qqhval2 - Mathbox for Thierry Arnoux

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Theorem qqhval2 34145

Description: Value of the canonical homormorphism from the rational number when the target ring is a division ring. (Contributed by Thierry Arnoux, 26-Oct-2017.)

**Hypotheses**
Ref	Expression
qqhval2.0	⊢ 𝐵 = (Base‘𝑅)
qqhval2.1	⊢ / = (/_r‘𝑅)
qqhval2.2	⊢ 𝐿 = (ℤRHom‘𝑅)

**Assertion**
Ref	Expression
qqhval2	⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) = (𝑞 ∈ ℚ ↦ ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))

Distinct variable groups: / ,𝑞 𝐵,𝑞 𝐿,𝑞 𝑅,𝑞

Proof of Theorem qqhval2 Dummy variables 𝑒 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3451	. . . 4 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ V)
2	1	adantr 480	. . 3 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → 𝑅 ∈ V)
3		qqhval2.1	. . . 4 ⊢ / = (/_r‘𝑅)
4		eqid 2737	. . . 4 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
5		qqhval2.2	. . . 4 ⊢ 𝐿 = (ℤRHom‘𝑅)
6	3, 4, 5	qqhval 34135	. . 3 ⊢ (𝑅 ∈ V → (ℚHom‘𝑅) = ran (𝑥 ∈ ℤ, 𝑦 ∈ (^◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩))
7	2, 6	syl 17	. 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) = ran (𝑥 ∈ ℤ, 𝑦 ∈ (^◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩))
8		eqid 2737	. . . 4 ⊢ ℤ = ℤ
9		qqhval2.0	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
10		eqid 2737	. . . . 5 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
11	9, 5, 10	zrhunitpreima 34139	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (^◡𝐿 “ (Unit‘𝑅)) = (ℤ ∖ {0}))
12		mpoeq12 7434	. . . 4 ⊢ ((ℤ = ℤ ∧ (^◡𝐿 “ (Unit‘𝑅)) = (ℤ ∖ {0})) → (𝑥 ∈ ℤ, 𝑦 ∈ (^◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) = (𝑥 ∈ ℤ, 𝑦 ∈ (ℤ ∖ {0}) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩))
13	8, 11, 12	sylancr 588	. . 3 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (𝑥 ∈ ℤ, 𝑦 ∈ (^◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) = (𝑥 ∈ ℤ, 𝑦 ∈ (ℤ ∖ {0}) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩))
14	13	rneqd 5888	. 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ran (𝑥 ∈ ℤ, 𝑦 ∈ (^◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) = ran (𝑥 ∈ ℤ, 𝑦 ∈ (ℤ ∖ {0}) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩))
15		nfv 1916	. . . 4 ⊢ Ⅎ𝑒(𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0)
16		nfab1 2901	. . . 4 ⊢ Ⅎ𝑒{𝑒 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩}
17		nfcv 2899	. . . 4 ⊢ Ⅎ𝑒{⟨𝑞, 𝑠⟩ ∣ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))}
18		simpr 484	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩)
19		zssq 12900	. . . . . . . . . . . 12 ⊢ ℤ ⊆ ℚ
20		simplrl 777	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑥 ∈ ℤ)
21	19, 20	sselid 3920	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑥 ∈ ℚ)
22		simplrr 778	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑦 ∈ (ℤ ∖ {0}))
23	22	eldifad 3902	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑦 ∈ ℤ)
24	19, 23	sselid 3920	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑦 ∈ ℚ)
25	22	eldifbd 3903	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → ¬ 𝑦 ∈ {0})
26		velsn 4584	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {0} ↔ 𝑦 = 0)
27	26	necon3bbii 2980	. . . . . . . . . . . 12 ⊢ (¬ 𝑦 ∈ {0} ↔ 𝑦 ≠ 0)
28	25, 27	sylib 218	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑦 ≠ 0)
29		qdivcl 12914	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → (𝑥 / 𝑦) ∈ ℚ)
30	21, 24, 28, 29	syl3anc 1374	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → (𝑥 / 𝑦) ∈ ℚ)
31		simplll 775	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑅 ∈ DivRing)
32		simpllr 776	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → (chr‘𝑅) = 0)
33	9, 3, 5	qqhval2lem 34144	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑦 ≠ 0)) → ((𝐿‘(numer‘(𝑥 / 𝑦))) / (𝐿‘(denom‘(𝑥 / 𝑦)))) = ((𝐿‘𝑥) / (𝐿‘𝑦)))
34	33	eqcomd 2743	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑦 ≠ 0)) → ((𝐿‘𝑥) / (𝐿‘𝑦)) = ((𝐿‘(numer‘(𝑥 / 𝑦))) / (𝐿‘(denom‘(𝑥 / 𝑦)))))
35	31, 32, 20, 23, 28, 34	syl23anc 1380	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → ((𝐿‘𝑥) / (𝐿‘𝑦)) = ((𝐿‘(numer‘(𝑥 / 𝑦))) / (𝐿‘(denom‘(𝑥 / 𝑦)))))
36		ovex 7394	. . . . . . . . . . 11 ⊢ (𝑥 / 𝑦) ∈ V
37		ovex 7394	. . . . . . . . . . 11 ⊢ ((𝐿‘𝑥) / (𝐿‘𝑦)) ∈ V
38		opeq12 4819	. . . . . . . . . . . . 13 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → ⟨𝑞, 𝑠⟩ = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩)
39	38	eqeq2d 2748	. . . . . . . . . . . 12 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → (𝑒 = ⟨𝑞, 𝑠⟩ ↔ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩))
40		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → 𝑞 = (𝑥 / 𝑦))
41	40	eleq1d 2822	. . . . . . . . . . . . 13 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → (𝑞 ∈ ℚ ↔ (𝑥 / 𝑦) ∈ ℚ))
42		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦)))
43	40	fveq2d 6839	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → (numer‘𝑞) = (numer‘(𝑥 / 𝑦)))
44	43	fveq2d 6839	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → (𝐿‘(numer‘𝑞)) = (𝐿‘(numer‘(𝑥 / 𝑦))))
45	40	fveq2d 6839	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → (denom‘𝑞) = (denom‘(𝑥 / 𝑦)))
46	45	fveq2d 6839	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → (𝐿‘(denom‘𝑞)) = (𝐿‘(denom‘(𝑥 / 𝑦))))
47	44, 46	oveq12d 7379	. . . . . . . . . . . . . 14 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))) = ((𝐿‘(numer‘(𝑥 / 𝑦))) / (𝐿‘(denom‘(𝑥 / 𝑦)))))
48	42, 47	eqeq12d 2753	. . . . . . . . . . . . 13 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → (𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))) ↔ ((𝐿‘𝑥) / (𝐿‘𝑦)) = ((𝐿‘(numer‘(𝑥 / 𝑦))) / (𝐿‘(denom‘(𝑥 / 𝑦))))))
49	41, 48	anbi12d 633	. . . . . . . . . . . 12 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → ((𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))) ↔ ((𝑥 / 𝑦) ∈ ℚ ∧ ((𝐿‘𝑥) / (𝐿‘𝑦)) = ((𝐿‘(numer‘(𝑥 / 𝑦))) / (𝐿‘(denom‘(𝑥 / 𝑦)))))))
50	39, 49	anbi12d 633	. . . . . . . . . . 11 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → ((𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))) ↔ (𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩ ∧ ((𝑥 / 𝑦) ∈ ℚ ∧ ((𝐿‘𝑥) / (𝐿‘𝑦)) = ((𝐿‘(numer‘(𝑥 / 𝑦))) / (𝐿‘(denom‘(𝑥 / 𝑦))))))))
51	36, 37, 50	spc2ev 3550	. . . . . . . . . 10 ⊢ ((𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩ ∧ ((𝑥 / 𝑦) ∈ ℚ ∧ ((𝐿‘𝑥) / (𝐿‘𝑦)) = ((𝐿‘(numer‘(𝑥 / 𝑦))) / (𝐿‘(denom‘(𝑥 / 𝑦)))))) → ∃𝑞∃𝑠(𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))))
52	18, 30, 35, 51	syl12anc 837	. . . . . . . . 9 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → ∃𝑞∃𝑠(𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))))
53	52	ex 412	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) → (𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩ → ∃𝑞∃𝑠(𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))))
54	53	rexlimdvva 3195	. . . . . . 7 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩ → ∃𝑞∃𝑠(𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))))
55	54	imp 406	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → ∃𝑞∃𝑠(𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))))
56		19.42vv 1959	. . . . . . 7 ⊢ (∃𝑞∃𝑠((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) ↔ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ ∃𝑞∃𝑠(𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))))
57		simprrl 781	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → 𝑞 ∈ ℚ)
58		qnumcl 16704	. . . . . . . . . 10 ⊢ (𝑞 ∈ ℚ → (numer‘𝑞) ∈ ℤ)
59	57, 58	syl 17	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → (numer‘𝑞) ∈ ℤ)
60		qdencl 16705	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ ℚ → (denom‘𝑞) ∈ ℕ)
61	57, 60	syl 17	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → (denom‘𝑞) ∈ ℕ)
62	61	nnzd 12544	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → (denom‘𝑞) ∈ ℤ)
63		nnne0 12205	. . . . . . . . . . 11 ⊢ ((denom‘𝑞) ∈ ℕ → (denom‘𝑞) ≠ 0)
64		nelsn 4611	. . . . . . . . . . 11 ⊢ ((denom‘𝑞) ≠ 0 → ¬ (denom‘𝑞) ∈ {0})
65	61, 63, 64	3syl 18	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → ¬ (denom‘𝑞) ∈ {0})
66	62, 65	eldifd 3901	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → (denom‘𝑞) ∈ (ℤ ∖ {0}))
67		simprl 771	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → 𝑒 = ⟨𝑞, 𝑠⟩)
68		qeqnumdivden 16710	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ ℚ → 𝑞 = ((numer‘𝑞) / (denom‘𝑞)))
69	57, 68	syl 17	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → 𝑞 = ((numer‘𝑞) / (denom‘𝑞)))
70		simprrr 782	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))
71	69, 70	opeq12d 4825	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → ⟨𝑞, 𝑠⟩ = ⟨((numer‘𝑞) / (denom‘𝑞)), ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))⟩)
72	67, 71	eqtrd 2772	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → 𝑒 = ⟨((numer‘𝑞) / (denom‘𝑞)), ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))⟩)
73		oveq1 7368	. . . . . . . . . . . 12 ⊢ (𝑥 = (numer‘𝑞) → (𝑥 / 𝑦) = ((numer‘𝑞) / 𝑦))
74		fveq2 6835	. . . . . . . . . . . . 13 ⊢ (𝑥 = (numer‘𝑞) → (𝐿‘𝑥) = (𝐿‘(numer‘𝑞)))
75	74	oveq1d 7376	. . . . . . . . . . . 12 ⊢ (𝑥 = (numer‘𝑞) → ((𝐿‘𝑥) / (𝐿‘𝑦)) = ((𝐿‘(numer‘𝑞)) / (𝐿‘𝑦)))
76	73, 75	opeq12d 4825	. . . . . . . . . . 11 ⊢ (𝑥 = (numer‘𝑞) → ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩ = ⟨((numer‘𝑞) / 𝑦), ((𝐿‘(numer‘𝑞)) / (𝐿‘𝑦))⟩)
77	76	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑥 = (numer‘𝑞) → (𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩ ↔ 𝑒 = ⟨((numer‘𝑞) / 𝑦), ((𝐿‘(numer‘𝑞)) / (𝐿‘𝑦))⟩))
78		oveq2 7369	. . . . . . . . . . . 12 ⊢ (𝑦 = (denom‘𝑞) → ((numer‘𝑞) / 𝑦) = ((numer‘𝑞) / (denom‘𝑞)))
79		fveq2 6835	. . . . . . . . . . . . 13 ⊢ (𝑦 = (denom‘𝑞) → (𝐿‘𝑦) = (𝐿‘(denom‘𝑞)))
80	79	oveq2d 7377	. . . . . . . . . . . 12 ⊢ (𝑦 = (denom‘𝑞) → ((𝐿‘(numer‘𝑞)) / (𝐿‘𝑦)) = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))
81	78, 80	opeq12d 4825	. . . . . . . . . . 11 ⊢ (𝑦 = (denom‘𝑞) → ⟨((numer‘𝑞) / 𝑦), ((𝐿‘(numer‘𝑞)) / (𝐿‘𝑦))⟩ = ⟨((numer‘𝑞) / (denom‘𝑞)), ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))⟩)
82	81	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑦 = (denom‘𝑞) → (𝑒 = ⟨((numer‘𝑞) / 𝑦), ((𝐿‘(numer‘𝑞)) / (𝐿‘𝑦))⟩ ↔ 𝑒 = ⟨((numer‘𝑞) / (denom‘𝑞)), ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))⟩))
83	77, 82	rspc2ev 3578	. . . . . . . . 9 ⊢ (((numer‘𝑞) ∈ ℤ ∧ (denom‘𝑞) ∈ (ℤ ∖ {0}) ∧ 𝑒 = ⟨((numer‘𝑞) / (denom‘𝑞)), ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))⟩) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩)
84	59, 66, 72, 83	syl3anc 1374	. . . . . . . 8 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩)
85	84	exlimivv 1934	. . . . . . 7 ⊢ (∃𝑞∃𝑠((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩)
86	56, 85	sylbir 235	. . . . . 6 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ ∃𝑞∃𝑠(𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩)
87	55, 86	impbida 801	. . . . 5 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩ ↔ ∃𝑞∃𝑠(𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))))
88		abid 2719	. . . . 5 ⊢ (𝑒 ∈ {𝑒 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩} ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩)
89		elopab 5476	. . . . 5 ⊢ (𝑒 ∈ {⟨𝑞, 𝑠⟩ ∣ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))} ↔ ∃𝑞∃𝑠(𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))))
90	87, 88, 89	3bitr4g 314	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (𝑒 ∈ {𝑒 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩} ↔ 𝑒 ∈ {⟨𝑞, 𝑠⟩ ∣ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))}))
91	15, 16, 17, 90	eqrd 3942	. . 3 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → {𝑒 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩} = {⟨𝑞, 𝑠⟩ ∣ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))})
92		eqid 2737	. . . 4 ⊢ (𝑥 ∈ ℤ, 𝑦 ∈ (ℤ ∖ {0}) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) = (𝑥 ∈ ℤ, 𝑦 ∈ (ℤ ∖ {0}) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩)
93	92	rnmpo 7494	. . 3 ⊢ ran (𝑥 ∈ ℤ, 𝑦 ∈ (ℤ ∖ {0}) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) = {𝑒 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩}
94		df-mpt 5168	. . 3 ⊢ (𝑞 ∈ ℚ ↦ ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))) = {⟨𝑞, 𝑠⟩ ∣ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))}
95	91, 93, 94	3eqtr4g 2797	. 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ran (𝑥 ∈ ℤ, 𝑦 ∈ (ℤ ∖ {0}) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) = (𝑞 ∈ ℚ ↦ ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))
96	7, 14, 95	3eqtrd 2776	1 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) = (𝑞 ∈ ℚ ↦ ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∃wex 1781 ∈ wcel 2114 {cab 2715 ≠ wne 2933 ∃wrex 3062 Vcvv 3430 ∖ cdif 3887 {csn 4568 ⟨cop 4574 {copab 5148 ↦ cmpt 5167 ^◡ccnv 5624 ran crn 5626 “ cima 5628 ‘cfv 6493 (class class class)co 7361 ∈ cmpo 7363 0cc0 11032 / cdiv 11801 ℕcn 12168 ℤcz 12518 ℚcq 12892 numercnumer 16697 denomcdenom 16698 Basecbs 17173 0_gc0g 17396 1_rcur 20156 Unitcui 20329 /_rcdvr 20374 DivRingcdr 20700 ℤRHomczrh 21492 chrcchr 21494 ℚHomcqqh 34133

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-pre-sup 11110 ax-addf 11111 ax-mulf 11112

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-1st 7936 df-2nd 7937 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-inf 9350 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-div 11802 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-7 12243 df-8 12244 df-9 12245 df-n0 12432 df-z 12519 df-dec 12639 df-uz 12783 df-q 12893 df-rp 12937 df-fz 13456 df-fl 13745 df-mod 13823 df-seq 13958 df-exp 14018 df-cj 15055 df-re 15056 df-im 15057 df-sqrt 15191 df-abs 15192 df-dvds 16216 df-gcd 16458 df-numer 16699 df-denom 16700 df-gz 16895 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-starv 17229 df-tset 17233 df-ple 17234 df-ds 17236 df-unif 17237 df-0g 17398 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-mhm 18745 df-grp 18906 df-minusg 18907 df-sbg 18908 df-mulg 19038 df-subg 19093 df-ghm 19182 df-od 19497 df-cmn 19751 df-abl 19752 df-mgp 20116 df-rng 20128 df-ur 20157 df-ring 20210 df-cring 20211 df-oppr 20311 df-dvdsr 20331 df-unit 20332 df-invr 20362 df-dvr 20375 df-rhm 20446 df-subrng 20517 df-subrg 20541 df-drng 20702 df-cnfld 21348 df-zring 21440 df-zrh 21496 df-chr 21498 df-qqh 34134

This theorem is referenced by: qqhvval 34146 qqhf 34149

W3C validator