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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 6891 | . . . 4 β’ (π = π· β (Pell14QRβπ) = (Pell14QRβπ·)) | |
2 | rabeq 3446 | . . . 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 9471 | . 2 β’ (π = π· β inf({π₯ β (Pell14QRβπ) β£ 1 < π₯}, β, < ) = inf({π₯ β (Pell14QRβπ·) β£ 1 < π₯}, β, < )) |
5 | df-pellfund 41573 | . 2 β’ PellFund = (π β (β β β»NN) β¦ inf({π₯ β (Pell14QRβπ) β£ 1 < π₯}, β, < )) | |
6 | ltso 11293 | . . 3 β’ < Or β | |
7 | 6 | infex 9487 | . 2 β’ inf({π₯ β (Pell14QRβπ·) β£ 1 < π₯}, β, < ) β V |
8 | 4, 5, 7 | fvmpt 6998 | 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 1541 β wcel 2106 {crab 3432 β cdif 3945 class class class wbr 5148 βcfv 6543 infcinf 9435 βcr 11108 1c1 11110 < clt 11247 βcn 12211 β»NNcsquarenn 41564 Pell14QRcpell14qr 41567 PellFundcpellfund 41568 |
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 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-resscn 11166 ax-pre-lttri 11183 ax-pre-lttrn 11184 |
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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-inf 9437 df-pnf 11249 df-mnf 11250 df-ltxr 11252 df-pellfund 41573 |
This theorem is referenced by: pellfundre 41609 pellfundge 41610 pellfundlb 41612 pellfundglb 41613 |
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