qqhval2 - Mathbox for Thierry Arnoux

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Theorem qqhval2 34614

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 3472	. . . 4 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ V)
2	1	adantr 486	. . 3 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → 𝑅 ∈ V)
3		qqhval2.1	. . . 4 ⊢ / = (/_r‘𝑅)
4		eqid 2761	. . . 4 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
5		qqhval2.2	. . . 4 ⊢ 𝐿 = (ℤRHom‘𝑅)
6	3, 4, 5	qqhval 34604	. . 3 ⊢ (𝑅 ∈ V → (ℚHom‘𝑅) = ran (𝑥 ∈ ℤ, 𝑦 ∈ (^◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩))
7	2, 6	syl 18	. 2 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) = ran (𝑥 ∈ ℤ, 𝑦 ∈ (^◡𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩))
8		eqid 2761	. . . 4 ⊢ ℤ = ℤ
9		qqhval2.0	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
10		eqid 2761	. . . . 5 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
11	9, 5, 10	zrhunitpreima 34608	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (^◡𝐿 “ (Unit‘𝑅)) = (ℤ ∖ {0}))
12		mpoeq12 7493	. . . 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 599	. . 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 5920	. 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 1947	. . . 4 ⊢ Ⅎ𝑒(𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0)
16		nfab1 2925	. . . 4 ⊢ Ⅎ𝑒{𝑒 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩}
17		nfcv 2923	. . . 4 ⊢ Ⅎ𝑒{⟨𝑞, 𝑠⟩ ∣ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))}
18		simpr 490	. . . . . . . . . 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 13083	. . . . . . . . . . . 12 ⊢ ℤ ⊆ ℚ
20		simplrl 789	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑥 ∈ ℤ)
21	19, 20	sselid 3929	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑥 ∈ ℚ)
22		simplrr 790	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑦 ∈ (ℤ ∖ {0}))
23	22	eldifad 3911	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑦 ∈ ℤ)
24	19, 23	sselid 3929	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑦 ∈ ℚ)
25	22	eldifbd 3912	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → ¬ 𝑦 ∈ {0})
26		velsn 4600	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {0} ↔ 𝑦 = 0)
27	26	necon3bbii 3003	. . . . . . . . . . . 12 ⊢ (¬ 𝑦 ∈ {0} ↔ 𝑦 ≠ 0)
28	25, 27	sylib 221	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑦 ≠ 0)
29		qdivcl 13098	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → (𝑥 / 𝑦) ∈ ℚ)
30	21, 24, 28, 29	syl3anc 1398	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → (𝑥 / 𝑦) ∈ ℚ)
31		simplll 787	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑅 ∈ DivRing)
32		simpllr 788	. . . . . . . . . . 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 34613	. . . . . . . . . . . 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 2767	. . . . . . . . . . 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 1404	. . . . . . . . . 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 7453	. . . . . . . . . . 11 ⊢ (𝑥 / 𝑦) ∈ V
37		ovex 7453	. . . . . . . . . . 11 ⊢ ((𝐿‘𝑥) / (𝐿‘𝑦)) ∈ V
38		opeq12 4835	. . . . . . . . . . . . 13 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → ⟨𝑞, 𝑠⟩ = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩)
39	38	eqeq2d 2772	. . . . . . . . . . . 12 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → (𝑒 = ⟨𝑞, 𝑠⟩ ↔ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩))
40		simpl 488	. . . . . . . . . . . . . 14 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → 𝑞 = (𝑥 / 𝑦))
41	40	eleq1d 2846	. . . . . . . . . . . . 13 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → (𝑞 ∈ ℚ ↔ (𝑥 / 𝑦) ∈ ℚ))
42		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦)))
43	40	fveq2d 6889	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → (numer‘𝑞) = (numer‘(𝑥 / 𝑦)))
44	43	fveq2d 6889	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → (𝐿‘(numer‘𝑞)) = (𝐿‘(numer‘(𝑥 / 𝑦))))
45	40	fveq2d 6889	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → (denom‘𝑞) = (denom‘(𝑥 / 𝑦)))
46	45	fveq2d 6889	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → (𝐿‘(denom‘𝑞)) = (𝐿‘(denom‘(𝑥 / 𝑦))))
47	44, 46	oveq12d 7438	. . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . 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 644	. . . . . . . . . . . 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 644	. . . . . . . . . . 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 3562	. . . . . . . . . 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 850	. . . . . . . . 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 418	. . . . . . . 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 3220	. . . . . . 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 412	. . . . . 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 1990	. . . . . . 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 793	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → 𝑞 ∈ ℚ)
58		qnumcl 16916	. . . . . . . . . 10 ⊢ (𝑞 ∈ ℚ → (numer‘𝑞) ∈ ℤ)
59	57, 58	syl 18	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → (numer‘𝑞) ∈ ℤ)
60		qdencl 16917	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ ℚ → (denom‘𝑞) ∈ ℕ)
61	57, 60	syl 18	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → (denom‘𝑞) ∈ ℕ)
62	61	nnzd 12719	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → (denom‘𝑞) ∈ ℤ)
63		nnne0 12372	. . . . . . . . . . 11 ⊢ ((denom‘𝑞) ∈ ℕ → (denom‘𝑞) ≠ 0)
64		nelsn 4627	. . . . . . . . . . 11 ⊢ ((denom‘𝑞) ≠ 0 → ¬ (denom‘𝑞) ∈ {0})
65	61, 63, 64	3syl 19	. . . . . . . . . 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 3910	. . . . . . . . 9 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → (denom‘𝑞) ∈ (ℤ ∖ {0}))
67		simprl 783	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → 𝑒 = ⟨𝑞, 𝑠⟩)
68		qeqnumdivden 16922	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ ℚ → 𝑞 = ((numer‘𝑞) / (denom‘𝑞)))
69	57, 68	syl 18	. . . . . . . . . . 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 794	. . . . . . . . . . 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 4841	. . . . . . . . . 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 2796	. . . . . . . . 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 7427	. . . . . . . . . . . 12 ⊢ (𝑥 = (numer‘𝑞) → (𝑥 / 𝑦) = ((numer‘𝑞) / 𝑦))
74		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑥 = (numer‘𝑞) → (𝐿‘𝑥) = (𝐿‘(numer‘𝑞)))
75	74	oveq1d 7435	. . . . . . . . . . . 12 ⊢ (𝑥 = (numer‘𝑞) → ((𝐿‘𝑥) / (𝐿‘𝑦)) = ((𝐿‘(numer‘𝑞)) / (𝐿‘𝑦)))
76	73, 75	opeq12d 4841	. . . . . . . . . . 11 ⊢ (𝑥 = (numer‘𝑞) → ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩ = ⟨((numer‘𝑞) / 𝑦), ((𝐿‘(numer‘𝑞)) / (𝐿‘𝑦))⟩)
77	76	eqeq2d 2772	. . . . . . . . . 10 ⊢ (𝑥 = (numer‘𝑞) → (𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩ ↔ 𝑒 = ⟨((numer‘𝑞) / 𝑦), ((𝐿‘(numer‘𝑞)) / (𝐿‘𝑦))⟩))
78		oveq2 7428	. . . . . . . . . . . 12 ⊢ (𝑦 = (denom‘𝑞) → ((numer‘𝑞) / 𝑦) = ((numer‘𝑞) / (denom‘𝑞)))
79		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑦 = (denom‘𝑞) → (𝐿‘𝑦) = (𝐿‘(denom‘𝑞)))
80	79	oveq2d 7436	. . . . . . . . . . . 12 ⊢ (𝑦 = (denom‘𝑞) → ((𝐿‘(numer‘𝑞)) / (𝐿‘𝑦)) = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))
81	78, 80	opeq12d 4841	. . . . . . . . . . 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 2772	. . . . . . . . . 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 3589	. . . . . . . . 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 1398	. . . . . . . 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 1965	. . . . . . 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 238	. . . . . 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 813	. . . . 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 2743	. . . . 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 5501	. . . . 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 317	. . . 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 3950	. . 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 2761	. . . 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 7553	. . 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 5187	. . 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 2821	. 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 2800	1 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) = (𝑞 ∈ ℚ ↦ ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∃wex 1812 ∈ wcel 2145 {cab 2739 ≠ wne 2956 ∃wrex 3087 Vcvv 3451 ∖ cdif 3896 {csn 4584 ⟨cop 4590 {copab 5167 ↦ cmpt 5186 ^◡ccnv 5650 ran crn 5652 “ cima 5654 ‘cfv 6538 (class class class)co 7420 ∈ cmpo 7422 0cc0 11200 / cdiv 11973 ℕcn 12335 ℤcz 12693 ℚcq 13075 numercnumer 16909 denomcdenom 16910 Basecbs 17387 0_gc0g 17610 1_rcur 20407 Unitcui 20585 /_rcdvr 20630 DivRingcdr 20980 ℤRHomczrh 21805 chrcchr 21807 ℚHomcqqh 34602

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279 ax-mulf 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-tpos 8243 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-er 8717 df-map 8849 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-sup 9434 df-inf 9435 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-fz 13640 df-fl 13932 df-mod 14010 df-seq 14145 df-exp 14205 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-dvds 16423 df-gcd 16665 df-numer 16911 df-denom 16912 df-gz 17108 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-0g 17612 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-mhm 18978 df-grp 19147 df-minusg 19148 df-sbg 19149 df-mulg 19278 df-subg 19333 df-ghm 19428 df-od 19742 df-cmn 19996 df-abl 19997 df-mgp 20361 df-rng 20375 df-ur 20408 df-ring 20461 df-cring 20462 df-oppr 20567 df-dvdsr 20587 df-unit 20588 df-invr 20618 df-dvr 20631 df-rhm 20702 df-subrng 20798 df-subrg 20822 df-drng 20982 df-cnfld 21679 df-zring 21753 df-zrh 21809 df-chr 21811 df-qqh 34603

This theorem is used by: qqhvval 34615 qqhf 34618

W3C validator