Theorem pellfundval 43326

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.

Step	Hyp	Ref	Expression
1		fveq2 6834	. . . 4 ⊢ (𝑎 = 𝐷 → (Pell14QR‘𝑎) = (Pell14QR‘𝐷))
2		rabeq 3404	. . . 4 ⊢ ((Pell14QR‘𝑎) = (Pell14QR‘𝐷) → {𝑥 ∈ (Pell14QR‘𝑎) ∣ 1 < 𝑥} = {𝑥 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑥})
3	1, 2	syl 17	. . 3 ⊢ (𝑎 = 𝐷 → {𝑥 ∈ (Pell14QR‘𝑎) ∣ 1 < 𝑥} = {𝑥 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑥})
4	3	infeq1d 9384	. 2 ⊢ (𝑎 = 𝐷 → inf({𝑥 ∈ (Pell14QR‘𝑎) ∣ 1 < 𝑥}, ℝ, < ) = inf({𝑥 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑥}, ℝ, < ))
5		df-pellfund 43291	. 2 ⊢ PellFund = (𝑎 ∈ (ℕ ∖ ◻NN) ↦ inf({𝑥 ∈ (Pell14QR‘𝑎) ∣ 1 < 𝑥}, ℝ, < ))
6		ltso 11217	. . 3 ⊢ < Or ℝ
7	6	infex 9401	. 2 ⊢ inf({𝑥 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑥}, ℝ, < ) ∈ V
8	4, 5, 7	fvmpt 6941	1 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) = inf({𝑥 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑥}, ℝ, < ))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {crab 3390   ∖ cdif 3887   class class class wbr 5086  ‘cfv 6492  infcinf 9347  ℝcr 11028  1c1 11030   < clt 11170  ℕcn 12165  ◻_NNcsquarenn 43282  Pell14QRcpell14qr 43285  PellFundcpellfund 43286

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-resscn 11086  ax-pre-lttri 11103  ax-pre-lttrn 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-po 5532  df-so 5533  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-sup 9348  df-inf 9349  df-pnf 11172  df-mnf 11173  df-ltxr 11175  df-pellfund 43291

This theorem is referenced by:  pellfundre  43327  pellfundge  43328  pellfundlb  43330  pellfundglb  43331