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Theorem qqhval 31289
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 2823 . . . 4 (𝑓 = 𝑅 → ℤ = ℤ)
2 fveq2 6652 . . . . . . 7 (𝑓 = 𝑅 → (ℤRHom‘𝑓) = (ℤRHom‘𝑅))
3 qqhval.3 . . . . . . 7 𝐿 = (ℤRHom‘𝑅)
42, 3eqtr4di 2875 . . . . . 6 (𝑓 = 𝑅 → (ℤRHom‘𝑓) = 𝐿)
54cnveqd 5723 . . . . 5 (𝑓 = 𝑅(ℤRHom‘𝑓) = 𝐿)
6 fveq2 6652 . . . . 5 (𝑓 = 𝑅 → (Unit‘𝑓) = (Unit‘𝑅))
75, 6imaeq12d 5908 . . . 4 (𝑓 = 𝑅 → ((ℤRHom‘𝑓) “ (Unit‘𝑓)) = (𝐿 “ (Unit‘𝑅)))
8 fveq2 6652 . . . . . . 7 (𝑓 = 𝑅 → (/r𝑓) = (/r𝑅))
9 qqhval.1 . . . . . . 7 / = (/r𝑅)
108, 9eqtr4di 2875 . . . . . 6 (𝑓 = 𝑅 → (/r𝑓) = / )
114fveq1d 6654 . . . . . 6 (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑥) = (𝐿𝑥))
124fveq1d 6654 . . . . . 6 (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑦) = (𝐿𝑦))
1310, 11, 12oveq123d 7161 . . . . 5 (𝑓 = 𝑅 → (((ℤRHom‘𝑓)‘𝑥)(/r𝑓)((ℤRHom‘𝑓)‘𝑦)) = ((𝐿𝑥) / (𝐿𝑦)))
1413opeq2d 4785 . . . 4 (𝑓 = 𝑅 → ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r𝑓)((ℤRHom‘𝑓)‘𝑦))⟩ = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩)
151, 7, 14mpoeq123dv 7213 . . 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 5785 . 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 31288 . 2 ℚHom = (𝑓 ∈ V ↦ ran (𝑥 ∈ ℤ, 𝑦 ∈ ((ℤRHom‘𝑓) “ (Unit‘𝑓)) ↦ ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r𝑓)((ℤRHom‘𝑓)‘𝑦))⟩))
18 zex 11978 . . . 4 ℤ ∈ V
193fvexi 6666 . . . . . 6 𝐿 ∈ V
2019cnvex 7616 . . . . 5 𝐿 ∈ V
21 imaexg 7606 . . . . 5 (𝐿 ∈ V → (𝐿 “ (Unit‘𝑅)) ∈ V)
2220, 21ax-mp 5 . . . 4 (𝐿 “ (Unit‘𝑅)) ∈ V
2318, 22mpoex 7764 . . 3 (𝑥 ∈ ℤ, 𝑦 ∈ (𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) ∈ V
2423rnex 7603 . 2 ran (𝑥 ∈ ℤ, 𝑦 ∈ (𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) ∈ V
2516, 17, 24fvmpt 6750 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 1538  wcel 2114  Vcvv 3469  cop 4545  ccnv 5531  ran crn 5533  cima 5535  cfv 6334  (class class class)co 7140  cmpo 7142   / cdiv 11286  cz 11969  1rcur 19242  Unitcui 19383  /rcdvr 19426  ℤRHomczrh 20191  ℚHomcqqh 31287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-neg 10862  df-z 11970  df-qqh 31288
This theorem is referenced by:  qqhval2  31297
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