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Theorem qqhval 29346
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 2611 . . . 4 (𝑓 = 𝑅 → ℤ = ℤ)
2 fveq2 6103 . . . . . . 7 (𝑓 = 𝑅 → (ℤRHom‘𝑓) = (ℤRHom‘𝑅))
3 qqhval.3 . . . . . . 7 𝐿 = (ℤRHom‘𝑅)
42, 3syl6eqr 2662 . . . . . 6 (𝑓 = 𝑅 → (ℤRHom‘𝑓) = 𝐿)
54cnveqd 5220 . . . . 5 (𝑓 = 𝑅(ℤRHom‘𝑓) = 𝐿)
6 fveq2 6103 . . . . 5 (𝑓 = 𝑅 → (Unit‘𝑓) = (Unit‘𝑅))
75, 6imaeq12d 5386 . . . 4 (𝑓 = 𝑅 → ((ℤRHom‘𝑓) “ (Unit‘𝑓)) = (𝐿 “ (Unit‘𝑅)))
8 fveq2 6103 . . . . . . 7 (𝑓 = 𝑅 → (/r𝑓) = (/r𝑅))
9 qqhval.1 . . . . . . 7 / = (/r𝑅)
108, 9syl6eqr 2662 . . . . . 6 (𝑓 = 𝑅 → (/r𝑓) = / )
114fveq1d 6105 . . . . . 6 (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑥) = (𝐿𝑥))
124fveq1d 6105 . . . . . 6 (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑦) = (𝐿𝑦))
1310, 11, 12oveq123d 6570 . . . . 5 (𝑓 = 𝑅 → (((ℤRHom‘𝑓)‘𝑥)(/r𝑓)((ℤRHom‘𝑓)‘𝑦)) = ((𝐿𝑥) / (𝐿𝑦)))
1413opeq2d 4347 . . . 4 (𝑓 = 𝑅 → ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r𝑓)((ℤRHom‘𝑓)‘𝑦))⟩ = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩)
151, 7, 14mpt2eq123dv 6615 . . 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 5274 . 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 29345 . 2 ℚHom = (𝑓 ∈ V ↦ ran (𝑥 ∈ ℤ, 𝑦 ∈ ((ℤRHom‘𝑓) “ (Unit‘𝑓)) ↦ ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r𝑓)((ℤRHom‘𝑓)‘𝑦))⟩))
18 zex 11263 . . . 4 ℤ ∈ V
19 fvex 6113 . . . . . . 7 (ℤRHom‘𝑅) ∈ V
203, 19eqeltri 2684 . . . . . 6 𝐿 ∈ V
2120cnvex 7006 . . . . 5 𝐿 ∈ V
22 imaexg 6995 . . . . 5 (𝐿 ∈ V → (𝐿 “ (Unit‘𝑅)) ∈ V)
2321, 22ax-mp 5 . . . 4 (𝐿 “ (Unit‘𝑅)) ∈ V
2418, 23mpt2ex 7136 . . 3 (𝑥 ∈ ℤ, 𝑦 ∈ (𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) ∈ V
2524rnex 6992 . 2 ran (𝑥 ∈ ℤ, 𝑦 ∈ (𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) ∈ V
2616, 17, 25fvmpt 6191 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 1475  wcel 1977  Vcvv 3173  cop 4131  ccnv 5037  ran crn 5039  cima 5041  cfv 5804  (class class class)co 6549  cmpt2 6551   / cdiv 10563  cz 11254  1rcur 18324  Unitcui 18462  /rcdvr 18505  ℤRHomczrh 19667  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-cnex 9871  ax-resscn 9872
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-ral 2901  df-rex 2902  df-reu 2903  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-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  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-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-neg 10148  df-z 11255  df-qqh 29345
This theorem is referenced by:  qqhval2  29354
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