qqhval2 - Mathbox for Thierry Arnoux

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Theorem qqhval2 32563

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 3463	. . . 4 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ V)
2	1	adantr 481	. . 3 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → 𝑅 ∈ V)
3		qqhval2.1	. . . 4 ⊢ / = (/_r‘𝑅)
4		eqid 2736	. . . 4 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
5		qqhval2.2	. . . 4 ⊢ 𝐿 = (ℤRHom‘𝑅)
6	3, 4, 5	qqhval 32555	. . 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		eqidd 2737	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ℤ = ℤ)
9		qqhval2.0	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
10		eqid 2736	. . . . 5 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
11	9, 5, 10	zrhunitpreima 32559	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (^◡𝐿 “ (Unit‘𝑅)) = (ℤ ∖ {0}))
12		mpoeq12 7430	. . . 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	syl2anc 584	. . 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 5893	. 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 1917	. . . 4 ⊢ Ⅎ𝑒(𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0)
16		nfab1 2909	. . . 4 ⊢ Ⅎ𝑒{𝑒 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩}
17		nfcv 2907	. . . 4 ⊢ Ⅎ𝑒{⟨𝑞, 𝑠⟩ ∣ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))}
18		simpr 485	. . . . . . . . . 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 12881	. . . . . . . . . . . 12 ⊢ ℤ ⊆ ℚ
20		simplrl 775	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑥 ∈ ℤ)
21	19, 20	sselid 3942	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑥 ∈ ℚ)
22		simplrr 776	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑦 ∈ (ℤ ∖ {0}))
23	22	eldifad 3922	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑦 ∈ ℤ)
24	19, 23	sselid 3942	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑦 ∈ ℚ)
25	22	eldifbd 3923	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → ¬ 𝑦 ∈ {0})
26		velsn 4602	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {0} ↔ 𝑦 = 0)
27	26	necon3bbii 2991	. . . . . . . . . . . 12 ⊢ (¬ 𝑦 ∈ {0} ↔ 𝑦 ≠ 0)
28	25, 27	sylib 217	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑦 ≠ 0)
29		qdivcl 12895	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → (𝑥 / 𝑦) ∈ ℚ)
30	21, 24, 28, 29	syl3anc 1371	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → (𝑥 / 𝑦) ∈ ℚ)
31		simplll 773	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑅 ∈ DivRing)
32		simpllr 774	. . . . . . . . . . 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 32562	. . . . . . . . . . . 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 2742	. . . . . . . . . . 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 1377	. . . . . . . . . 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 7390	. . . . . . . . . . 11 ⊢ (𝑥 / 𝑦) ∈ V
37		ovex 7390	. . . . . . . . . . 11 ⊢ ((𝐿‘𝑥) / (𝐿‘𝑦)) ∈ V
38		opeq12 4832	. . . . . . . . . . . . 13 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → ⟨𝑞, 𝑠⟩ = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩)
39	38	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → (𝑒 = ⟨𝑞, 𝑠⟩ ↔ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩))
40		simpl 483	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . 14 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦)))
43	40	fveq2d 6846	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → (numer‘𝑞) = (numer‘(𝑥 / 𝑦)))
44	43	fveq2d 6846	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → (𝐿‘(numer‘𝑞)) = (𝐿‘(numer‘(𝑥 / 𝑦))))
45	40	fveq2d 6846	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → (denom‘𝑞) = (denom‘(𝑥 / 𝑦)))
46	45	fveq2d 6846	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → (𝐿‘(denom‘𝑞)) = (𝐿‘(denom‘(𝑥 / 𝑦))))
47	44, 46	oveq12d 7375	. . . . . . . . . . . . . 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 2752	. . . . . . . . . . . . 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 631	. . . . . . . . . . . 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 631	. . . . . . . . . . 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 3566	. . . . . . . . . 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 835	. . . . . . . . 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 413	. . . . . . . 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 3205	. . . . . . 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 407	. . . . . 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 1961	. . . . . . 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 779	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → 𝑞 ∈ ℚ)
58		qnumcl 16615	. . . . . . . . . 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 16616	. . . . . . . . . . . 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 12526	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → (denom‘𝑞) ∈ ℤ)
63		nnne0 12187	. . . . . . . . . . 11 ⊢ ((denom‘𝑞) ∈ ℕ → (denom‘𝑞) ≠ 0)
64		nelsn 4626	. . . . . . . . . . 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 3921	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → (denom‘𝑞) ∈ (ℤ ∖ {0}))
67		simprl 769	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → 𝑒 = ⟨𝑞, 𝑠⟩)
68		qeqnumdivden 16621	. . . . . . . . . . . 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 780	. . . . . . . . . . 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 4838	. . . . . . . . . 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 2776	. . . . . . . . 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 7364	. . . . . . . . . . . 12 ⊢ (𝑥 = (numer‘𝑞) → (𝑥 / 𝑦) = ((numer‘𝑞) / 𝑦))
74		fveq2 6842	. . . . . . . . . . . . 13 ⊢ (𝑥 = (numer‘𝑞) → (𝐿‘𝑥) = (𝐿‘(numer‘𝑞)))
75	74	oveq1d 7372	. . . . . . . . . . . 12 ⊢ (𝑥 = (numer‘𝑞) → ((𝐿‘𝑥) / (𝐿‘𝑦)) = ((𝐿‘(numer‘𝑞)) / (𝐿‘𝑦)))
76	73, 75	opeq12d 4838	. . . . . . . . . . 11 ⊢ (𝑥 = (numer‘𝑞) → ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩ = ⟨((numer‘𝑞) / 𝑦), ((𝐿‘(numer‘𝑞)) / (𝐿‘𝑦))⟩)
77	76	eqeq2d 2747	. . . . . . . . . 10 ⊢ (𝑥 = (numer‘𝑞) → (𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩ ↔ 𝑒 = ⟨((numer‘𝑞) / 𝑦), ((𝐿‘(numer‘𝑞)) / (𝐿‘𝑦))⟩))
78		oveq2 7365	. . . . . . . . . . . 12 ⊢ (𝑦 = (denom‘𝑞) → ((numer‘𝑞) / 𝑦) = ((numer‘𝑞) / (denom‘𝑞)))
79		fveq2 6842	. . . . . . . . . . . . 13 ⊢ (𝑦 = (denom‘𝑞) → (𝐿‘𝑦) = (𝐿‘(denom‘𝑞)))
80	79	oveq2d 7373	. . . . . . . . . . . 12 ⊢ (𝑦 = (denom‘𝑞) → ((𝐿‘(numer‘𝑞)) / (𝐿‘𝑦)) = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))
81	78, 80	opeq12d 4838	. . . . . . . . . . 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 2747	. . . . . . . . . 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 3592	. . . . . . . . 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 1371	. . . . . . . 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 1935	. . . . . . 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 234	. . . . . 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 799	. . . . 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 2717	. . . . 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 5484	. . . . 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 313	. . . 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 3963	. . 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 2736	. . . 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 7489	. . 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 5189	. . 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 2801	. 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 2780	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 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 {cab 2713 ≠ wne 2943 ∃wrex 3073 Vcvv 3445 ∖ cdif 3907 {csn 4586 ⟨cop 4592 {copab 5167 ↦ cmpt 5188 ^◡ccnv 5632 ran crn 5634 “ cima 5636 ‘cfv 6496 (class class class)co 7357 ∈ cmpo 7359 0cc0 11051 / cdiv 11812 ℕcn 12153 ℤcz 12499 ℚcq 12873 numercnumer 16608 denomcdenom 16609 Basecbs 17083 0_gc0g 17321 1_rcur 19913 Unitcui 20068 /_rcdvr 20111 DivRingcdr 20185 ℤRHomczrh 20900 chrcchr 20902 ℚHomcqqh 32553

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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-fz 13425 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-dvds 16137 df-gcd 16375 df-numer 16610 df-denom 16611 df-gz 16802 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-0g 17323 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-ghm 19006 df-od 19310 df-cmn 19564 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-rnghom 20146 df-drng 20187 df-subrg 20220 df-cnfld 20797 df-zring 20870 df-zrh 20904 df-chr 20906 df-qqh 32554

This theorem is referenced by: qqhvval 32564 qqhf 32567

W3C validator