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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 | β’ (π· β (β β β»NN) β (PellFundβπ·) = inf({π₯ β (Pell14QRβπ·) β£ 1 < π₯}, β, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6846 | . . . 4 β’ (π = π· β (Pell14QRβπ) = (Pell14QRβπ·)) | |
2 | rabeq 3420 | . . . 4 β’ ((Pell14QRβπ) = (Pell14QRβπ·) β {π₯ β (Pell14QRβπ) β£ 1 < π₯} = {π₯ β (Pell14QRβπ·) β£ 1 < π₯}) | |
3 | 1, 2 | syl 17 | . . 3 β’ (π = π· β {π₯ β (Pell14QRβπ) β£ 1 < π₯} = {π₯ β (Pell14QRβπ·) β£ 1 < π₯}) |
4 | 3 | infeq1d 9421 | . 2 β’ (π = π· β inf({π₯ β (Pell14QRβπ) β£ 1 < π₯}, β, < ) = inf({π₯ β (Pell14QRβπ·) β£ 1 < π₯}, β, < )) |
5 | df-pellfund 41215 | . 2 β’ PellFund = (π β (β β β»NN) β¦ inf({π₯ β (Pell14QRβπ) β£ 1 < π₯}, β, < )) | |
6 | ltso 11243 | . . 3 β’ < Or β | |
7 | 6 | infex 9437 | . 2 β’ inf({π₯ β (Pell14QRβπ·) β£ 1 < π₯}, β, < ) β V |
8 | 4, 5, 7 | fvmpt 6952 | 1 β’ (π· β (β β β»NN) β (PellFundβπ·) = inf({π₯ β (Pell14QRβπ·) β£ 1 < π₯}, β, < )) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 {crab 3406 β cdif 3911 class class class wbr 5109 βcfv 6500 infcinf 9385 βcr 11058 1c1 11060 < clt 11197 βcn 12161 β»NNcsquarenn 41206 Pell14QRcpell14qr 41209 PellFundcpellfund 41210 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-resscn 11116 ax-pre-lttri 11133 ax-pre-lttrn 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-po 5549 df-so 5550 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-sup 9386 df-inf 9387 df-pnf 11199 df-mnf 11200 df-ltxr 11202 df-pellfund 41215 |
This theorem is referenced by: pellfundre 41251 pellfundge 41252 pellfundlb 41254 pellfundglb 41255 |
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