 𝑀 ∈ (LIdealβ€˜π‘‚))
1511, 12, 14syl2anc 583 . . . . . 6 (πœ‘ β†’ 𝑀 ∈ (LIdealβ€˜π‘‚))
168, 15elind 4194 . . . . 5 (πœ‘ β†’ 𝑀 ∈ ((LIdealβ€˜π‘…) ∩ (LIdealβ€˜π‘‚)))
17 eqid 2731 . . . . . 6 (LIdealβ€˜π‘…) = (LIdealβ€˜π‘…)
18 eqid 2731 . . . . . 6 (LIdealβ€˜π‘‚) = (LIdealβ€˜π‘‚)
19 eqid 2731 . . . . . 6 (2Idealβ€˜π‘…) = (2Idealβ€˜π‘…)
2017, 9, 18, 192idlval 21008 . . . . 5 (2Idealβ€˜π‘…) = ((LIdealβ€˜π‘…) ∩ (LIdealβ€˜π‘‚))
2116, 20eleqtrrdi 2843 . . . 4 (πœ‘ β†’ 𝑀 ∈ (2Idealβ€˜π‘…))
222mxidlnr 32855 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) β†’ 𝑀 β‰  (Baseβ€˜π‘…))
235, 6, 22syl2anc 583 . . . 4 (πœ‘ β†’ 𝑀 β‰  (Baseβ€˜π‘…))
241, 2, 5, 3, 21, 23qsnzr 32849 . . 3 (πœ‘ β†’ 𝑄 ∈ NzRing)
25 eqid 2731 . . . 4 (1rβ€˜π‘„) = (1rβ€˜π‘„)
26 eqid 2731 . . . 4 (0gβ€˜π‘„) = (0gβ€˜π‘„)
2725, 26nzrnz 20407 . . 3 (𝑄 ∈ NzRing β†’ (1rβ€˜π‘„) β‰  (0gβ€˜π‘„))
2824, 27syl 17 . 2 (πœ‘ β†’ (1rβ€˜π‘„) β‰  (0gβ€˜π‘„))
29 eqid 2731 . . . . . . . . . . . . . 14 (Baseβ€˜π‘„) = (Baseβ€˜π‘„)
30 eqid 2731 . . . . . . . . . . . . . 14 (.rβ€˜π‘„) = (.rβ€˜π‘„)
31 eqid 2731 . . . . . . . . . . . . . 14 (Unitβ€˜π‘„) = (Unitβ€˜π‘„)
321, 19qusring 21024 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (2Idealβ€˜π‘…)) β†’ 𝑄 ∈ Ring)
335, 21, 32syl2anc 583 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ 𝑄 ∈ Ring)
3433ad10antr 741 . . . . . . . . . . . . . . 15 (((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) β†’ 𝑄 ∈ Ring)
3534adantr 480 . . . . . . . . . . . . . 14 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ 𝑄 ∈ Ring)
36 eldifi 4126 . . . . . . . . . . . . . . . 16 (𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)}) β†’ 𝑒 ∈ (Baseβ€˜π‘„))
3736adantl 481 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) β†’ 𝑒 ∈ (Baseβ€˜π‘„))
3837ad10antr 741 . . . . . . . . . . . . . 14 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ 𝑒 ∈ (Baseβ€˜π‘„))
39 ovex 7445 . . . . . . . . . . . . . . . . 17 (𝑅 ~QG 𝑀) ∈ V
4039ecelqsi 8770 . . . . . . . . . . . . . . . 16 (π‘Ÿ ∈ (Baseβ€˜π‘…) β†’ [π‘Ÿ](𝑅 ~QG 𝑀) ∈ ((Baseβ€˜π‘…) / (𝑅 ~QG 𝑀)))
4140ad4antlr 730 . . . . . . . . . . . . . . 15 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ [π‘Ÿ](𝑅 ~QG 𝑀) ∈ ((Baseβ€˜π‘…) / (𝑅 ~QG 𝑀)))
421a1i 11 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ 𝑄 = (𝑅 /s (𝑅 ~QG 𝑀)))
43 eqidd 2732 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ (Baseβ€˜π‘…) = (Baseβ€˜π‘…))
44 ovexd 7447 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ (𝑅 ~QG 𝑀) ∈ V)
4542, 43, 44, 3qusbas 17496 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ ((Baseβ€˜π‘…) / (𝑅 ~QG 𝑀)) = (Baseβ€˜π‘„))
4645adantr 480 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) β†’ ((Baseβ€˜π‘…) / (𝑅 ~QG 𝑀)) = (Baseβ€˜π‘„))
4746ad10antr 741 . . . . . . . . . . . . . . 15 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ ((Baseβ€˜π‘…) / (𝑅 ~QG 𝑀)) = (Baseβ€˜π‘„))
4841, 47eleqtrd 2834 . . . . . . . . . . . . . 14 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ [π‘Ÿ](𝑅 ~QG 𝑀) ∈ (Baseβ€˜π‘„))
4939ecelqsi 8770 . . . . . . . . . . . . . . . 16 (𝑠 ∈ (Baseβ€˜π‘…) β†’ [𝑠](𝑅 ~QG 𝑀) ∈ ((Baseβ€˜π‘…) / (𝑅 ~QG 𝑀)))
5049ad2antlr 724 . . . . . . . . . . . . . . 15 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ [𝑠](𝑅 ~QG 𝑀) ∈ ((Baseβ€˜π‘…) / (𝑅 ~QG 𝑀)))
5150, 47eleqtrd 2834 . . . . . . . . . . . . . 14 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ [𝑠](𝑅 ~QG 𝑀) ∈ (Baseβ€˜π‘„))
52 simpllr 773 . . . . . . . . . . . . . . . 16 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀))
53 simp-9r 791 . . . . . . . . . . . . . . . . 17 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ 𝑒 = [π‘₯](𝑅 ~QG 𝑀))
5453eqcomd 2737 . . . . . . . . . . . . . . . 16 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ [π‘₯](𝑅 ~QG 𝑀) = 𝑒)
5552, 54oveq12d 7430 . . . . . . . . . . . . . . 15 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = ([π‘Ÿ](𝑅 ~QG 𝑀)(.rβ€˜π‘„)𝑒))
56 simp-7r 787 . . . . . . . . . . . . . . 15 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„))
5755, 56eqtr3d 2773 . . . . . . . . . . . . . 14 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ ([π‘Ÿ](𝑅 ~QG 𝑀)(.rβ€˜π‘„)𝑒) = (1rβ€˜π‘„))
58 eqid 2731 . . . . . . . . . . . . . . . 16 (opprβ€˜π‘„) = (opprβ€˜π‘„)
59 eqid 2731 . . . . . . . . . . . . . . . 16 (.rβ€˜(opprβ€˜π‘„)) = (.rβ€˜(opprβ€˜π‘„))
6029, 30, 58, 59opprmul 20229 . . . . . . . . . . . . . . 15 ([𝑠](𝑅 ~QG 𝑀)(.rβ€˜(opprβ€˜π‘„))𝑒) = (𝑒(.rβ€˜π‘„)[𝑠](𝑅 ~QG 𝑀))
61 simp-5r 783 . . . . . . . . . . . . . . . 16 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀))))
625ad3antrrr 727 . . . . . . . . . . . . . . . . . . 19 ((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) β†’ 𝑅 ∈ Ring)
6362ad8antr 737 . . . . . . . . . . . . . . . . . 18 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ 𝑅 ∈ Ring)
6421ad3antrrr 727 . . . . . . . . . . . . . . . . . . 19 ((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) β†’ 𝑀 ∈ (2Idealβ€˜π‘…))
6564ad8antr 737 . . . . . . . . . . . . . . . . . 18 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ 𝑀 ∈ (2Idealβ€˜π‘…))
662, 9, 1, 63, 65, 29, 51, 38opprqusmulr 32880 . . . . . . . . . . . . . . . . 17 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ ([𝑠](𝑅 ~QG 𝑀)(.rβ€˜(opprβ€˜π‘„))𝑒) = ([𝑠](𝑅 ~QG 𝑀)(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))𝑒))
67 simpr 484 . . . . . . . . . . . . . . . . . 18 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ 𝑀 = [𝑠](𝑅 ~QG 𝑀))
682, 17lidlss 20979 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑀 ∈ (LIdealβ€˜π‘…) β†’ 𝑀 βŠ† (Baseβ€˜π‘…))
698, 68syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (πœ‘ β†’ 𝑀 βŠ† (Baseβ€˜π‘…))
709, 2oppreqg 32872 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ Ring ∧ 𝑀 βŠ† (Baseβ€˜π‘…)) β†’ (𝑅 ~QG 𝑀) = (𝑂 ~QG 𝑀))
715, 69, 70syl2anc 583 . . . . . . . . . . . . . . . . . . . . . 22 (πœ‘ β†’ (𝑅 ~QG 𝑀) = (𝑂 ~QG 𝑀))
7271ad10antr 741 . . . . . . . . . . . . . . . . . . . . 21 (((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) β†’ (𝑅 ~QG 𝑀) = (𝑂 ~QG 𝑀))
7372adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ (𝑅 ~QG 𝑀) = (𝑂 ~QG 𝑀))
7473eceq2d 8748 . . . . . . . . . . . . . . . . . . 19 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ [π‘₯](𝑅 ~QG 𝑀) = [π‘₯](𝑂 ~QG 𝑀))
7553, 74eqtr2d 2772 . . . . . . . . . . . . . . . . . 18 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ [π‘₯](𝑂 ~QG 𝑀) = 𝑒)
7667, 75oveq12d 7430 . . . . . . . . . . . . . . . . 17 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = ([𝑠](𝑅 ~QG 𝑀)(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))𝑒))
7766, 76eqtr4d 2774 . . . . . . . . . . . . . . . 16 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ ([𝑠](𝑅 ~QG 𝑀)(.rβ€˜(opprβ€˜π‘„))𝑒) = (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)))
7858, 25oppr1 20242 . . . . . . . . . . . . . . . . . . 19 (1rβ€˜π‘„) = (1rβ€˜(opprβ€˜π‘„))
792, 9, 1, 5, 21opprqus1r 32881 . . . . . . . . . . . . . . . . . . 19 (πœ‘ β†’ (1rβ€˜(opprβ€˜π‘„)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀))))
8078, 79eqtrid 2783 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ (1rβ€˜π‘„) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀))))
8180ad10antr 741 . . . . . . . . . . . . . . . . 17 (((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) β†’ (1rβ€˜π‘„) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀))))
8281adantr 480 . . . . . . . . . . . . . . . 16 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ (1rβ€˜π‘„) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀))))
8361, 77, 823eqtr4d 2781 . . . . . . . . . . . . . . 15 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ ([𝑠](𝑅 ~QG 𝑀)(.rβ€˜(opprβ€˜π‘„))𝑒) = (1rβ€˜π‘„))
8460, 83eqtr3id 2785 . . . . . . . . . . . . . 14 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ (𝑒(.rβ€˜π‘„)[𝑠](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„))
8529, 26, 25, 30, 31, 35, 38, 48, 51, 57, 84ringinveu 32665 . . . . . . . . . . . . 13 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ [𝑠](𝑅 ~QG 𝑀) = [π‘Ÿ](𝑅 ~QG 𝑀))
8685, 67, 523eqtr4rd 2782 . . . . . . . . . . . 12 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ 𝑣 = 𝑀)
8786oveq2d 7428 . . . . . . . . . . 11 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ (𝑒(.rβ€˜π‘„)𝑣) = (𝑒(.rβ€˜π‘„)𝑀))
8867oveq2d 7428 . . . . . . . . . . 11 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ (𝑒(.rβ€˜π‘„)𝑀) = (𝑒(.rβ€˜π‘„)[𝑠](𝑅 ~QG 𝑀)))
8987, 88, 843eqtrd 2775 . . . . . . . . . 10 ((((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) ∧ 𝑠 ∈ (Baseβ€˜π‘…)) ∧ 𝑀 = [𝑠](𝑅 ~QG 𝑀)) β†’ (𝑒(.rβ€˜π‘„)𝑣) = (1rβ€˜π‘„))
90 simp-4r 781 . . . . . . . . . . . 12 ((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) β†’ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀))))
9171qseq2d 8763 . . . . . . . . . . . . . 14 (πœ‘ β†’ ((Baseβ€˜π‘…) / (𝑅 ~QG 𝑀)) = ((Baseβ€˜π‘…) / (𝑂 ~QG 𝑀)))
9291ad9antr 739 . . . . . . . . . . . . 13 ((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) β†’ ((Baseβ€˜π‘…) / (𝑅 ~QG 𝑀)) = ((Baseβ€˜π‘…) / (𝑂 ~QG 𝑀)))
93 eqidd 2732 . . . . . . . . . . . . . 14 ((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) β†’ (𝑂 /s (𝑂 ~QG 𝑀)) = (𝑂 /s (𝑂 ~QG 𝑀)))
949, 2opprbas 20233 . . . . . . . . . . . . . . 15 (Baseβ€˜π‘…) = (Baseβ€˜π‘‚)
9594a1i 11 . . . . . . . . . . . . . 14 ((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) β†’ (Baseβ€˜π‘…) = (Baseβ€˜π‘‚))
96 ovexd 7447 . . . . . . . . . . . . . 14 ((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) β†’ (𝑂 ~QG 𝑀) ∈ V)
979fvexi 6905 . . . . . . . . . . . . . . 15 𝑂 ∈ V
9897a1i 11 . . . . . . . . . . . . . 14 ((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) β†’ 𝑂 ∈ V)
9993, 95, 96, 98qusbas 17496 . . . . . . . . . . . . 13 ((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) β†’ ((Baseβ€˜π‘…) / (𝑂 ~QG 𝑀)) = (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀))))
10092, 99eqtr2d 2772 . . . . . . . . . . . 12 ((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) β†’ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀))) = ((Baseβ€˜π‘…) / (𝑅 ~QG 𝑀)))
10190, 100eleqtrd 2834 . . . . . . . . . . 11 ((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) β†’ 𝑀 ∈ ((Baseβ€˜π‘…) / (𝑅 ~QG 𝑀)))
102 elqsi 8767 . . . . . . . . . . 11 (𝑀 ∈ ((Baseβ€˜π‘…) / (𝑅 ~QG 𝑀)) β†’ βˆƒπ‘  ∈ (Baseβ€˜π‘…)𝑀 = [𝑠](𝑅 ~QG 𝑀))
103101, 102syl 17 . . . . . . . . . 10 ((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) β†’ βˆƒπ‘  ∈ (Baseβ€˜π‘…)𝑀 = [𝑠](𝑅 ~QG 𝑀))
10489, 103r19.29a 3161 . . . . . . . . 9 ((((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ π‘Ÿ ∈ (Baseβ€˜π‘…)) ∧ 𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀)) β†’ (𝑒(.rβ€˜π‘„)𝑣) = (1rβ€˜π‘„))
105 simp-4r 781 . . . . . . . . . . 11 ((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) β†’ 𝑣 ∈ (Baseβ€˜π‘„))
10646ad6antr 733 . . . . . . . . . . 11 ((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) β†’ ((Baseβ€˜π‘…) / (𝑅 ~QG 𝑀)) = (Baseβ€˜π‘„))
107105, 106eleqtrrd 2835 . . . . . . . . . 10 ((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) β†’ 𝑣 ∈ ((Baseβ€˜π‘…) / (𝑅 ~QG 𝑀)))
108 elqsi 8767 . . . . . . . . . 10 (𝑣 ∈ ((Baseβ€˜π‘…) / (𝑅 ~QG 𝑀)) β†’ βˆƒπ‘Ÿ ∈ (Baseβ€˜π‘…)𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀))
109107, 108syl 17 . . . . . . . . 9 ((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) β†’ βˆƒπ‘Ÿ ∈ (Baseβ€˜π‘…)𝑣 = [π‘Ÿ](𝑅 ~QG 𝑀))
110104, 109r19.29a 3161 . . . . . . . 8 ((((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) ∧ 𝑀 ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) ∧ (𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))) β†’ (𝑒(.rβ€˜π‘„)𝑣) = (1rβ€˜π‘„))
111 eqid 2731 . . . . . . . . . 10 (opprβ€˜π‘‚) = (opprβ€˜π‘‚)
112 eqid 2731 . . . . . . . . . 10 (𝑂 /s (𝑂 ~QG 𝑀)) = (𝑂 /s (𝑂 ~QG 𝑀))
1133ad3antrrr 727 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) β†’ 𝑅 ∈ NzRing)
1149opprnzr 20412 . . . . . . . . . . 11 (𝑅 ∈ NzRing β†’ 𝑂 ∈ NzRing)
115113, 114syl 17 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) β†’ 𝑂 ∈ NzRing)
11612ad3antrrr 727 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) β†’ 𝑀 ∈ (MaxIdealβ€˜π‘‚))
1176ad3antrrr 727 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) β†’ 𝑀 ∈ (MaxIdealβ€˜π‘…))
1189, 62, 117opprmxidlabs 32876 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) β†’ 𝑀 ∈ (MaxIdealβ€˜(opprβ€˜π‘‚)))
119 simplr 766 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) β†’ π‘₯ ∈ (Baseβ€˜π‘…))
12094a1i 11 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) β†’ (Baseβ€˜π‘…) = (Baseβ€˜π‘‚))
121119, 120eleqtrd 2834 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) β†’ π‘₯ ∈ (Baseβ€˜π‘‚))
122 simplr 766 . . . . . . . . . . . 12 (((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ π‘₯ ∈ 𝑀) β†’ 𝑒 = [π‘₯](𝑅 ~QG 𝑀))
1235ringgrpd 20137 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝑅 ∈ Grp)
124123ad4antr 729 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ π‘₯ ∈ 𝑀) β†’ 𝑅 ∈ Grp)
125 lidlnsg 32839 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (LIdealβ€˜π‘…)) β†’ 𝑀 ∈ (NrmSGrpβ€˜π‘…))
1265, 8, 125syl2anc 583 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ 𝑀 ∈ (NrmSGrpβ€˜π‘…))
127 nsgsubg 19075 . . . . . . . . . . . . . . . 16 (𝑀 ∈ (NrmSGrpβ€˜π‘…) β†’ 𝑀 ∈ (SubGrpβ€˜π‘…))
128126, 127syl 17 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝑀 ∈ (SubGrpβ€˜π‘…))
129128ad4antr 729 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ π‘₯ ∈ 𝑀) β†’ 𝑀 ∈ (SubGrpβ€˜π‘…))
130 simpr 484 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ π‘₯ ∈ 𝑀) β†’ π‘₯ ∈ 𝑀)
131 eqid 2731 . . . . . . . . . . . . . . . 16 (𝑅 ~QG 𝑀) = (𝑅 ~QG 𝑀)
132131eqg0el 32748 . . . . . . . . . . . . . . 15 ((𝑅 ∈ Grp ∧ 𝑀 ∈ (SubGrpβ€˜π‘…)) β†’ ([π‘₯](𝑅 ~QG 𝑀) = 𝑀 ↔ π‘₯ ∈ 𝑀))
133132biimpar 477 . . . . . . . . . . . . . 14 (((𝑅 ∈ Grp ∧ 𝑀 ∈ (SubGrpβ€˜π‘…)) ∧ π‘₯ ∈ 𝑀) β†’ [π‘₯](𝑅 ~QG 𝑀) = 𝑀)
134124, 129, 130, 133syl21anc 835 . . . . . . . . . . . . 13 (((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ π‘₯ ∈ 𝑀) β†’ [π‘₯](𝑅 ~QG 𝑀) = 𝑀)
135 eqid 2731 . . . . . . . . . . . . . . 15 (0gβ€˜π‘…) = (0gβ€˜π‘…)
1362, 131, 135eqgid 19097 . . . . . . . . . . . . . 14 (𝑀 ∈ (SubGrpβ€˜π‘…) β†’ [(0gβ€˜π‘…)](𝑅 ~QG 𝑀) = 𝑀)
137129, 136syl 17 . . . . . . . . . . . . 13 (((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ π‘₯ ∈ 𝑀) β†’ [(0gβ€˜π‘…)](𝑅 ~QG 𝑀) = 𝑀)
138134, 137eqtr4d 2774 . . . . . . . . . . . 12 (((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ π‘₯ ∈ 𝑀) β†’ [π‘₯](𝑅 ~QG 𝑀) = [(0gβ€˜π‘…)](𝑅 ~QG 𝑀))
1391, 135qus0 19105 . . . . . . . . . . . . . 14 (𝑀 ∈ (NrmSGrpβ€˜π‘…) β†’ [(0gβ€˜π‘…)](𝑅 ~QG 𝑀) = (0gβ€˜π‘„))
140126, 139syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ [(0gβ€˜π‘…)](𝑅 ~QG 𝑀) = (0gβ€˜π‘„))
141140ad4antr 729 . . . . . . . . . . . 12 (((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ π‘₯ ∈ 𝑀) β†’ [(0gβ€˜π‘…)](𝑅 ~QG 𝑀) = (0gβ€˜π‘„))
142122, 138, 1413eqtrd 2775 . . . . . . . . . . 11 (((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ π‘₯ ∈ 𝑀) β†’ 𝑒 = (0gβ€˜π‘„))
143 eldifsnneq 4794 . . . . . . . . . . . 12 (𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)}) β†’ Β¬ 𝑒 = (0gβ€˜π‘„))
144143ad4antlr 730 . . . . . . . . . . 11 (((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ π‘₯ ∈ 𝑀) β†’ Β¬ 𝑒 = (0gβ€˜π‘„))
145142, 144pm2.65da 814 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) β†’ Β¬ π‘₯ ∈ 𝑀)
146111, 112, 115, 116, 118, 121, 145qsdrngilem 32883 . . . . . . . . 9 ((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) β†’ βˆƒπ‘€ ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))(𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀))))
147146ad2antrr 723 . . . . . . . 8 ((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) β†’ βˆƒπ‘€ ∈ (Baseβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))(𝑀(.rβ€˜(𝑂 /s (𝑂 ~QG 𝑀)))[π‘₯](𝑂 ~QG 𝑀)) = (1rβ€˜(𝑂 /s (𝑂 ~QG 𝑀))))
148110, 147r19.29a 3161 . . . . . . 7 ((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) β†’ (𝑒(.rβ€˜π‘„)𝑣) = (1rβ€˜π‘„))
149 simpllr 773 . . . . . . . . 9 ((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) β†’ 𝑒 = [π‘₯](𝑅 ~QG 𝑀))
150149oveq2d 7428 . . . . . . . 8 ((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) β†’ (𝑣(.rβ€˜π‘„)𝑒) = (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)))
151 simpr 484 . . . . . . . 8 ((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) β†’ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„))
152150, 151eqtrd 2771 . . . . . . 7 ((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) β†’ (𝑣(.rβ€˜π‘„)𝑒) = (1rβ€˜π‘„))
153148, 152jca 511 . . . . . 6 ((((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ 𝑣 ∈ (Baseβ€˜π‘„)) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„)) β†’ ((𝑒(.rβ€˜π‘„)𝑣) = (1rβ€˜π‘„) ∧ (𝑣(.rβ€˜π‘„)𝑒) = (1rβ€˜π‘„)))
154153anasss 466 . . . . 5 (((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) ∧ (𝑣 ∈ (Baseβ€˜π‘„) ∧ (𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„))) β†’ ((𝑒(.rβ€˜π‘„)𝑣) = (1rβ€˜π‘„) ∧ (𝑣(.rβ€˜π‘„)𝑒) = (1rβ€˜π‘„)))
1559, 1, 113, 117, 116, 119, 145qsdrngilem 32883 . . . . 5 ((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) β†’ βˆƒπ‘£ ∈ (Baseβ€˜π‘„)(𝑣(.rβ€˜π‘„)[π‘₯](𝑅 ~QG 𝑀)) = (1rβ€˜π‘„))
156154, 155reximddv 3170 . . . 4 ((((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) ∧ π‘₯ ∈ (Baseβ€˜π‘…)) ∧ 𝑒 = [π‘₯](𝑅 ~QG 𝑀)) β†’ βˆƒπ‘£ ∈ (Baseβ€˜π‘„)((𝑒(.rβ€˜π‘„)𝑣) = (1rβ€˜π‘„) ∧ (𝑣(.rβ€˜π‘„)𝑒) = (1rβ€˜π‘„)))
15737, 46eleqtrrd 2835 . . . . 5 ((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) β†’ 𝑒 ∈ ((Baseβ€˜π‘…) / (𝑅 ~QG 𝑀)))
158 elqsi 8767 . . . . 5 (𝑒 ∈ ((Baseβ€˜π‘…) / (𝑅 ~QG 𝑀)) β†’ βˆƒπ‘₯ ∈ (Baseβ€˜π‘…)𝑒 = [π‘₯](𝑅 ~QG 𝑀))
159157, 158syl 17 . . . 4 ((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) β†’ βˆƒπ‘₯ ∈ (Baseβ€˜π‘…)𝑒 = [π‘₯](𝑅 ~QG 𝑀))
160156, 159r19.29a 3161 . . 3 ((πœ‘ ∧ 𝑒 ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})) β†’ βˆƒπ‘£ ∈ (Baseβ€˜π‘„)((𝑒(.rβ€˜π‘„)𝑣) = (1rβ€˜π‘„) ∧ (𝑣(.rβ€˜π‘„)𝑒) = (1rβ€˜π‘„)))
161160ralrimiva 3145 . 2 (πœ‘ β†’ βˆ€π‘’ ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})βˆƒπ‘£ ∈ (Baseβ€˜π‘„)((𝑒(.rβ€˜π‘„)𝑣) = (1rβ€˜π‘„) ∧ (𝑣(.rβ€˜π‘„)𝑒) = (1rβ€˜π‘„)))
16229, 26, 25, 30, 31, 33isdrng4 32666 . 2 (πœ‘ β†’ (𝑄 ∈ DivRing ↔ ((1rβ€˜π‘„) β‰  (0gβ€˜π‘„) ∧ βˆ€π‘’ ∈ ((Baseβ€˜π‘„) βˆ– {(0gβ€˜π‘„)})βˆƒπ‘£ ∈ (Baseβ€˜π‘„)((𝑒(.rβ€˜π‘„)𝑣) = (1rβ€˜π‘„) ∧ (𝑣(.rβ€˜π‘„)𝑒) = (1rβ€˜π‘„)))))
16328, 161, 162mpbir2and 710 1 (πœ‘ β†’ 𝑄 ∈ DivRing)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  βˆ€wral 3060  βˆƒwrex 3069  Vcvv 3473   βˆ– cdif 3945   ∩ cin 3947   βŠ† wss 3948  {csn 4628  β€˜cfv 6543  (class class class)co 7412  [cec 8704   / cqs 8705  Basecbs 17149  .rcmulr 17203  0gc0g 17390   /s cqus 17456  Grpcgrp 18856  SubGrpcsubg 19037  NrmSGrpcnsg 19038   ~QG cqg 19039  1rcur 20076  Ringcrg 20128  opprcoppr 20225  Unitcui 20247  NzRingcnzr 20404  DivRingcdr 20501  LIdealclidl 20929  2Idealc2idl 21006  MaxIdealcmxidl 32850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7673  df-om 7859  df-1st 7978  df-2nd 7979  df-supp 8150  df-tpos 8214  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-2o 8470  df-er 8706  df-ec 8708  df-qs 8712  df-map 8825  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fsupp 9365  df-sup 9440  df-inf 9441  df-oi 9508  df-card 9937  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12478  df-z 12564  df-dec 12683  df-uz 12828  df-fz 13490  df-fzo 13633  df-seq 13972  df-hash 14296  df-struct 17085  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-ress 17179  df-plusg 17215  df-mulr 17216  df-sca 17218  df-vsca 17219  df-ip 17220  df-tset 17221  df-ple 17222  df-ds 17224  df-hom 17226  df-cco 17227  df-0g 17392  df-gsum 17393  df-prds 17398  df-pws 17400  df-imas 17459  df-qus 17460  df-mre 17535  df-mrc 17536  df-acs 17538  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-mhm 18706  df-submnd 18707  df-grp 18859  df-minusg 18860  df-sbg 18861  df-mulg 18988  df-subg 19040  df-nsg 19041  df-eqg 19042  df-ghm 19129  df-cntz 19223  df-cmn 19692  df-abl 19693  df-mgp 20030  df-rng 20048  df-ur 20077  df-ring 20130  df-oppr 20226  df-dvdsr 20249  df-unit 20250  df-invr 20280  df-nzr 20405  df-subrg 20460  df-drng 20503  df-lmod 20617  df-lss 20688  df-lsp 20728  df-lmhm 20778  df-lbs 20831  df-sra 20931  df-rgmod 20932  df-lidl 20933  df-rsp 20934  df-2idl 21007  df-dsmm 21507  df-frlm 21522  df-uvc 21558  df-mxidl 32851
This theorem is referenced by:  qsdrng  32886  algextdeglem4  33066
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