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Theorem qqhval 33935
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 2736 . . . 4 (𝑓 = 𝑅 → ℤ = ℤ)
2 fveq2 6907 . . . . . . 7 (𝑓 = 𝑅 → (ℤRHom‘𝑓) = (ℤRHom‘𝑅))
3 qqhval.3 . . . . . . 7 𝐿 = (ℤRHom‘𝑅)
42, 3eqtr4di 2793 . . . . . 6 (𝑓 = 𝑅 → (ℤRHom‘𝑓) = 𝐿)
54cnveqd 5889 . . . . 5 (𝑓 = 𝑅(ℤRHom‘𝑓) = 𝐿)
6 fveq2 6907 . . . . 5 (𝑓 = 𝑅 → (Unit‘𝑓) = (Unit‘𝑅))
75, 6imaeq12d 6081 . . . 4 (𝑓 = 𝑅 → ((ℤRHom‘𝑓) “ (Unit‘𝑓)) = (𝐿 “ (Unit‘𝑅)))
8 fveq2 6907 . . . . . . 7 (𝑓 = 𝑅 → (/r𝑓) = (/r𝑅))
9 qqhval.1 . . . . . . 7 / = (/r𝑅)
108, 9eqtr4di 2793 . . . . . 6 (𝑓 = 𝑅 → (/r𝑓) = / )
114fveq1d 6909 . . . . . 6 (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑥) = (𝐿𝑥))
124fveq1d 6909 . . . . . 6 (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑦) = (𝐿𝑦))
1310, 11, 12oveq123d 7452 . . . . 5 (𝑓 = 𝑅 → (((ℤRHom‘𝑓)‘𝑥)(/r𝑓)((ℤRHom‘𝑓)‘𝑦)) = ((𝐿𝑥) / (𝐿𝑦)))
1413opeq2d 4885 . . . 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 5952 . 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 33934 . 2 ℚHom = (𝑓 ∈ V ↦ ran (𝑥 ∈ ℤ, 𝑦 ∈ ((ℤRHom‘𝑓) “ (Unit‘𝑓)) ↦ ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r𝑓)((ℤRHom‘𝑓)‘𝑦))⟩))
18 zex 12620 . . . 4 ℤ ∈ V
193fvexi 6921 . . . . . 6 𝐿 ∈ V
2019cnvex 7948 . . . . 5 𝐿 ∈ V
21 imaexg 7936 . . . . 5 (𝐿 ∈ V → (𝐿 “ (Unit‘𝑅)) ∈ V)
2220, 21ax-mp 5 . . . 4 (𝐿 “ (Unit‘𝑅)) ∈ V
2318, 22mpoex 8103 . . 3 (𝑥 ∈ ℤ, 𝑦 ∈ (𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) ∈ V
2423rnex 7933 . 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 1537  wcel 2106  Vcvv 3478  cop 4637  ccnv 5688  ran crn 5690  cima 5692  cfv 6563  (class class class)co 7431  cmpo 7433   / cdiv 11918  cz 12611  1rcur 20199  Unitcui 20372  /rcdvr 20417  ℤRHomczrh 21528  ℚHomcqqh 33933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-neg 11493  df-z 12612  df-qqh 33934
This theorem is referenced by:  qqhval2  33945
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