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Theorem pellfundval 41250
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 6846 . . . 4 (π‘Ž = 𝐷 β†’ (Pell14QRβ€˜π‘Ž) = (Pell14QRβ€˜π·))
2 rabeq 3420 . . . 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 9421 . 2 (π‘Ž = 𝐷 β†’ inf({π‘₯ ∈ (Pell14QRβ€˜π‘Ž) ∣ 1 < π‘₯}, ℝ, < ) = inf({π‘₯ ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘₯}, ℝ, < ))
5 df-pellfund 41215 . 2 PellFund = (π‘Ž ∈ (β„• βˆ– β—»NN) ↦ inf({π‘₯ ∈ (Pell14QRβ€˜π‘Ž) ∣ 1 < π‘₯}, ℝ, < ))
6 ltso 11243 . . 3 < Or ℝ
76infex 9437 . 2 inf({π‘₯ ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘₯}, ℝ, < ) ∈ V
84, 5, 7fvmpt 6952 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 1542   ∈ wcel 2107  {crab 3406   βˆ– cdif 3911   class class class wbr 5109  β€˜cfv 6500  infcinf 9385  β„cr 11058  1c1 11060   < clt 11197  β„•cn 12161  β—»NNcsquarenn 41206  Pell14QRcpell14qr 41209  PellFundcpellfund 41210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-resscn 11116  ax-pre-lttri 11133  ax-pre-lttrn 11134
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-po 5549  df-so 5550  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-sup 9386  df-inf 9387  df-pnf 11199  df-mnf 11200  df-ltxr 11202  df-pellfund 41215
This theorem is referenced by:  pellfundre  41251  pellfundge  41252  pellfundlb  41254  pellfundglb  41255
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