Theorem pellfundval 38230

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 6411	. . . 4 ⊢ (𝑎 = 𝐷 → (Pell14QR‘𝑎) = (Pell14QR‘𝐷))
2		rabeq 3376	. . . 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 8625	. 2 ⊢ (𝑎 = 𝐷 → inf({𝑥 ∈ (Pell14QR‘𝑎) ∣ 1 < 𝑥}, ℝ, < ) = inf({𝑥 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑥}, ℝ, < ))
5		df-pellfund 38195	. 2 ⊢ PellFund = (𝑎 ∈ (ℕ ∖ ◻NN) ↦ inf({𝑥 ∈ (Pell14QR‘𝑎) ∣ 1 < 𝑥}, ℝ, < ))
6		ltso 10408	. . 3 ⊢ < Or ℝ
7	6	infex 8641	. 2 ⊢ inf({𝑥 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑥}, ℝ, < ) ∈ V
8	4, 5, 7	fvmpt 6507	1 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) = inf({𝑥 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑥}, ℝ, < ))

Syntax hints:   → wi 4   = wceq 1653   ∈ wcel 2157  {crab 3093   ∖ cdif 3766   class class class wbr 4843  ‘cfv 6101  infcinf 8589  ℝcr 10223  1c1 10225   < clt 10363  ℕcn 11312  ◻_NNcsquarenn 38186  Pell14QRcpell14qr 38189  PellFundcpellfund 38190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-resscn 10281  ax-pre-lttri 10298  ax-pre-lttrn 10299

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-po 5233  df-so 5234  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-sup 8590  df-inf 8591  df-pnf 10365  df-mnf 10366  df-ltxr 10368  df-pellfund 38195

This theorem is referenced by:  pellfundre  38231  pellfundge  38232  pellfundlb  38234  pellfundglb  38235