Theorem pellfundval 43118

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 3413	. . . 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 9381	. 2 ⊢ (𝑎 = 𝐷 → inf({𝑥 ∈ (Pell14QR‘𝑎) ∣ 1 < 𝑥}, ℝ, < ) = inf({𝑥 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑥}, ℝ, < ))
5		df-pellfund 43083	. 2 ⊢ PellFund = (𝑎 ∈ (ℕ ∖ ◻NN) ↦ inf({𝑥 ∈ (Pell14QR‘𝑎) ∣ 1 < 𝑥}, ℝ, < ))
6		ltso 11213	. . 3 ⊢ < Or ℝ
7	6	infex 9398	. 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 1541   ∈ wcel 2113  {crab 3399   ∖ cdif 3898   class class class wbr 5098  ‘cfv 6492  infcinf 9344  ℝcr 11025  1c1 11027   < clt 11166  ℕcn 12145  ◻_NNcsquarenn 43074  Pell14QRcpell14qr 43077  PellFundcpellfund 43078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-resscn 11083  ax-pre-lttri 11100  ax-pre-lttrn 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  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 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9345  df-inf 9346  df-pnf 11168  df-mnf 11169  df-ltxr 11171  df-pellfund 43083

This theorem is referenced by:  pellfundre  43119  pellfundge  43120  pellfundlb  43122  pellfundglb  43123