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Theorem qqhval 30564
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 2827 . . . 4 (𝑓 = 𝑅 → ℤ = ℤ)
2 fveq2 6434 . . . . . . 7 (𝑓 = 𝑅 → (ℤRHom‘𝑓) = (ℤRHom‘𝑅))
3 qqhval.3 . . . . . . 7 𝐿 = (ℤRHom‘𝑅)
42, 3syl6eqr 2880 . . . . . 6 (𝑓 = 𝑅 → (ℤRHom‘𝑓) = 𝐿)
54cnveqd 5531 . . . . 5 (𝑓 = 𝑅(ℤRHom‘𝑓) = 𝐿)
6 fveq2 6434 . . . . 5 (𝑓 = 𝑅 → (Unit‘𝑓) = (Unit‘𝑅))
75, 6imaeq12d 5709 . . . 4 (𝑓 = 𝑅 → ((ℤRHom‘𝑓) “ (Unit‘𝑓)) = (𝐿 “ (Unit‘𝑅)))
8 fveq2 6434 . . . . . . 7 (𝑓 = 𝑅 → (/r𝑓) = (/r𝑅))
9 qqhval.1 . . . . . . 7 / = (/r𝑅)
108, 9syl6eqr 2880 . . . . . 6 (𝑓 = 𝑅 → (/r𝑓) = / )
114fveq1d 6436 . . . . . 6 (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑥) = (𝐿𝑥))
124fveq1d 6436 . . . . . 6 (𝑓 = 𝑅 → ((ℤRHom‘𝑓)‘𝑦) = (𝐿𝑦))
1310, 11, 12oveq123d 6927 . . . . 5 (𝑓 = 𝑅 → (((ℤRHom‘𝑓)‘𝑥)(/r𝑓)((ℤRHom‘𝑓)‘𝑦)) = ((𝐿𝑥) / (𝐿𝑦)))
1413opeq2d 4631 . . . 4 (𝑓 = 𝑅 → ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r𝑓)((ℤRHom‘𝑓)‘𝑦))⟩ = ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩)
151, 7, 14mpt2eq123dv 6978 . . 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 5586 . 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 30563 . 2 ℚHom = (𝑓 ∈ V ↦ ran (𝑥 ∈ ℤ, 𝑦 ∈ ((ℤRHom‘𝑓) “ (Unit‘𝑓)) ↦ ⟨(𝑥 / 𝑦), (((ℤRHom‘𝑓)‘𝑥)(/r𝑓)((ℤRHom‘𝑓)‘𝑦))⟩))
18 zex 11714 . . . 4 ℤ ∈ V
193fvexi 6448 . . . . . 6 𝐿 ∈ V
2019cnvex 7376 . . . . 5 𝐿 ∈ V
21 imaexg 7366 . . . . 5 (𝐿 ∈ V → (𝐿 “ (Unit‘𝑅)) ∈ V)
2220, 21ax-mp 5 . . . 4 (𝐿 “ (Unit‘𝑅)) ∈ V
2318, 22mpt2ex 7511 . . 3 (𝑥 ∈ ℤ, 𝑦 ∈ (𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) ∈ V
2423rnex 7363 . 2 ran (𝑥 ∈ ℤ, 𝑦 ∈ (𝐿 “ (Unit‘𝑅)) ↦ ⟨(𝑥 / 𝑦), ((𝐿𝑥) / (𝐿𝑦))⟩) ∈ V
2516, 17, 24fvmpt 6530 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 1658  wcel 2166  Vcvv 3415  cop 4404  ccnv 5342  ran crn 5344  cima 5346  cfv 6124  (class class class)co 6906  cmpt2 6908   / cdiv 11010  cz 11705  1rcur 18856  Unitcui 18994  /rcdvr 19037  ℤRHomczrh 20209  ℚHomcqqh 30562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-cnex 10309  ax-resscn 10310
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-1st 7429  df-2nd 7430  df-neg 10589  df-z 11706  df-qqh 30563
This theorem is referenced by:  qqhval2  30572
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