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Theorem qqhval2 32620
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 𝐡 = (Baseβ€˜π‘…)
qqhval2.1 / = (/rβ€˜π‘…)
qqhval2.2 𝐿 = (β„€RHomβ€˜π‘…)
Assertion
Ref Expression
qqhval2 ((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) β†’ (β„šHomβ€˜π‘…) = (π‘ž ∈ β„š ↦ ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))))
Distinct variable groups:   / ,π‘ž   𝐡,π‘ž   𝐿,π‘ž   𝑅,π‘ž

Proof of Theorem qqhval2
Dummy variables 𝑒 𝑠 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3462 . . . 4 (𝑅 ∈ DivRing β†’ 𝑅 ∈ V)
21adantr 482 . . 3 ((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) β†’ 𝑅 ∈ V)
3 qqhval2.1 . . . 4 / = (/rβ€˜π‘…)
4 eqid 2733 . . . 4 (1rβ€˜π‘…) = (1rβ€˜π‘…)
5 qqhval2.2 . . . 4 𝐿 = (β„€RHomβ€˜π‘…)
63, 4, 5qqhval 32612 . . 3 (𝑅 ∈ V β†’ (β„šHomβ€˜π‘…) = ran (π‘₯ ∈ β„€, 𝑦 ∈ (◑𝐿 β€œ (Unitβ€˜π‘…)) ↦ ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩))
72, 6syl 17 . 2 ((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) β†’ (β„šHomβ€˜π‘…) = ran (π‘₯ ∈ β„€, 𝑦 ∈ (◑𝐿 β€œ (Unitβ€˜π‘…)) ↦ ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩))
8 eqid 2733 . . . 4 β„€ = β„€
9 qqhval2.0 . . . . 5 𝐡 = (Baseβ€˜π‘…)
10 eqid 2733 . . . . 5 (0gβ€˜π‘…) = (0gβ€˜π‘…)
119, 5, 10zrhunitpreima 32616 . . . 4 ((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) β†’ (◑𝐿 β€œ (Unitβ€˜π‘…)) = (β„€ βˆ– {0}))
12 mpoeq12 7431 . . . 4 ((β„€ = β„€ ∧ (◑𝐿 β€œ (Unitβ€˜π‘…)) = (β„€ βˆ– {0})) β†’ (π‘₯ ∈ β„€, 𝑦 ∈ (◑𝐿 β€œ (Unitβ€˜π‘…)) ↦ ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩) = (π‘₯ ∈ β„€, 𝑦 ∈ (β„€ βˆ– {0}) ↦ ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩))
138, 11, 12sylancr 588 . . 3 ((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) β†’ (π‘₯ ∈ β„€, 𝑦 ∈ (◑𝐿 β€œ (Unitβ€˜π‘…)) ↦ ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩) = (π‘₯ ∈ β„€, 𝑦 ∈ (β„€ βˆ– {0}) ↦ ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩))
1413rneqd 5894 . 2 ((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) β†’ ran (π‘₯ ∈ β„€, 𝑦 ∈ (◑𝐿 β€œ (Unitβ€˜π‘…)) ↦ ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩) = ran (π‘₯ ∈ β„€, 𝑦 ∈ (β„€ βˆ– {0}) ↦ ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩))
15 nfv 1918 . . . 4 Ⅎ𝑒(𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0)
16 nfab1 2906 . . . 4 Ⅎ𝑒{𝑒 ∣ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ (β„€ βˆ– {0})𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩}
17 nfcv 2904 . . . 4 Ⅎ𝑒{βŸ¨π‘ž, π‘ βŸ© ∣ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž))))}
18 simpr 486 . . . . . . . . . 10 ((((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ (β„€ βˆ– {0}))) ∧ 𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩) β†’ 𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩)
19 zssq 12886 . . . . . . . . . . . 12 β„€ βŠ† β„š
20 simplrl 776 . . . . . . . . . . . 12 ((((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ (β„€ βˆ– {0}))) ∧ 𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩) β†’ π‘₯ ∈ β„€)
2119, 20sselid 3943 . . . . . . . . . . 11 ((((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ (β„€ βˆ– {0}))) ∧ 𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩) β†’ π‘₯ ∈ β„š)
22 simplrr 777 . . . . . . . . . . . . 13 ((((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ (β„€ βˆ– {0}))) ∧ 𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩) β†’ 𝑦 ∈ (β„€ βˆ– {0}))
2322eldifad 3923 . . . . . . . . . . . 12 ((((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ (β„€ βˆ– {0}))) ∧ 𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩) β†’ 𝑦 ∈ β„€)
2419, 23sselid 3943 . . . . . . . . . . 11 ((((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ (β„€ βˆ– {0}))) ∧ 𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩) β†’ 𝑦 ∈ β„š)
2522eldifbd 3924 . . . . . . . . . . . 12 ((((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ (β„€ βˆ– {0}))) ∧ 𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩) β†’ Β¬ 𝑦 ∈ {0})
26 velsn 4603 . . . . . . . . . . . . 13 (𝑦 ∈ {0} ↔ 𝑦 = 0)
2726necon3bbii 2988 . . . . . . . . . . . 12 (Β¬ 𝑦 ∈ {0} ↔ 𝑦 β‰  0)
2825, 27sylib 217 . . . . . . . . . . 11 ((((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ (β„€ βˆ– {0}))) ∧ 𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩) β†’ 𝑦 β‰  0)
29 qdivcl 12900 . . . . . . . . . . 11 ((π‘₯ ∈ β„š ∧ 𝑦 ∈ β„š ∧ 𝑦 β‰  0) β†’ (π‘₯ / 𝑦) ∈ β„š)
3021, 24, 28, 29syl3anc 1372 . . . . . . . . . 10 ((((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ (β„€ βˆ– {0}))) ∧ 𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩) β†’ (π‘₯ / 𝑦) ∈ β„š)
31 simplll 774 . . . . . . . . . . 11 ((((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ (β„€ βˆ– {0}))) ∧ 𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩) β†’ 𝑅 ∈ DivRing)
32 simpllr 775 . . . . . . . . . . 11 ((((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ (β„€ βˆ– {0}))) ∧ 𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩) β†’ (chrβ€˜π‘…) = 0)
339, 3, 5qqhval2lem 32619 . . . . . . . . . . . 12 (((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€ ∧ 𝑦 β‰  0)) β†’ ((πΏβ€˜(numerβ€˜(π‘₯ / 𝑦))) / (πΏβ€˜(denomβ€˜(π‘₯ / 𝑦)))) = ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦)))
3433eqcomd 2739 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ β„€ ∧ 𝑦 β‰  0)) β†’ ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦)) = ((πΏβ€˜(numerβ€˜(π‘₯ / 𝑦))) / (πΏβ€˜(denomβ€˜(π‘₯ / 𝑦)))))
3531, 32, 20, 23, 28, 34syl23anc 1378 . . . . . . . . . 10 ((((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ (β„€ βˆ– {0}))) ∧ 𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩) β†’ ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦)) = ((πΏβ€˜(numerβ€˜(π‘₯ / 𝑦))) / (πΏβ€˜(denomβ€˜(π‘₯ / 𝑦)))))
36 ovex 7391 . . . . . . . . . . 11 (π‘₯ / 𝑦) ∈ V
37 ovex 7391 . . . . . . . . . . 11 ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦)) ∈ V
38 opeq12 4833 . . . . . . . . . . . . 13 ((π‘ž = (π‘₯ / 𝑦) ∧ 𝑠 = ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))) β†’ βŸ¨π‘ž, π‘ βŸ© = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩)
3938eqeq2d 2744 . . . . . . . . . . . 12 ((π‘ž = (π‘₯ / 𝑦) ∧ 𝑠 = ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))) β†’ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ↔ 𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩))
40 simpl 484 . . . . . . . . . . . . . 14 ((π‘ž = (π‘₯ / 𝑦) ∧ 𝑠 = ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))) β†’ π‘ž = (π‘₯ / 𝑦))
4140eleq1d 2819 . . . . . . . . . . . . 13 ((π‘ž = (π‘₯ / 𝑦) ∧ 𝑠 = ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))) β†’ (π‘ž ∈ β„š ↔ (π‘₯ / 𝑦) ∈ β„š))
42 simpr 486 . . . . . . . . . . . . . 14 ((π‘ž = (π‘₯ / 𝑦) ∧ 𝑠 = ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))) β†’ 𝑠 = ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦)))
4340fveq2d 6847 . . . . . . . . . . . . . . . 16 ((π‘ž = (π‘₯ / 𝑦) ∧ 𝑠 = ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))) β†’ (numerβ€˜π‘ž) = (numerβ€˜(π‘₯ / 𝑦)))
4443fveq2d 6847 . . . . . . . . . . . . . . 15 ((π‘ž = (π‘₯ / 𝑦) ∧ 𝑠 = ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))) β†’ (πΏβ€˜(numerβ€˜π‘ž)) = (πΏβ€˜(numerβ€˜(π‘₯ / 𝑦))))
4540fveq2d 6847 . . . . . . . . . . . . . . . 16 ((π‘ž = (π‘₯ / 𝑦) ∧ 𝑠 = ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))) β†’ (denomβ€˜π‘ž) = (denomβ€˜(π‘₯ / 𝑦)))
4645fveq2d 6847 . . . . . . . . . . . . . . 15 ((π‘ž = (π‘₯ / 𝑦) ∧ 𝑠 = ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))) β†’ (πΏβ€˜(denomβ€˜π‘ž)) = (πΏβ€˜(denomβ€˜(π‘₯ / 𝑦))))
4744, 46oveq12d 7376 . . . . . . . . . . . . . 14 ((π‘ž = (π‘₯ / 𝑦) ∧ 𝑠 = ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))) β†’ ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž))) = ((πΏβ€˜(numerβ€˜(π‘₯ / 𝑦))) / (πΏβ€˜(denomβ€˜(π‘₯ / 𝑦)))))
4842, 47eqeq12d 2749 . . . . . . . . . . . . 13 ((π‘ž = (π‘₯ / 𝑦) ∧ 𝑠 = ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))) β†’ (𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž))) ↔ ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦)) = ((πΏβ€˜(numerβ€˜(π‘₯ / 𝑦))) / (πΏβ€˜(denomβ€˜(π‘₯ / 𝑦))))))
4941, 48anbi12d 632 . . . . . . . . . . . 12 ((π‘ž = (π‘₯ / 𝑦) ∧ 𝑠 = ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))) β†’ ((π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))) ↔ ((π‘₯ / 𝑦) ∈ β„š ∧ ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦)) = ((πΏβ€˜(numerβ€˜(π‘₯ / 𝑦))) / (πΏβ€˜(denomβ€˜(π‘₯ / 𝑦)))))))
5039, 49anbi12d 632 . . . . . . . . . . 11 ((π‘ž = (π‘₯ / 𝑦) ∧ 𝑠 = ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))) β†’ ((𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž))))) ↔ (𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩ ∧ ((π‘₯ / 𝑦) ∈ β„š ∧ ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦)) = ((πΏβ€˜(numerβ€˜(π‘₯ / 𝑦))) / (πΏβ€˜(denomβ€˜(π‘₯ / 𝑦))))))))
5136, 37, 50spc2ev 3565 . . . . . . . . . 10 ((𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩ ∧ ((π‘₯ / 𝑦) ∈ β„š ∧ ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦)) = ((πΏβ€˜(numerβ€˜(π‘₯ / 𝑦))) / (πΏβ€˜(denomβ€˜(π‘₯ / 𝑦)))))) β†’ βˆƒπ‘žβˆƒπ‘ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž))))))
5218, 30, 35, 51syl12anc 836 . . . . . . . . 9 ((((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ (β„€ βˆ– {0}))) ∧ 𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩) β†’ βˆƒπ‘žβˆƒπ‘ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž))))))
5352ex 414 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (π‘₯ ∈ β„€ ∧ 𝑦 ∈ (β„€ βˆ– {0}))) β†’ (𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩ β†’ βˆƒπ‘žβˆƒπ‘ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))))))
5453rexlimdvva 3202 . . . . . . 7 ((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) β†’ (βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ (β„€ βˆ– {0})𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩ β†’ βˆƒπ‘žβˆƒπ‘ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))))))
5554imp 408 . . . . . 6 (((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ (β„€ βˆ– {0})𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩) β†’ βˆƒπ‘žβˆƒπ‘ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž))))))
56 19.42vv 1962 . . . . . . 7 (βˆƒπ‘žβˆƒπ‘ ((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))))) ↔ ((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ βˆƒπ‘žβˆƒπ‘ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))))))
57 simprrl 780 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))))) β†’ π‘ž ∈ β„š)
58 qnumcl 16620 . . . . . . . . . 10 (π‘ž ∈ β„š β†’ (numerβ€˜π‘ž) ∈ β„€)
5957, 58syl 17 . . . . . . . . 9 (((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))))) β†’ (numerβ€˜π‘ž) ∈ β„€)
60 qdencl 16621 . . . . . . . . . . . 12 (π‘ž ∈ β„š β†’ (denomβ€˜π‘ž) ∈ β„•)
6157, 60syl 17 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))))) β†’ (denomβ€˜π‘ž) ∈ β„•)
6261nnzd 12531 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))))) β†’ (denomβ€˜π‘ž) ∈ β„€)
63 nnne0 12192 . . . . . . . . . . 11 ((denomβ€˜π‘ž) ∈ β„• β†’ (denomβ€˜π‘ž) β‰  0)
64 nelsn 4627 . . . . . . . . . . 11 ((denomβ€˜π‘ž) β‰  0 β†’ Β¬ (denomβ€˜π‘ž) ∈ {0})
6561, 63, 643syl 18 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))))) β†’ Β¬ (denomβ€˜π‘ž) ∈ {0})
6662, 65eldifd 3922 . . . . . . . . 9 (((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))))) β†’ (denomβ€˜π‘ž) ∈ (β„€ βˆ– {0}))
67 simprl 770 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))))) β†’ 𝑒 = βŸ¨π‘ž, π‘ βŸ©)
68 qeqnumdivden 16626 . . . . . . . . . . . 12 (π‘ž ∈ β„š β†’ π‘ž = ((numerβ€˜π‘ž) / (denomβ€˜π‘ž)))
6957, 68syl 17 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))))) β†’ π‘ž = ((numerβ€˜π‘ž) / (denomβ€˜π‘ž)))
70 simprrr 781 . . . . . . . . . . 11 (((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))))) β†’ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž))))
7169, 70opeq12d 4839 . . . . . . . . . 10 (((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))))) β†’ βŸ¨π‘ž, π‘ βŸ© = ⟨((numerβ€˜π‘ž) / (denomβ€˜π‘ž)), ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))⟩)
7267, 71eqtrd 2773 . . . . . . . . 9 (((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))))) β†’ 𝑒 = ⟨((numerβ€˜π‘ž) / (denomβ€˜π‘ž)), ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))⟩)
73 oveq1 7365 . . . . . . . . . . . 12 (π‘₯ = (numerβ€˜π‘ž) β†’ (π‘₯ / 𝑦) = ((numerβ€˜π‘ž) / 𝑦))
74 fveq2 6843 . . . . . . . . . . . . 13 (π‘₯ = (numerβ€˜π‘ž) β†’ (πΏβ€˜π‘₯) = (πΏβ€˜(numerβ€˜π‘ž)))
7574oveq1d 7373 . . . . . . . . . . . 12 (π‘₯ = (numerβ€˜π‘ž) β†’ ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦)) = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜π‘¦)))
7673, 75opeq12d 4839 . . . . . . . . . . 11 (π‘₯ = (numerβ€˜π‘ž) β†’ ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩ = ⟨((numerβ€˜π‘ž) / 𝑦), ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜π‘¦))⟩)
7776eqeq2d 2744 . . . . . . . . . 10 (π‘₯ = (numerβ€˜π‘ž) β†’ (𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩ ↔ 𝑒 = ⟨((numerβ€˜π‘ž) / 𝑦), ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜π‘¦))⟩))
78 oveq2 7366 . . . . . . . . . . . 12 (𝑦 = (denomβ€˜π‘ž) β†’ ((numerβ€˜π‘ž) / 𝑦) = ((numerβ€˜π‘ž) / (denomβ€˜π‘ž)))
79 fveq2 6843 . . . . . . . . . . . . 13 (𝑦 = (denomβ€˜π‘ž) β†’ (πΏβ€˜π‘¦) = (πΏβ€˜(denomβ€˜π‘ž)))
8079oveq2d 7374 . . . . . . . . . . . 12 (𝑦 = (denomβ€˜π‘ž) β†’ ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜π‘¦)) = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž))))
8178, 80opeq12d 4839 . . . . . . . . . . 11 (𝑦 = (denomβ€˜π‘ž) β†’ ⟨((numerβ€˜π‘ž) / 𝑦), ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜π‘¦))⟩ = ⟨((numerβ€˜π‘ž) / (denomβ€˜π‘ž)), ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))⟩)
8281eqeq2d 2744 . . . . . . . . . 10 (𝑦 = (denomβ€˜π‘ž) β†’ (𝑒 = ⟨((numerβ€˜π‘ž) / 𝑦), ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜π‘¦))⟩ ↔ 𝑒 = ⟨((numerβ€˜π‘ž) / (denomβ€˜π‘ž)), ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))⟩))
8377, 82rspc2ev 3591 . . . . . . . . 9 (((numerβ€˜π‘ž) ∈ β„€ ∧ (denomβ€˜π‘ž) ∈ (β„€ βˆ– {0}) ∧ 𝑒 = ⟨((numerβ€˜π‘ž) / (denomβ€˜π‘ž)), ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))⟩) β†’ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ (β„€ βˆ– {0})𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩)
8459, 66, 72, 83syl3anc 1372 . . . . . . . 8 (((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))))) β†’ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ (β„€ βˆ– {0})𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩)
8584exlimivv 1936 . . . . . . 7 (βˆƒπ‘žβˆƒπ‘ ((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))))) β†’ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ (β„€ βˆ– {0})𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩)
8656, 85sylbir 234 . . . . . 6 (((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) ∧ βˆƒπ‘žβˆƒπ‘ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))))) β†’ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ (β„€ βˆ– {0})𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩)
8755, 86impbida 800 . . . . 5 ((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) β†’ (βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ (β„€ βˆ– {0})𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩ ↔ βˆƒπ‘žβˆƒπ‘ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))))))
88 abid 2714 . . . . 5 (𝑒 ∈ {𝑒 ∣ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ (β„€ βˆ– {0})𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩} ↔ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ (β„€ βˆ– {0})𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩)
89 elopab 5485 . . . . 5 (𝑒 ∈ {βŸ¨π‘ž, π‘ βŸ© ∣ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž))))} ↔ βˆƒπ‘žβˆƒπ‘ (𝑒 = βŸ¨π‘ž, π‘ βŸ© ∧ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž))))))
9087, 88, 893bitr4g 314 . . . 4 ((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) β†’ (𝑒 ∈ {𝑒 ∣ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ (β„€ βˆ– {0})𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩} ↔ 𝑒 ∈ {βŸ¨π‘ž, π‘ βŸ© ∣ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž))))}))
9115, 16, 17, 90eqrd 3964 . . 3 ((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) β†’ {𝑒 ∣ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ (β„€ βˆ– {0})𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩} = {βŸ¨π‘ž, π‘ βŸ© ∣ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž))))})
92 eqid 2733 . . . 4 (π‘₯ ∈ β„€, 𝑦 ∈ (β„€ βˆ– {0}) ↦ ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩) = (π‘₯ ∈ β„€, 𝑦 ∈ (β„€ βˆ– {0}) ↦ ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩)
9392rnmpo 7490 . . 3 ran (π‘₯ ∈ β„€, 𝑦 ∈ (β„€ βˆ– {0}) ↦ ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩) = {𝑒 ∣ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ (β„€ βˆ– {0})𝑒 = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩}
94 df-mpt 5190 . . 3 (π‘ž ∈ β„š ↦ ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))) = {βŸ¨π‘ž, π‘ βŸ© ∣ (π‘ž ∈ β„š ∧ 𝑠 = ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž))))}
9591, 93, 943eqtr4g 2798 . 2 ((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) β†’ ran (π‘₯ ∈ β„€, 𝑦 ∈ (β„€ βˆ– {0}) ↦ ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩) = (π‘ž ∈ β„š ↦ ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))))
967, 14, 953eqtrd 2777 1 ((𝑅 ∈ DivRing ∧ (chrβ€˜π‘…) = 0) β†’ (β„šHomβ€˜π‘…) = (π‘ž ∈ β„š ↦ ((πΏβ€˜(numerβ€˜π‘ž)) / (πΏβ€˜(denomβ€˜π‘ž)))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710   β‰  wne 2940  βˆƒwrex 3070  Vcvv 3444   βˆ– cdif 3908  {csn 4587  βŸ¨cop 4593  {copab 5168   ↦ cmpt 5189  β—‘ccnv 5633  ran crn 5635   β€œ cima 5637  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  0cc0 11056   / cdiv 11817  β„•cn 12158  β„€cz 12504  β„šcq 12878  numercnumer 16613  denomcdenom 16614  Basecbs 17088  0gc0g 17326  1rcur 19918  Unitcui 20073  /rcdvr 20116  DivRingcdr 20197  β„€RHomczrh 20916  chrcchr 20918  β„šHomcqqh 32610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134  ax-addf 11135  ax-mulf 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-tpos 8158  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9383  df-inf 9384  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-q 12879  df-rp 12921  df-fz 13431  df-fl 13703  df-mod 13781  df-seq 13913  df-exp 13974  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-dvds 16142  df-gcd 16380  df-numer 16615  df-denom 16616  df-gz 16807  df-struct 17024  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-mulr 17152  df-starv 17153  df-tset 17157  df-ple 17158  df-ds 17160  df-unif 17161  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-mhm 18606  df-grp 18756  df-minusg 18757  df-sbg 18758  df-mulg 18878  df-subg 18930  df-ghm 19011  df-od 19315  df-cmn 19569  df-mgp 19902  df-ur 19919  df-ring 19971  df-cring 19972  df-oppr 20054  df-dvdsr 20075  df-unit 20076  df-invr 20106  df-dvr 20117  df-rnghom 20153  df-drng 20199  df-subrg 20234  df-cnfld 20813  df-zring 20886  df-zrh 20920  df-chr 20922  df-qqh 32611
This theorem is referenced by:  qqhvval  32621  qqhf  32624
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