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Theorem pellfundlb 41925
Description: A nontrivial first quadrant solution is at least as large as the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Proof shortened by AV, 15-Sep-2020.)
Assertion
Ref Expression
pellfundlb ((𝐷 ∈ (β„• βˆ– β—»NN) ∧ 𝐴 ∈ (Pell14QRβ€˜π·) ∧ 1 < 𝐴) β†’ (PellFundβ€˜π·) ≀ 𝐴)
Proof of Theorem pellfundlb
Dummy variables π‘Ž 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pellfundval 41921 . . 3 (𝐷 ∈ (β„• βˆ– β—»NN) β†’ (PellFundβ€˜π·) = inf({π‘Ž ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘Ž}, ℝ, < ))
213ad2ant1 1132 . 2 ((𝐷 ∈ (β„• βˆ– β—»NN) ∧ 𝐴 ∈ (Pell14QRβ€˜π·) ∧ 1 < 𝐴) β†’ (PellFundβ€˜π·) = inf({π‘Ž ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘Ž}, ℝ, < ))
3 ssrab2 4078 . . . . 5 {π‘Ž ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘Ž} βŠ† (Pell14QRβ€˜π·)
4 pell14qrre 41898 . . . . . . 7 ((𝐷 ∈ (β„• βˆ– β—»NN) ∧ 𝑑 ∈ (Pell14QRβ€˜π·)) β†’ 𝑑 ∈ ℝ)
54ex 412 . . . . . 6 (𝐷 ∈ (β„• βˆ– β—»NN) β†’ (𝑑 ∈ (Pell14QRβ€˜π·) β†’ 𝑑 ∈ ℝ))
65ssrdv 3989 . . . . 5 (𝐷 ∈ (β„• βˆ– β—»NN) β†’ (Pell14QRβ€˜π·) βŠ† ℝ)
73, 6sstrid 3994 . . . 4 (𝐷 ∈ (β„• βˆ– β—»NN) β†’ {π‘Ž ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘Ž} βŠ† ℝ)
873ad2ant1 1132 . . 3 ((𝐷 ∈ (β„• βˆ– β—»NN) ∧ 𝐴 ∈ (Pell14QRβ€˜π·) ∧ 1 < 𝐴) β†’ {π‘Ž ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘Ž} βŠ† ℝ)
9 1re 11219 . . . 4 1 ∈ ℝ
10 breq2 5153 . . . . . . . 8 (π‘Ž = 𝑐 β†’ (1 < π‘Ž ↔ 1 < 𝑐))
1110elrab 3684 . . . . . . 7 (𝑐 ∈ {π‘Ž ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘Ž} ↔ (𝑐 ∈ (Pell14QRβ€˜π·) ∧ 1 < 𝑐))
12 pell14qrre 41898 . . . . . . . . 9 ((𝐷 ∈ (β„• βˆ– β—»NN) ∧ 𝑐 ∈ (Pell14QRβ€˜π·)) β†’ 𝑐 ∈ ℝ)
13 ltle 11307 . . . . . . . . 9 ((1 ∈ ℝ ∧ 𝑐 ∈ ℝ) β†’ (1 < 𝑐 β†’ 1 ≀ 𝑐))
149, 12, 13sylancr 586 . . . . . . . 8 ((𝐷 ∈ (β„• βˆ– β—»NN) ∧ 𝑐 ∈ (Pell14QRβ€˜π·)) β†’ (1 < 𝑐 β†’ 1 ≀ 𝑐))
1514expimpd 453 . . . . . . 7 (𝐷 ∈ (β„• βˆ– β—»NN) β†’ ((𝑐 ∈ (Pell14QRβ€˜π·) ∧ 1 < 𝑐) β†’ 1 ≀ 𝑐))
1611, 15biimtrid 241 . . . . . 6 (𝐷 ∈ (β„• βˆ– β—»NN) β†’ (𝑐 ∈ {π‘Ž ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘Ž} β†’ 1 ≀ 𝑐))
1716ralrimiv 3144 . . . . 5 (𝐷 ∈ (β„• βˆ– β—»NN) β†’ βˆ€π‘ ∈ {π‘Ž ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘Ž}1 ≀ 𝑐)
18173ad2ant1 1132 . . . 4 ((𝐷 ∈ (β„• βˆ– β—»NN) ∧ 𝐴 ∈ (Pell14QRβ€˜π·) ∧ 1 < 𝐴) β†’ βˆ€π‘ ∈ {π‘Ž ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘Ž}1 ≀ 𝑐)
19 breq1 5152 . . . . . 6 (𝑏 = 1 β†’ (𝑏 ≀ 𝑐 ↔ 1 ≀ 𝑐))
2019ralbidv 3176 . . . . 5 (𝑏 = 1 β†’ (βˆ€π‘ ∈ {π‘Ž ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘Ž}𝑏 ≀ 𝑐 ↔ βˆ€π‘ ∈ {π‘Ž ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘Ž}1 ≀ 𝑐))
2120rspcev 3613 . . . 4 ((1 ∈ ℝ ∧ βˆ€π‘ ∈ {π‘Ž ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘Ž}1 ≀ 𝑐) β†’ βˆƒπ‘ ∈ ℝ βˆ€π‘ ∈ {π‘Ž ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘Ž}𝑏 ≀ 𝑐)
229, 18, 21sylancr 586 . . 3 ((𝐷 ∈ (β„• βˆ– β—»NN) ∧ 𝐴 ∈ (Pell14QRβ€˜π·) ∧ 1 < 𝐴) β†’ βˆƒπ‘ ∈ ℝ βˆ€π‘ ∈ {π‘Ž ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘Ž}𝑏 ≀ 𝑐)
23 simp2 1136 . . . 4 ((𝐷 ∈ (β„• βˆ– β—»NN) ∧ 𝐴 ∈ (Pell14QRβ€˜π·) ∧ 1 < 𝐴) β†’ 𝐴 ∈ (Pell14QRβ€˜π·))
24 simp3 1137 . . . 4 ((𝐷 ∈ (β„• βˆ– β—»NN) ∧ 𝐴 ∈ (Pell14QRβ€˜π·) ∧ 1 < 𝐴) β†’ 1 < 𝐴)
25 breq2 5153 . . . . 5 (π‘Ž = 𝐴 β†’ (1 < π‘Ž ↔ 1 < 𝐴))
2625elrab 3684 . . . 4 (𝐴 ∈ {π‘Ž ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘Ž} ↔ (𝐴 ∈ (Pell14QRβ€˜π·) ∧ 1 < 𝐴))
2723, 24, 26sylanbrc 582 . . 3 ((𝐷 ∈ (β„• βˆ– β—»NN) ∧ 𝐴 ∈ (Pell14QRβ€˜π·) ∧ 1 < 𝐴) β†’ 𝐴 ∈ {π‘Ž ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘Ž})
28 infrelb 12204 . . 3 (({π‘Ž ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘Ž} βŠ† ℝ ∧ βˆƒπ‘ ∈ ℝ βˆ€π‘ ∈ {π‘Ž ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘Ž}𝑏 ≀ 𝑐 ∧ 𝐴 ∈ {π‘Ž ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘Ž}) β†’ inf({π‘Ž ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘Ž}, ℝ, < ) ≀ 𝐴)
298, 22, 27, 28syl3anc 1370 . 2 ((𝐷 ∈ (β„• βˆ– β—»NN) ∧ 𝐴 ∈ (Pell14QRβ€˜π·) ∧ 1 < 𝐴) β†’ inf({π‘Ž ∈ (Pell14QRβ€˜π·) ∣ 1 < π‘Ž}, ℝ, < ) ≀ 𝐴)
302, 29eqbrtrd 5171 1 ((𝐷 ∈ (β„• βˆ– β—»NN) ∧ 𝐴 ∈ (Pell14QRβ€˜π·) ∧ 1 < 𝐴) β†’ (PellFundβ€˜π·) ≀ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  βˆƒwrex 3069  {crab 3431   βˆ– cdif 3946   βŠ† wss 3949   class class class wbr 5149  β€˜cfv 6544  infcinf 9439  β„cr 11112  1c1 11114   < clt 11253   ≀ cle 11254  β„•cn 12217  β—»NNcsquarenn 41877  Pell14QRcpell14qr 41880  PellFundcpellfund 41881
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190  ax-pre-sup 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-er 8706  df-en 8943  df-dom 8944  df-sdom 8945  df-sup 9440  df-inf 9441  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-n0 12478  df-z 12564  df-pell14qr 41884  df-pell1234qr 41885  df-pellfund 41886
This theorem is referenced by:  pellfundglb  41926  pellfund14gap  41928  rmspecfund  41950
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