Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  qqhval Structured version   Visualization version   GIF version

Theorem qqhval 32367
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 / = (/rβ€˜π‘…)
qqhval.2 1 = (1rβ€˜π‘…)
qqhval.3 𝐿 = (β„€RHomβ€˜π‘…)
Assertion
Ref Expression
qqhval (𝑅 ∈ V β†’ (β„šHomβ€˜π‘…) = ran (π‘₯ ∈ β„€, 𝑦 ∈ (◑𝐿 β€œ (Unitβ€˜π‘…)) ↦ ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩))
Distinct variable groups:   π‘₯,𝑦,𝑅   𝑦,𝐿
Allowed substitution hints:   / (π‘₯,𝑦)   1 (π‘₯,𝑦)   𝐿(π‘₯)

Proof of Theorem qqhval
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 eqidd 2738 . . . 4 (𝑓 = 𝑅 β†’ β„€ = β„€)
2 fveq2 6839 . . . . . . 7 (𝑓 = 𝑅 β†’ (β„€RHomβ€˜π‘“) = (β„€RHomβ€˜π‘…))
3 qqhval.3 . . . . . . 7 𝐿 = (β„€RHomβ€˜π‘…)
42, 3eqtr4di 2795 . . . . . 6 (𝑓 = 𝑅 β†’ (β„€RHomβ€˜π‘“) = 𝐿)
54cnveqd 5829 . . . . 5 (𝑓 = 𝑅 β†’ β—‘(β„€RHomβ€˜π‘“) = ◑𝐿)
6 fveq2 6839 . . . . 5 (𝑓 = 𝑅 β†’ (Unitβ€˜π‘“) = (Unitβ€˜π‘…))
75, 6imaeq12d 6012 . . . 4 (𝑓 = 𝑅 β†’ (β—‘(β„€RHomβ€˜π‘“) β€œ (Unitβ€˜π‘“)) = (◑𝐿 β€œ (Unitβ€˜π‘…)))
8 fveq2 6839 . . . . . . 7 (𝑓 = 𝑅 β†’ (/rβ€˜π‘“) = (/rβ€˜π‘…))
9 qqhval.1 . . . . . . 7 / = (/rβ€˜π‘…)
108, 9eqtr4di 2795 . . . . . 6 (𝑓 = 𝑅 β†’ (/rβ€˜π‘“) = / )
114fveq1d 6841 . . . . . 6 (𝑓 = 𝑅 β†’ ((β„€RHomβ€˜π‘“)β€˜π‘₯) = (πΏβ€˜π‘₯))
124fveq1d 6841 . . . . . 6 (𝑓 = 𝑅 β†’ ((β„€RHomβ€˜π‘“)β€˜π‘¦) = (πΏβ€˜π‘¦))
1310, 11, 12oveq123d 7372 . . . . 5 (𝑓 = 𝑅 β†’ (((β„€RHomβ€˜π‘“)β€˜π‘₯)(/rβ€˜π‘“)((β„€RHomβ€˜π‘“)β€˜π‘¦)) = ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦)))
1413opeq2d 4835 . . . 4 (𝑓 = 𝑅 β†’ ⟨(π‘₯ / 𝑦), (((β„€RHomβ€˜π‘“)β€˜π‘₯)(/rβ€˜π‘“)((β„€RHomβ€˜π‘“)β€˜π‘¦))⟩ = ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩)
151, 7, 14mpoeq123dv 7426 . . 3 (𝑓 = 𝑅 β†’ (π‘₯ ∈ β„€, 𝑦 ∈ (β—‘(β„€RHomβ€˜π‘“) β€œ (Unitβ€˜π‘“)) ↦ ⟨(π‘₯ / 𝑦), (((β„€RHomβ€˜π‘“)β€˜π‘₯)(/rβ€˜π‘“)((β„€RHomβ€˜π‘“)β€˜π‘¦))⟩) = (π‘₯ ∈ β„€, 𝑦 ∈ (◑𝐿 β€œ (Unitβ€˜π‘…)) ↦ ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩))
1615rneqd 5891 . 2 (𝑓 = 𝑅 β†’ ran (π‘₯ ∈ β„€, 𝑦 ∈ (β—‘(β„€RHomβ€˜π‘“) β€œ (Unitβ€˜π‘“)) ↦ ⟨(π‘₯ / 𝑦), (((β„€RHomβ€˜π‘“)β€˜π‘₯)(/rβ€˜π‘“)((β„€RHomβ€˜π‘“)β€˜π‘¦))⟩) = ran (π‘₯ ∈ β„€, 𝑦 ∈ (◑𝐿 β€œ (Unitβ€˜π‘…)) ↦ ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩))
17 df-qqh 32366 . 2 β„šHom = (𝑓 ∈ V ↦ ran (π‘₯ ∈ β„€, 𝑦 ∈ (β—‘(β„€RHomβ€˜π‘“) β€œ (Unitβ€˜π‘“)) ↦ ⟨(π‘₯ / 𝑦), (((β„€RHomβ€˜π‘“)β€˜π‘₯)(/rβ€˜π‘“)((β„€RHomβ€˜π‘“)β€˜π‘¦))⟩))
18 zex 12466 . . . 4 β„€ ∈ V
193fvexi 6853 . . . . . 6 𝐿 ∈ V
2019cnvex 7854 . . . . 5 ◑𝐿 ∈ V
21 imaexg 7844 . . . . 5 (◑𝐿 ∈ V β†’ (◑𝐿 β€œ (Unitβ€˜π‘…)) ∈ V)
2220, 21ax-mp 5 . . . 4 (◑𝐿 β€œ (Unitβ€˜π‘…)) ∈ V
2318, 22mpoex 8004 . . 3 (π‘₯ ∈ β„€, 𝑦 ∈ (◑𝐿 β€œ (Unitβ€˜π‘…)) ↦ ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩) ∈ V
2423rnex 7841 . 2 ran (π‘₯ ∈ β„€, 𝑦 ∈ (◑𝐿 β€œ (Unitβ€˜π‘…)) ↦ ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩) ∈ V
2516, 17, 24fvmpt 6945 1 (𝑅 ∈ V β†’ (β„šHomβ€˜π‘…) = ran (π‘₯ ∈ β„€, 𝑦 ∈ (◑𝐿 β€œ (Unitβ€˜π‘…)) ↦ ⟨(π‘₯ / 𝑦), ((πΏβ€˜π‘₯) / (πΏβ€˜π‘¦))⟩))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3443  βŸ¨cop 4590  β—‘ccnv 5630  ran crn 5632   β€œ cima 5634  β€˜cfv 6493  (class class class)co 7351   ∈ cmpo 7353   / cdiv 11770  β„€cz 12457  1rcur 19872  Unitcui 20021  /rcdvr 20064  β„€RHomczrh 20853  β„šHomcqqh 32365
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-cnex 11065  ax-resscn 11066
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-neg 11346  df-z 12458  df-qqh 32366
This theorem is referenced by:  qqhval2  32375
  Copyright terms: Public domain W3C validator