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Theorem pellfundval 37946
 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 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) = inf({𝑥 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑥}, ℝ, < ))
Distinct variable group:   𝑥,𝐷

Proof of Theorem pellfundval
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6352 . . . 4 (𝑎 = 𝐷 → (Pell14QR‘𝑎) = (Pell14QR‘𝐷))
2 rabeq 3332 . . . 4 ((Pell14QR‘𝑎) = (Pell14QR‘𝐷) → {𝑥 ∈ (Pell14QR‘𝑎) ∣ 1 < 𝑥} = {𝑥 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑥})
31, 2syl 17 . . 3 (𝑎 = 𝐷 → {𝑥 ∈ (Pell14QR‘𝑎) ∣ 1 < 𝑥} = {𝑥 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑥})
43infeq1d 8548 . 2 (𝑎 = 𝐷 → inf({𝑥 ∈ (Pell14QR‘𝑎) ∣ 1 < 𝑥}, ℝ, < ) = inf({𝑥 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑥}, ℝ, < ))
5 df-pellfund 37911 . 2 PellFund = (𝑎 ∈ (ℕ ∖ ◻NN) ↦ inf({𝑥 ∈ (Pell14QR‘𝑎) ∣ 1 < 𝑥}, ℝ, < ))
6 ltso 10310 . . 3 < Or ℝ
76infex 8564 . 2 inf({𝑥 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑥}, ℝ, < ) ∈ V
84, 5, 7fvmpt 6444 1 (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) = inf({𝑥 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑥}, ℝ, < ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139  {crab 3054   ∖ cdif 3712   class class class wbr 4804  ‘cfv 6049  infcinf 8512  ℝcr 10127  1c1 10129   < clt 10266  ℕcn 11212  ◻NNcsquarenn 37902  Pell14QRcpell14qr 37905  PellFundcpellfund 37906 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-resscn 10185  ax-pre-lttri 10202  ax-pre-lttrn 10203 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-sup 8513  df-inf 8514  df-pnf 10268  df-mnf 10269  df-ltxr 10271  df-pellfund 37911 This theorem is referenced by:  pellfundre  37947  pellfundge  37948  pellfundlb  37950  pellfundglb  37951
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