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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pellfundval | Structured version Visualization version GIF version |
Description: Value of the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 18-Sep-2014.) (Revised by AV, 17-Sep-2020.) |
Ref | Expression |
---|---|
pellfundval | ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) = inf({𝑥 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑥}, ℝ, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6888 | . . . 4 ⊢ (𝑎 = 𝐷 → (Pell14QR‘𝑎) = (Pell14QR‘𝐷)) | |
2 | rabeq 3446 | . . . 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 9468 | . 2 ⊢ (𝑎 = 𝐷 → inf({𝑥 ∈ (Pell14QR‘𝑎) ∣ 1 < 𝑥}, ℝ, < ) = inf({𝑥 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑥}, ℝ, < )) |
5 | df-pellfund 41568 | . 2 ⊢ PellFund = (𝑎 ∈ (ℕ ∖ ◻NN) ↦ inf({𝑥 ∈ (Pell14QR‘𝑎) ∣ 1 < 𝑥}, ℝ, < )) | |
6 | ltso 11290 | . . 3 ⊢ < Or ℝ | |
7 | 6 | infex 9484 | . 2 ⊢ inf({𝑥 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑥}, ℝ, < ) ∈ V |
8 | 4, 5, 7 | fvmpt 6995 | 1 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) = inf({𝑥 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑥}, ℝ, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {crab 3432 ∖ cdif 3944 class class class wbr 5147 ‘cfv 6540 infcinf 9432 ℝcr 11105 1c1 11107 < clt 11244 ℕcn 12208 ◻NNcsquarenn 41559 Pell14QRcpell14qr 41562 PellFundcpellfund 41563 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-resscn 11163 ax-pre-lttri 11180 ax-pre-lttrn 11181 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-ltxr 11249 df-pellfund 41568 |
This theorem is referenced by: pellfundre 41604 pellfundge 41605 pellfundlb 41607 pellfundglb 41608 |
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