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| Mirrors > Home > MPE Home > Th. List > Mathboxes > pellfundlb | Structured version Visualization version GIF version | ||
| 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.) |
| Ref | Expression |
|---|---|
| pellfundlb | ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → (PellFund‘𝐷) ≤ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pellfundval 42891 | . . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) = inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < )) | |
| 2 | 1 | 3ad2ant1 1134 | . 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → (PellFund‘𝐷) = inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < )) |
| 3 | ssrab2 4080 | . . . . 5 ⊢ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ (Pell14QR‘𝐷) | |
| 4 | pell14qrre 42868 | . . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑑 ∈ (Pell14QR‘𝐷)) → 𝑑 ∈ ℝ) | |
| 5 | 4 | ex 412 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝑑 ∈ (Pell14QR‘𝐷) → 𝑑 ∈ ℝ)) |
| 6 | 5 | ssrdv 3989 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) ⊆ ℝ) |
| 7 | 3, 6 | sstrid 3995 | . . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ ℝ) |
| 8 | 7 | 3ad2ant1 1134 | . . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ ℝ) |
| 9 | 1re 11261 | . . . 4 ⊢ 1 ∈ ℝ | |
| 10 | breq2 5147 | . . . . . . . 8 ⊢ (𝑎 = 𝑐 → (1 < 𝑎 ↔ 1 < 𝑐)) | |
| 11 | 10 | elrab 3692 | . . . . . . 7 ⊢ (𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ↔ (𝑐 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑐)) |
| 12 | pell14qrre 42868 | . . . . . . . . 9 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑐 ∈ (Pell14QR‘𝐷)) → 𝑐 ∈ ℝ) | |
| 13 | ltle 11349 | . . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ 𝑐 ∈ ℝ) → (1 < 𝑐 → 1 ≤ 𝑐)) | |
| 14 | 9, 12, 13 | sylancr 587 | . . . . . . . 8 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑐 ∈ (Pell14QR‘𝐷)) → (1 < 𝑐 → 1 ≤ 𝑐)) |
| 15 | 14 | expimpd 453 | . . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ((𝑐 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑐) → 1 ≤ 𝑐)) |
| 16 | 11, 15 | biimtrid 242 | . . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} → 1 ≤ 𝑐)) |
| 17 | 16 | ralrimiv 3145 | . . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ∀𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}1 ≤ 𝑐) |
| 18 | 17 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → ∀𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}1 ≤ 𝑐) |
| 19 | breq1 5146 | . . . . . 6 ⊢ (𝑏 = 1 → (𝑏 ≤ 𝑐 ↔ 1 ≤ 𝑐)) | |
| 20 | 19 | ralbidv 3178 | . . . . 5 ⊢ (𝑏 = 1 → (∀𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}𝑏 ≤ 𝑐 ↔ ∀𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}1 ≤ 𝑐)) |
| 21 | 20 | rspcev 3622 | . . . 4 ⊢ ((1 ∈ ℝ ∧ ∀𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}1 ≤ 𝑐) → ∃𝑏 ∈ ℝ ∀𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}𝑏 ≤ 𝑐) |
| 22 | 9, 18, 21 | sylancr 587 | . . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → ∃𝑏 ∈ ℝ ∀𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}𝑏 ≤ 𝑐) |
| 23 | simp2 1138 | . . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → 𝐴 ∈ (Pell14QR‘𝐷)) | |
| 24 | simp3 1139 | . . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → 1 < 𝐴) | |
| 25 | breq2 5147 | . . . . 5 ⊢ (𝑎 = 𝐴 → (1 < 𝑎 ↔ 1 < 𝐴)) | |
| 26 | 25 | elrab 3692 | . . . 4 ⊢ (𝐴 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ↔ (𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴)) |
| 27 | 23, 24, 26 | sylanbrc 583 | . . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → 𝐴 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}) |
| 28 | infrelb 12253 | . . 3 ⊢ (({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ ℝ ∧ ∃𝑏 ∈ ℝ ∀𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}𝑏 ≤ 𝑐 ∧ 𝐴 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}) → inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ) ≤ 𝐴) | |
| 29 | 8, 22, 27, 28 | syl3anc 1373 | . 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ) ≤ 𝐴) |
| 30 | 2, 29 | eqbrtrd 5165 | 1 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → (PellFund‘𝐷) ≤ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 {crab 3436 ∖ cdif 3948 ⊆ wss 3951 class class class wbr 5143 ‘cfv 6561 infcinf 9481 ℝcr 11154 1c1 11156 < clt 11295 ≤ cle 11296 ℕcn 12266 ◻NNcsquarenn 42847 Pell14QRcpell14qr 42850 PellFundcpellfund 42851 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-pell14qr 42854 df-pell1234qr 42855 df-pellfund 42856 |
| This theorem is referenced by: pellfundglb 42896 pellfund14gap 42898 rmspecfund 42920 |
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