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Theorem pellfundval 41608
Description: Value of the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 18-Sep-2014.) (Revised by AV, 17-Sep-2020.)
Assertion
Ref Expression
pellfundval (𝐷 ∈ (β„• βˆ– β—»NN) β†’ (PellFundβ€˜π·) = inf({π‘₯ ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘₯}, ℝ, < ))
Distinct variable group:   π‘₯,𝐷

Proof of Theorem pellfundval
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 fveq2 6891 . . . 4 (π‘Ž = 𝐷 β†’ (Pell14QRβ€˜π‘Ž) = (Pell14QRβ€˜π·))
2 rabeq 3446 . . . 4 ((Pell14QRβ€˜π‘Ž) = (Pell14QRβ€˜π·) β†’ {π‘₯ ∈ (Pell14QRβ€˜π‘Ž) ∣ 1 < π‘₯} = {π‘₯ ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘₯})
31, 2syl 17 . . 3 (π‘Ž = 𝐷 β†’ {π‘₯ ∈ (Pell14QRβ€˜π‘Ž) ∣ 1 < π‘₯} = {π‘₯ ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘₯})
43infeq1d 9471 . 2 (π‘Ž = 𝐷 β†’ inf({π‘₯ ∈ (Pell14QRβ€˜π‘Ž) ∣ 1 < π‘₯}, ℝ, < ) = inf({π‘₯ ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘₯}, ℝ, < ))
5 df-pellfund 41573 . 2 PellFund = (π‘Ž ∈ (β„• βˆ– β—»NN) ↦ inf({π‘₯ ∈ (Pell14QRβ€˜π‘Ž) ∣ 1 < π‘₯}, ℝ, < ))
6 ltso 11293 . . 3 < Or ℝ
76infex 9487 . 2 inf({π‘₯ ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘₯}, ℝ, < ) ∈ V
84, 5, 7fvmpt 6998 1 (𝐷 ∈ (β„• βˆ– β—»NN) β†’ (PellFundβ€˜π·) = inf({π‘₯ ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘₯}, ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  {crab 3432   βˆ– cdif 3945   class class class wbr 5148  β€˜cfv 6543  infcinf 9435  β„cr 11108  1c1 11110   < clt 11247  β„•cn 12211  β—»NNcsquarenn 41564  Pell14QRcpell14qr 41567  PellFundcpellfund 41568
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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-resscn 11166  ax-pre-lttri 11183  ax-pre-lttrn 11184
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-sup 9436  df-inf 9437  df-pnf 11249  df-mnf 11250  df-ltxr 11252  df-pellfund 41573
This theorem is referenced by:  pellfundre  41609  pellfundge  41610  pellfundlb  41612  pellfundglb  41613
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