Theorem pellfundlb 43001

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 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → (PellFund‘𝐷) ≤ 𝐴)

Proof of Theorem pellfundlb

Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pellfundval 42997	. . 3 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) = inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ))
2	1	3ad2ant1 1133	. 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → (PellFund‘𝐷) = inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ))
3		ssrab2 4029	. . . . 5 ⊢ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ (Pell14QR‘𝐷)
4		pell14qrre 42974	. . . . . . 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 3936	. . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) ⊆ ℝ)
7	3, 6	sstrid 3942	. . . 4 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ ℝ)
8	7	3ad2ant1 1133	. . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ ℝ)
9		1re 11119	. . . 4 ⊢ 1 ∈ ℝ
10		breq2 5097	. . . . . . . 8 ⊢ (𝑎 = 𝑐 → (1 < 𝑎 ↔ 1 < 𝑐))
11	10	elrab 3643	. . . . . . 7 ⊢ (𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ↔ (𝑐 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑐))
12		pell14qrre 42974	. . . . . . . . 9 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑐 ∈ (Pell14QR‘𝐷)) → 𝑐 ∈ ℝ)
13		ltle 11208	. . . . . . . . 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 3124	. . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ∀𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}1 ≤ 𝑐)
18	17	3ad2ant1 1133	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → ∀𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}1 ≤ 𝑐)
19		breq1 5096	. . . . . 6 ⊢ (𝑏 = 1 → (𝑏 ≤ 𝑐 ↔ 1 ≤ 𝑐))
20	19	ralbidv 3156	. . . . 5 ⊢ (𝑏 = 1 → (∀𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}𝑏 ≤ 𝑐 ↔ ∀𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}1 ≤ 𝑐))
21	20	rspcev 3573	. . . 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 1137	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → 𝐴 ∈ (Pell14QR‘𝐷))
24		simp3 1138	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → 1 < 𝐴)
25		breq2 5097	. . . . 5 ⊢ (𝑎 = 𝐴 → (1 < 𝑎 ↔ 1 < 𝐴))
26	25	elrab 3643	. . . 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 12114	. . 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 5115	1 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → (PellFund‘𝐷) ≤ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ∃wrex 3057  {crab 3396   ∖ cdif 3895   ⊆ wss 3898   class class class wbr 5093  ‘cfv 6486  infcinf 9332  ℝcr 11012  1c1 11014   < clt 11153   ≤ cle 11154  ℕcn 12132  ◻_NNcsquarenn 42953  Pell14QRcpell14qr 42956  PellFundcpellfund 42957

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090  ax-pre-sup 11091

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-sup 9333  df-inf 9334  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-n0 12389  df-z 12476  df-pell14qr 42960  df-pell1234qr 42961  df-pellfund 42962

This theorem is referenced by:  pellfundglb  43002  pellfund14gap  43004  rmspecfund  43026