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Theorem qqhval 33973
Description: Value of the canonical homormorphism from the rational number to a field. (Contributed by Thierry Arnoux, 22-Oct-2017.)
Hypotheses
Ref Expression
qqhval.1 / = (/r𝑅)
qqhval.2 1 = (1r𝑅)
qqhval.3 𝐿 = (ℤRHom‘𝑅)
Assertion
Ref Expression
qqhval (𝑅 ∈ V → (ℚHom‘𝑅) = ran (𝑥 ∈ ℤ, 𝑦 ∈ (𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩))
Distinct variable groups:   𝑥,𝑦,𝑅   𝑦,𝐿
Allowed substitution hints:   / (𝑥,𝑦)   1 (𝑥,𝑦)   𝐿(𝑥)

Proof of Theorem qqhval
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 eqidd 2738 . . . 4 (𝑓 = 𝑅 → ℤ = ℤ)
2 fveq2 6906 . . . . . . 7 (𝑓 = 𝑅 → (ℤRHom‘𝑓) = (ℤRHom‘𝑅))
3 qqhval.3 . . . . . . 7 𝐿 = (ℤRHom‘𝑅)
42, 3eqtr4di 2795 . . . . . 6 (𝑓 = 𝑅 → (ℤRHom‘𝑓) = 𝐿)
54cnveqd 5886 . . . . 5 (𝑓 = 𝑅(ℤRHom‘𝑓) = 𝐿)
6 fveq2 6906 . . . . 5 (𝑓 = 𝑅 → (Unit‘𝑓) = (Unit‘𝑅))
75, 6imaeq12d 6079 . . . 4 (𝑓 = 𝑅 → ((ℤRHom‘𝑓) “ (Unit‘𝑓)) = (𝐿 “ (Unit‘𝑅)))
8 fveq2 6906 . . . . . . 7 (𝑓 = 𝑅 → (/r𝑓) = (/r𝑅))
9 qqhval.1 . . . . . . 7 / = (/r𝑅)
108, 9eqtr4di 2795 . . . . . 6 (𝑓 = 𝑅 → (/r𝑓) = / )
114fveq1d 6908 . . . . . 6 (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑥) = (𝐿𝑥))
124fveq1d 6908 . . . . . 6 (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑦) = (𝐿𝑦))
1310, 11, 12oveq123d 7452 . . . . 5 (𝑓 = 𝑅 → (((ℤRHom‘𝑓)‘𝑥)(/r𝑓)((ℤRHom‘𝑓)‘𝑦)) = ((𝐿𝑥) / (𝐿𝑦)))
1413opeq2d 4880 . . . 4 (𝑓 = 𝑅 → ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r𝑓)((ℤRHom‘𝑓)‘𝑦))⟩ = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩)
151, 7, 14mpoeq123dv 7508 . . 3 (𝑓 = 𝑅 → (𝑥 ∈ ℤ, 𝑦 ∈ ((ℤRHom‘𝑓) “ (Unit‘𝑓)) ↦ ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r𝑓)((ℤRHom‘𝑓)‘𝑦))⟩) = (𝑥 ∈ ℤ, 𝑦 ∈ (𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩))
1615rneqd 5949 . 2 (𝑓 = 𝑅 → ran (𝑥 ∈ ℤ, 𝑦 ∈ ((ℤRHom‘𝑓) “ (Unit‘𝑓)) ↦ ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r𝑓)((ℤRHom‘𝑓)‘𝑦))⟩) = ran (𝑥 ∈ ℤ, 𝑦 ∈ (𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩))
17 df-qqh 33972 . 2 ℚHom = (𝑓 ∈ V ↦ ran (𝑥 ∈ ℤ, 𝑦 ∈ ((ℤRHom‘𝑓) “ (Unit‘𝑓)) ↦ ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r𝑓)((ℤRHom‘𝑓)‘𝑦))⟩))
18 zex 12622 . . . 4 ℤ ∈ V
193fvexi 6920 . . . . . 6 𝐿 ∈ V
2019cnvex 7947 . . . . 5 𝐿 ∈ V
21 imaexg 7935 . . . . 5 (𝐿 ∈ V → (𝐿 “ (Unit‘𝑅)) ∈ V)
2220, 21ax-mp 5 . . . 4 (𝐿 “ (Unit‘𝑅)) ∈ V
2318, 22mpoex 8104 . . 3 (𝑥 ∈ ℤ, 𝑦 ∈ (𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) ∈ V
2423rnex 7932 . 2 ran (𝑥 ∈ ℤ, 𝑦 ∈ (𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) ∈ V
2516, 17, 24fvmpt 7016 1 (𝑅 ∈ V → (ℚHom‘𝑅) = ran (𝑥 ∈ ℤ, 𝑦 ∈ (𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632  ccnv 5684  ran crn 5686  cima 5688  cfv 6561  (class class class)co 7431  cmpo 7433   / cdiv 11920  cz 12613  1rcur 20178  Unitcui 20355  /rcdvr 20400  ℤRHomczrh 21510  ℚHomcqqh 33971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-neg 11495  df-z 12614  df-qqh 33972
This theorem is referenced by:  qqhval2  33983
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