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

Theorem qqhval2 29354
 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 / = (/r𝑅)
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 𝑥 𝑦 𝑒 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3185 . . . 4 (𝑅 ∈ DivRing → 𝑅 ∈ V)
21adantr 480 . . 3 ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → 𝑅 ∈ V)
3 qqhval2.1 . . . 4 / = (/r𝑅)
4 eqid 2610 . . . 4 (1r𝑅) = (1r𝑅)
5 qqhval2.2 . . . 4 𝐿 = (ℤRHom‘𝑅)
63, 4, 5qqhval 29346 . . 3 (𝑅 ∈ V → (ℚHom‘𝑅) = ran (𝑥 ∈ ℤ, 𝑦 ∈ (𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩))
72, 6syl 17 . 2 ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (ℚHom‘𝑅) = ran (𝑥 ∈ ℤ, 𝑦 ∈ (𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩))
8 eqidd 2611 . . . 4 ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ℤ = ℤ)
9 qqhval2.0 . . . . 5 𝐵 = (Base‘𝑅)
10 eqid 2610 . . . . 5 (0g𝑅) = (0g𝑅)
119, 5, 10zrhunitpreima 29350 . . . 4 ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (𝐿 “ (Unit‘𝑅)) = (ℤ ∖ {0}))
12 mpt2eq12 6613 . . . 4 ((ℤ = ℤ ∧ (𝐿 “ (Unit‘𝑅)) = (ℤ ∖ {0})) → (𝑥 ∈ ℤ, 𝑦 ∈ (𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) = (𝑥 ∈ ℤ, 𝑦 ∈ (ℤ ∖ {0}) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩))
138, 11, 12syl2anc 691 . . 3 ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (𝑥 ∈ ℤ, 𝑦 ∈ (𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) = (𝑥 ∈ ℤ, 𝑦 ∈ (ℤ ∖ {0}) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩))
1413rneqd 5274 . 2 ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ran (𝑥 ∈ ℤ, 𝑦 ∈ (𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) = ran (𝑥 ∈ ℤ, 𝑦 ∈ (ℤ ∖ {0}) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩))
15 nfv 1830 . . . 4 𝑒(𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0)
16 nfab1 2753 . . . 4 𝑒{𝑒 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩}
17 nfcv 2751 . . . 4 𝑒{⟨𝑞, 𝑠⟩ ∣ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))}
18 simpr 476 . . . . . . . . . 10 ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) → 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩)
19 zssq 11671 . . . . . . . . . . . 12 ℤ ⊆ ℚ
20 simplrl 796 . . . . . . . . . . . 12 ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) → 𝑥 ∈ ℤ)
2119, 20sseldi 3566 . . . . . . . . . . 11 ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) → 𝑥 ∈ ℚ)
22 simplrr 797 . . . . . . . . . . . . 13 ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) → 𝑦 ∈ (ℤ ∖ {0}))
2322eldifad 3552 . . . . . . . . . . . 12 ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) → 𝑦 ∈ ℤ)
2419, 23sseldi 3566 . . . . . . . . . . 11 ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) → 𝑦 ∈ ℚ)
2522eldifbd 3553 . . . . . . . . . . . 12 ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) → ¬ 𝑦 ∈ {0})
26 velsn 4141 . . . . . . . . . . . . 13 (𝑦 ∈ {0} ↔ 𝑦 = 0)
2726necon3bbii 2829 . . . . . . . . . . . 12 𝑦 ∈ {0} ↔ 𝑦 ≠ 0)
2825, 27sylib 207 . . . . . . . . . . 11 ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) → 𝑦 ≠ 0)
29 qdivcl 11685 . . . . . . . . . . 11 ((𝑥 ∈ ℚ ∧ 𝑦 ∈ ℚ ∧ 𝑦 ≠ 0) → (𝑥 / 𝑦) ∈ ℚ)
3021, 24, 28, 29syl3anc 1318 . . . . . . . . . 10 ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) → (𝑥 / 𝑦) ∈ ℚ)
31 simplll 794 . . . . . . . . . . 11 ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) → 𝑅 ∈ DivRing)
32 simpllr 795 . . . . . . . . . . 11 ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) → (chr‘𝑅) = 0)
339, 3, 5qqhval2lem 29353 . . . . . . . . . . . 12 (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑦 ≠ 0)) → ((𝐿‘(numer‘(𝑥 / 𝑦))) / (𝐿‘(denom‘(𝑥 / 𝑦)))) = ((𝐿𝑥) / (𝐿𝑦)))
3433eqcomd 2616 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑦 ≠ 0)) → ((𝐿𝑥) / (𝐿𝑦)) = ((𝐿‘(numer‘(𝑥 / 𝑦))) / (𝐿‘(denom‘(𝑥 / 𝑦)))))
3531, 32, 20, 23, 28, 34syl23anc 1325 . . . . . . . . . 10 ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) → ((𝐿𝑥) / (𝐿𝑦)) = ((𝐿‘(numer‘(𝑥 / 𝑦))) / (𝐿‘(denom‘(𝑥 / 𝑦)))))
36 ovex 6577 . . . . . . . . . . 11 (𝑥 / 𝑦) ∈ V
37 ovex 6577 . . . . . . . . . . 11 ((𝐿𝑥) / (𝐿𝑦)) ∈ V
38 opeq12 4342 . . . . . . . . . . . . 13 ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿𝑥) / (𝐿𝑦))) → ⟨𝑞, 𝑠⟩ = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩)
3938eqeq2d 2620 . . . . . . . . . . . 12 ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿𝑥) / (𝐿𝑦))) → (𝑒 = ⟨𝑞, 𝑠⟩ ↔ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩))
40 simpl 472 . . . . . . . . . . . . . 14 ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿𝑥) / (𝐿𝑦))) → 𝑞 = (𝑥 / 𝑦))
4140eleq1d 2672 . . . . . . . . . . . . 13 ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿𝑥) / (𝐿𝑦))) → (𝑞 ∈ ℚ ↔ (𝑥 / 𝑦) ∈ ℚ))
42 simpr 476 . . . . . . . . . . . . . 14 ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿𝑥) / (𝐿𝑦))) → 𝑠 = ((𝐿𝑥) / (𝐿𝑦)))
4340fveq2d 6107 . . . . . . . . . . . . . . . 16 ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿𝑥) / (𝐿𝑦))) → (numer‘𝑞) = (numer‘(𝑥 / 𝑦)))
4443fveq2d 6107 . . . . . . . . . . . . . . 15 ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿𝑥) / (𝐿𝑦))) → (𝐿‘(numer‘𝑞)) = (𝐿‘(numer‘(𝑥 / 𝑦))))
4540fveq2d 6107 . . . . . . . . . . . . . . . 16 ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿𝑥) / (𝐿𝑦))) → (denom‘𝑞) = (denom‘(𝑥 / 𝑦)))
4645fveq2d 6107 . . . . . . . . . . . . . . 15 ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿𝑥) / (𝐿𝑦))) → (𝐿‘(denom‘𝑞)) = (𝐿‘(denom‘(𝑥 / 𝑦))))
4744, 46oveq12d 6567 . . . . . . . . . . . . . 14 ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿𝑥) / (𝐿𝑦))) → ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))) = ((𝐿‘(numer‘(𝑥 / 𝑦))) / (𝐿‘(denom‘(𝑥 / 𝑦)))))
4842, 47eqeq12d 2625 . . . . . . . . . . . . 13 ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿𝑥) / (𝐿𝑦))) → (𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))) ↔ ((𝐿𝑥) / (𝐿𝑦)) = ((𝐿‘(numer‘(𝑥 / 𝑦))) / (𝐿‘(denom‘(𝑥 / 𝑦))))))
4941, 48anbi12d 743 . . . . . . . . . . . 12 ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿𝑥) / (𝐿𝑦))) → ((𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))) ↔ ((𝑥 / 𝑦) ∈ ℚ ∧ ((𝐿𝑥) / (𝐿𝑦)) = ((𝐿‘(numer‘(𝑥 / 𝑦))) / (𝐿‘(denom‘(𝑥 / 𝑦)))))))
5039, 49anbi12d 743 . . . . . . . . . . 11 ((𝑞 = (𝑥 / 𝑦) ∧ 𝑠 = ((𝐿𝑥) / (𝐿𝑦))) → ((𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))) ↔ (𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩ ∧ ((𝑥 / 𝑦) ∈ ℚ ∧ ((𝐿𝑥) / (𝐿𝑦)) = ((𝐿‘(numer‘(𝑥 / 𝑦))) / (𝐿‘(denom‘(𝑥 / 𝑦))))))))
5136, 37, 50spc2ev 3274 . . . . . . . . . 10 ((𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩ ∧ ((𝑥 / 𝑦) ∈ ℚ ∧ ((𝐿𝑥) / (𝐿𝑦)) = ((𝐿‘(numer‘(𝑥 / 𝑦))) / (𝐿‘(denom‘(𝑥 / 𝑦)))))) → ∃𝑞𝑠(𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))))
5218, 30, 35, 51syl12anc 1316 . . . . . . . . 9 ((((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) ∧ 𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) → ∃𝑞𝑠(𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))))
5352ex 449 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ (ℤ ∖ {0}))) → (𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩ → ∃𝑞𝑠(𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))))
5453rexlimdvva 3020 . . . . . . 7 ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩ → ∃𝑞𝑠(𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))))
5554imp 444 . . . . . 6 (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ ∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) → ∃𝑞𝑠(𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))))
56 19.42vv 1907 . . . . . . 7 (∃𝑞𝑠((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) ↔ ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ ∃𝑞𝑠(𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))))
57 simprrl 800 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → 𝑞 ∈ ℚ)
58 qnumcl 15286 . . . . . . . . . 10 (𝑞 ∈ ℚ → (numer‘𝑞) ∈ ℤ)
5957, 58syl 17 . . . . . . . . 9 (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → (numer‘𝑞) ∈ ℤ)
60 qdencl 15287 . . . . . . . . . . . 12 (𝑞 ∈ ℚ → (denom‘𝑞) ∈ ℕ)
6157, 60syl 17 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → (denom‘𝑞) ∈ ℕ)
6261nnzd 11357 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → (denom‘𝑞) ∈ ℤ)
63 nnne0 10930 . . . . . . . . . . 11 ((denom‘𝑞) ∈ ℕ → (denom‘𝑞) ≠ 0)
64 nelsn 4159 . . . . . . . . . . 11 ((denom‘𝑞) ≠ 0 → ¬ (denom‘𝑞) ∈ {0})
6561, 63, 643syl 18 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → ¬ (denom‘𝑞) ∈ {0})
6662, 65eldifd 3551 . . . . . . . . 9 (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → (denom‘𝑞) ∈ (ℤ ∖ {0}))
67 simprl 790 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ (𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → 𝑒 = ⟨𝑞, 𝑠⟩)
68 qeqnumdivden 15292 . . . . . . . . . . . 12 (𝑞 ∈ ℚ → 𝑞 = ((numer‘𝑞) / (denom‘𝑞)))
6957, 68syl 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 801 . . . . . . . . . . 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))))
7169, 70opeq12d 4348 . . . . . . . . . 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)))⟩)
7267, 71eqtrd 2644 . . . . . . . . 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 6556 . . . . . . . . . . . 12 (𝑥 = (numer‘𝑞) → (𝑥 / 𝑦) = ((numer‘𝑞) / 𝑦))
74 fveq2 6103 . . . . . . . . . . . . 13 (𝑥 = (numer‘𝑞) → (𝐿𝑥) = (𝐿‘(numer‘𝑞)))
7574oveq1d 6564 . . . . . . . . . . . 12 (𝑥 = (numer‘𝑞) → ((𝐿𝑥) / (𝐿𝑦)) = ((𝐿‘(numer‘𝑞)) / (𝐿𝑦)))
7673, 75opeq12d 4348 . . . . . . . . . . 11 (𝑥 = (numer‘𝑞) → ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩ = ⟨((numer‘𝑞) / 𝑦), ((𝐿‘(numer‘𝑞)) / (𝐿𝑦))⟩)
7776eqeq2d 2620 . . . . . . . . . 10 (𝑥 = (numer‘𝑞) → (𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩ ↔ 𝑒 = ⟨((numer‘𝑞) / 𝑦), ((𝐿‘(numer‘𝑞)) / (𝐿𝑦))⟩))
78 oveq2 6557 . . . . . . . . . . . 12 (𝑦 = (denom‘𝑞) → ((numer‘𝑞) / 𝑦) = ((numer‘𝑞) / (denom‘𝑞)))
79 fveq2 6103 . . . . . . . . . . . . 13 (𝑦 = (denom‘𝑞) → (𝐿𝑦) = (𝐿‘(denom‘𝑞)))
8079oveq2d 6565 . . . . . . . . . . . 12 (𝑦 = (denom‘𝑞) → ((𝐿‘(numer‘𝑞)) / (𝐿𝑦)) = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))
8178, 80opeq12d 4348 . . . . . . . . . . 11 (𝑦 = (denom‘𝑞) → ⟨((numer‘𝑞) / 𝑦), ((𝐿‘(numer‘𝑞)) / (𝐿𝑦))⟩ = ⟨((numer‘𝑞) / (denom‘𝑞)), ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))⟩)
8281eqeq2d 2620 . . . . . . . . . 10 (𝑦 = (denom‘𝑞) → (𝑒 = ⟨((numer‘𝑞) / 𝑦), ((𝐿‘(numer‘𝑞)) / (𝐿𝑦))⟩ ↔ 𝑒 = ⟨((numer‘𝑞) / (denom‘𝑞)), ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))⟩))
8377, 82rspc2ev 3295 . . . . . . . . 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), ((𝐿𝑥) / (𝐿𝑦))⟩)
8459, 66, 72, 83syl3anc 1318 . . . . . . . 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), ((𝐿𝑥) / (𝐿𝑦))⟩)
8584exlimivv 1847 . . . . . . 7 (∃𝑞𝑠((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), ((𝐿𝑥) / (𝐿𝑦))⟩)
8656, 85sylbir 224 . . . . . 6 (((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) ∧ ∃𝑞𝑠(𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))) → ∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩)
8755, 86impbida 873 . . . . 5 ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩ ↔ ∃𝑞𝑠(𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))))
88 abid 2598 . . . . 5 (𝑒 ∈ {𝑒 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩} ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩)
89 elopab 4908 . . . . 5 (𝑒 ∈ {⟨𝑞, 𝑠⟩ ∣ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))} ↔ ∃𝑞𝑠(𝑒 = ⟨𝑞, 𝑠⟩ ∧ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))))
9087, 88, 893bitr4g 302 . . . 4 ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → (𝑒 ∈ {𝑒 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩} ↔ 𝑒 ∈ {⟨𝑞, 𝑠⟩ ∣ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))}))
9115, 16, 17, 90eqrd 3586 . . 3 ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → {𝑒 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩} = {⟨𝑞, 𝑠⟩ ∣ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))})
92 eqid 2610 . . . 4 (𝑥 ∈ ℤ, 𝑦 ∈ (ℤ ∖ {0}) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) = (𝑥 ∈ ℤ, 𝑦 ∈ (ℤ ∖ {0}) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩)
9392rnmpt2 6668 . . 3 ran (𝑥 ∈ ℤ, 𝑦 ∈ (ℤ ∖ {0}) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) = {𝑒 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ (ℤ ∖ {0})𝑒 = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩}
94 df-mpt 4645 . . 3 (𝑞 ∈ ℚ ↦ ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))) = {⟨𝑞, 𝑠⟩ ∣ (𝑞 ∈ ℚ ∧ 𝑠 = ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞))))}
9591, 93, 943eqtr4g 2669 . 2 ((𝑅 ∈ DivRing ∧ (chr‘𝑅) = 0) → ran (𝑥 ∈ ℤ, 𝑦 ∈ (ℤ ∖ {0}) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) = (𝑞 ∈ ℚ ↦ ((𝐿‘(numer‘𝑞)) / (𝐿‘(denom‘𝑞)))))
967, 14, 953eqtrd 2648 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 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537  {csn 4125  ⟨cop 4131  {copab 4642   ↦ cmpt 4643  ◡ccnv 5037  ran crn 5039   “ cima 5041  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815   / cdiv 10563  ℕcn 10897  ℤcz 11254  ℚcq 11664  numercnumer 15279  denomcdenom 15280  Basecbs 15695  0gc0g 15923  1rcur 18324  Unitcui 18462  /rcdvr 18505  DivRingcdr 18570  ℤRHomczrh 19667  chrcchr 19669  ℚHomcqqh 29344 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-fz 12198  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-dvds 14822  df-gcd 15055  df-numer 15281  df-denom 15282  df-gz 15472  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-od 17771  df-cmn 18018  df-mgp 18313  df-ur 18325  df-ring 18372  df-cring 18373  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-dvr 18506  df-rnghom 18538  df-drng 18572  df-subrg 18601  df-cnfld 19568  df-zring 19638  df-zrh 19671  df-chr 19673  df-qqh 29345 This theorem is referenced by:  qqhvval  29355  qqhf  29358
 Copyright terms: Public domain W3C validator