Theorem pellfundlb 42857

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 42853	. . 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 4033	. . . . 5 ⊢ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ (Pell14QR‘𝐷)
4		pell14qrre 42830	. . . . . . 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 3943	. . . . 5 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) ⊆ ℝ)
7	3, 6	sstrid 3949	. . . 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 11134	. . . 4 ⊢ 1 ∈ ℝ
10		breq2 5099	. . . . . . . 8 ⊢ (𝑎 = 𝑐 → (1 < 𝑎 ↔ 1 < 𝑐))
11	10	elrab 3650	. . . . . . 7 ⊢ (𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ↔ (𝑐 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑐))
12		pell14qrre 42830	. . . . . . . . 9 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑐 ∈ (Pell14QR‘𝐷)) → 𝑐 ∈ ℝ)
13		ltle 11222	. . . . . . . . 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 3120	. . . . 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 5098	. . . . . 6 ⊢ (𝑏 = 1 → (𝑏 ≤ 𝑐 ↔ 1 ≤ 𝑐))
20	19	ralbidv 3152	. . . . 5 ⊢ (𝑏 = 1 → (∀𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}𝑏 ≤ 𝑐 ↔ ∀𝑐 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}1 ≤ 𝑐))
21	20	rspcev 3579	. . . 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 5099	. . . . 5 ⊢ (𝑎 = 𝐴 → (1 < 𝑎 ↔ 1 < 𝐴))
26	25	elrab 3650	. . . 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 12128	. . 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 5117	1 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → (PellFund‘𝐷) ≤ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3396   ∖ cdif 3902   ⊆ wss 3905   class class class wbr 5095  ‘cfv 6486  infcinf 9350  ℝcr 11027  1c1 11029   < clt 11168   ≤ cle 11169  ℕcn 12146  ◻_NNcsquarenn 42809  Pell14QRcpell14qr 42812  PellFundcpellfund 42813

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105  ax-pre-sup 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  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 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-sup 9351  df-inf 9352  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-n0 12403  df-z 12490  df-pell14qr 42816  df-pell1234qr 42817  df-pellfund 42818

This theorem is referenced by:  pellfundglb  42858  pellfund14gap  42860  rmspecfund  42882