qqhval2 - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > qqhval2		Structured version Visualization version GIF version

Theorem qqhval2 32950

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 3492	. . . 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 2732	. . . 4 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
5		qqhval2.2	. . . 4 ⊢ 𝐿 = (ℤRHom‘𝑅)
6	3, 4, 5	qqhval 32942	. . 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 2732	. . . 4 ⊢ ℤ = ℤ
9		qqhval2.0	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
10		eqid 2732	. . . . 5 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
11	9, 5, 10	zrhunitpreima 32946	. . . 4 ⊢ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (^◡𝐿 “ (Unit‘𝑅)) = (ℤ ∖ {0}))
12		mpoeq12 7478	. . . 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 587	. . 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 5935	. 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 2905	. . . 4 ⊢ Ⅎ𝑒{𝑒 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩}
17		nfcv 2903	. . . 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 12936	. . . . . . . . . . . 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 3979	. . . . . . . . . . 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 3959	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑦 ∈ ℤ)
24	19, 23	sselid 3979	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → 𝑦 ∈ ℚ)
25	22	eldifbd 3960	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩) → ¬ 𝑦 ∈ {0})
26		velsn 4643	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {0} ↔ 𝑦 = 0)
27	26	necon3bbii 2988	. . . . . . . . . . . 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 12950	. . . . . . . . . . 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 32949	. . . . . . . . . . . 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 2738	. . . . . . . . . . 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 7438	. . . . . . . . . . 11 ⊢ (𝑥 / 𝑦) ∈ V
37		ovex 7438	. . . . . . . . . . 11 ⊢ ((𝐿‘𝑥) / (𝐿‘𝑦)) ∈ V
38		opeq12 4874	. . . . . . . . . . . . 13 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → ⟨𝑞, 𝑠⟩ = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩)
39	38	eqeq2d 2743	. . . . . . . . . . . 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 2818	. . . . . . . . . . . . 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 6892	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → (numer‘𝑞) = (numer‘(𝑥 / 𝑦)))
44	43	fveq2d 6892	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → (𝐿‘(numer‘𝑞)) = (𝐿‘(numer‘(𝑥 / 𝑦))))
45	40	fveq2d 6892	. . . . . . . . . . . . . . . 16 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → (denom‘𝑞) = (denom‘(𝑥 / 𝑦)))
46	45	fveq2d 6892	. . . . . . . . . . . . . . 15 ⊢ ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿‘𝑥) / (𝐿‘𝑦))) → (𝐿‘(denom‘𝑞)) = (𝐿‘(denom‘(𝑥 / 𝑦))))
47	44, 46	oveq12d 7423	. . . . . . . . . . . . . 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 2748	. . . . . . . . . . . . 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 3597	. . . . . . . . . 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 3211	. . . . . . 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 16672	. . . . . . . . . 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 16673	. . . . . . . . . . . 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 12581	. . . . . . . . . 10 ⊢ (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → (denom‘𝑞) ∈ ℤ)
63		nnne0 12242	. . . . . . . . . . 11 ⊢ ((denom‘𝑞) ∈ ℕ → (denom‘𝑞) ≠ 0)
64		nelsn 4667	. . . . . . . . . . 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 3958	. . . . . . . . 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 16678	. . . . . . . . . . . 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 4880	. . . . . . . . . 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 7412	. . . . . . . . . . . 12 ⊢ (𝑥 = (numer‘𝑞) → (𝑥 / 𝑦) = ((numer‘𝑞) / 𝑦))
74		fveq2 6888	. . . . . . . . . . . . 13 ⊢ (𝑥 = (numer‘𝑞) → (𝐿‘𝑥) = (𝐿‘(numer‘𝑞)))
75	74	oveq1d 7420	. . . . . . . . . . . 12 ⊢ (𝑥 = (numer‘𝑞) → ((𝐿‘𝑥) / (𝐿‘𝑦)) = ((𝐿‘(numer‘𝑞)) / (𝐿‘𝑦)))
76	73, 75	opeq12d 4880	. . . . . . . . . . 11 ⊢ (𝑥 = (numer‘𝑞) → ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩ = ⟨((numer‘𝑞) / 𝑦), ((𝐿‘(numer‘𝑞)) / (𝐿‘𝑦))⟩)
77	76	eqeq2d 2743	. . . . . . . . . 10 ⊢ (𝑥 = (numer‘𝑞) → (𝑒 = ⟨(𝑥 / 𝑦), ((𝐿‘𝑥) / (𝐿‘𝑦))⟩ ↔ 𝑒 = ⟨((numer‘𝑞) / 𝑦), ((𝐿‘(numer‘𝑞)) / (𝐿‘𝑦))⟩))
78		oveq2 7413	. . . . . . . . . . . 12 ⊢ (𝑦 = (denom‘𝑞) → ((numer‘𝑞) / 𝑦) = ((numer‘𝑞) / (denom‘𝑞)))
79		fveq2 6888	. . . . . . . . . . . . 13 ⊢ (𝑦 = (denom‘𝑞) → (𝐿‘𝑦) = (𝐿‘(denom‘𝑞)))
80	79	oveq2d 7421	. . . . . . . . . . . 12 ⊢ (𝑦 = (denom‘𝑞) → ((𝐿‘(numer‘𝑞)) / (𝐿‘𝑦)) = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))
81	78, 80	opeq12d 4880	. . . . . . . . . . 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 2743	. . . . . . . . . 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 3623	. . . . . . . . 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 2713	. . . . 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 5526	. . . . 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 4000	. . 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 2732	. . . 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 7538	. . 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 5231	. . 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 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 {cab 2709 ≠ wne 2940 ∃wrex 3070 Vcvv 3474 ∖ cdif 3944 {csn 4627 ⟨cop 4633 {copab 5209 ↦ cmpt 5230 ^◡ccnv 5674 ran crn 5676 “ cima 5678 ‘cfv 6540 (class class class)co 7405 ∈ cmpo 7407 0cc0 11106 / cdiv 11867 ℕcn 12208 ℤcz 12554 ℚcq 12928 numercnumer 16665 denomcdenom 16666 Basecbs 17140 0_gc0g 17381 1_rcur 19998 Unitcui 20161 /_rcdvr 20206 DivRingcdr 20307 ℤRHomczrh 21040 chrcchr 21042 ℚHomcqqh 32940

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 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-fz 13481 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-dvds 16194 df-gcd 16432 df-numer 16667 df-denom 16668 df-gz 16859 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-ghm 19084 df-od 19390 df-cmn 19644 df-mgp 19982 df-ur 19999 df-ring 20051 df-cring 20052 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-dvr 20207 df-rnghom 20243 df-drng 20309 df-subrg 20353 df-cnfld 20937 df-zring 21010 df-zrh 21044 df-chr 21046 df-qqh 32941

This theorem is referenced by: qqhvval 32951 qqhf 32954

W3C validator