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Theorem pellfundglb 36467
 Description: If a real is larger than the fundamental solution, there is a nontrivial solution less than it. (Contributed by Stefan O'Rear, 18-Sep-2014.)
Assertion
Ref Expression
pellfundglb ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ∃𝑥 ∈ (Pell1QR‘𝐷)((PellFund‘𝐷) ≤ 𝑥𝑥 < 𝐴))
Distinct variable groups:   𝑥,𝐷   𝑥,𝐴

Proof of Theorem pellfundglb
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 pellfundval 36462 . . . . . . 7 (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) = inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ))
213ad2ant1 1075 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → (PellFund‘𝐷) = inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ))
3 simp3 1056 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → (PellFund‘𝐷) < 𝐴)
42, 3eqbrtrrd 4607 . . . . 5 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ) < 𝐴)
5 pellfundre 36463 . . . . . . . 8 (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ ℝ)
653ad2ant1 1075 . . . . . . 7 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → (PellFund‘𝐷) ∈ ℝ)
72, 6eqeltrrd 2689 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ) ∈ ℝ)
8 simp2 1055 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → 𝐴 ∈ ℝ)
97, 8ltnled 10063 . . . . 5 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → (inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ) < 𝐴 ↔ ¬ 𝐴 ≤ inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < )))
104, 9mpbid 221 . . . 4 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ¬ 𝐴 ≤ inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ))
11 ssrab2 3650 . . . . . 6 {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ (Pell14QR‘𝐷)
12 pell14qrre 36439 . . . . . . . . 9 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑎 ∈ (Pell14QR‘𝐷)) → 𝑎 ∈ ℝ)
1312ex 449 . . . . . . . 8 (𝐷 ∈ (ℕ ∖ ◻NN) → (𝑎 ∈ (Pell14QR‘𝐷) → 𝑎 ∈ ℝ))
1413ssrdv 3574 . . . . . . 7 (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) ⊆ ℝ)
15143ad2ant1 1075 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → (Pell14QR‘𝐷) ⊆ ℝ)
1611, 15syl5ss 3579 . . . . 5 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ ℝ)
17 pell1qrss14 36450 . . . . . . . 8 (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷))
18173ad2ant1 1075 . . . . . . 7 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → (Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷))
19 pellqrex 36461 . . . . . . . 8 (𝐷 ∈ (ℕ ∖ ◻NN) → ∃𝑎 ∈ (Pell1QR‘𝐷)1 < 𝑎)
20193ad2ant1 1075 . . . . . . 7 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ∃𝑎 ∈ (Pell1QR‘𝐷)1 < 𝑎)
21 ssrexv 3630 . . . . . . 7 ((Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷) → (∃𝑎 ∈ (Pell1QR‘𝐷)1 < 𝑎 → ∃𝑎 ∈ (Pell14QR‘𝐷)1 < 𝑎))
2218, 20, 21sylc 63 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ∃𝑎 ∈ (Pell14QR‘𝐷)1 < 𝑎)
23 rabn0 3912 . . . . . 6 ({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ≠ ∅ ↔ ∃𝑎 ∈ (Pell14QR‘𝐷)1 < 𝑎)
2422, 23sylibr 223 . . . . 5 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ≠ ∅)
25 infmrgelbi 36460 . . . . . 6 ((({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ ℝ ∧ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ≠ ∅ ∧ 𝐴 ∈ ℝ) ∧ ∀𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}𝐴𝑥) → 𝐴 ≤ inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ))
2625ex 449 . . . . 5 (({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ ℝ ∧ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ≠ ∅ ∧ 𝐴 ∈ ℝ) → (∀𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}𝐴𝑥𝐴 ≤ inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < )))
2716, 24, 8, 26syl3anc 1318 . . . 4 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → (∀𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}𝐴𝑥𝐴 ≤ inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < )))
2810, 27mtod 188 . . 3 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ¬ ∀𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}𝐴𝑥)
29 rexnal 2978 . . 3 (∃𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ¬ 𝐴𝑥 ↔ ¬ ∀𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}𝐴𝑥)
3028, 29sylibr 223 . 2 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ∃𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ¬ 𝐴𝑥)
31 breq2 4587 . . . . . . . 8 (𝑎 = 𝑥 → (1 < 𝑎 ↔ 1 < 𝑥))
3231elrab 3331 . . . . . . 7 (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ↔ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥))
33 simprl 790 . . . . . . . . 9 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → 𝑥 ∈ (Pell14QR‘𝐷))
34 1red 9934 . . . . . . . . . 10 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → 1 ∈ ℝ)
35 simpl1 1057 . . . . . . . . . . 11 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → 𝐷 ∈ (ℕ ∖ ◻NN))
36 pell14qrre 36439 . . . . . . . . . . 11 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ (Pell14QR‘𝐷)) → 𝑥 ∈ ℝ)
3735, 33, 36syl2anc 691 . . . . . . . . . 10 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → 𝑥 ∈ ℝ)
38 simprr 792 . . . . . . . . . 10 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → 1 < 𝑥)
3934, 37, 38ltled 10064 . . . . . . . . 9 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → 1 ≤ 𝑥)
4033, 39jca 553 . . . . . . . 8 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 ≤ 𝑥))
41 elpell1qr2 36454 . . . . . . . . 9 (𝐷 ∈ (ℕ ∖ ◻NN) → (𝑥 ∈ (Pell1QR‘𝐷) ↔ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 ≤ 𝑥)))
4235, 41syl 17 . . . . . . . 8 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → (𝑥 ∈ (Pell1QR‘𝐷) ↔ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 ≤ 𝑥)))
4340, 42mpbird 246 . . . . . . 7 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → 𝑥 ∈ (Pell1QR‘𝐷))
4432, 43sylan2b 491 . . . . . 6 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ 𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}) → 𝑥 ∈ (Pell1QR‘𝐷))
4544adantrr 749 . . . . 5 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → 𝑥 ∈ (Pell1QR‘𝐷))
46 simpl1 1057 . . . . . . 7 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → 𝐷 ∈ (ℕ ∖ ◻NN))
47 simprl 790 . . . . . . . 8 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → 𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎})
4811, 47sseldi 3566 . . . . . . 7 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → 𝑥 ∈ (Pell14QR‘𝐷))
49 simpr 476 . . . . . . . . . . 11 ((𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥) → 1 < 𝑥)
5049a1i 11 . . . . . . . . . 10 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ((𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥) → 1 < 𝑥))
5132, 50syl5bi 231 . . . . . . . . 9 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} → 1 < 𝑥))
5251imp 444 . . . . . . . 8 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ 𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}) → 1 < 𝑥)
5352adantrr 749 . . . . . . 7 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → 1 < 𝑥)
54 pellfundlb 36466 . . . . . . 7 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥) → (PellFund‘𝐷) ≤ 𝑥)
5546, 48, 53, 54syl3anc 1318 . . . . . 6 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → (PellFund‘𝐷) ≤ 𝑥)
56 simprr 792 . . . . . . 7 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → ¬ 𝐴𝑥)
5715adantr 480 . . . . . . . . 9 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → (Pell14QR‘𝐷) ⊆ ℝ)
5857, 48sseldd 3569 . . . . . . . 8 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → 𝑥 ∈ ℝ)
59 simpl2 1058 . . . . . . . 8 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → 𝐴 ∈ ℝ)
6058, 59ltnled 10063 . . . . . . 7 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → (𝑥 < 𝐴 ↔ ¬ 𝐴𝑥))
6156, 60mpbird 246 . . . . . 6 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → 𝑥 < 𝐴)
6255, 61jca 553 . . . . 5 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → ((PellFund‘𝐷) ≤ 𝑥𝑥 < 𝐴))
6345, 62jca 553 . . . 4 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → (𝑥 ∈ (Pell1QR‘𝐷) ∧ ((PellFund‘𝐷) ≤ 𝑥𝑥 < 𝐴)))
6463ex 449 . . 3 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ((𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥) → (𝑥 ∈ (Pell1QR‘𝐷) ∧ ((PellFund‘𝐷) ≤ 𝑥𝑥 < 𝐴))))
6564reximdv2 2997 . 2 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → (∃𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ¬ 𝐴𝑥 → ∃𝑥 ∈ (Pell1QR‘𝐷)((PellFund‘𝐷) ≤ 𝑥𝑥 < 𝐴)))
6630, 65mpd 15 1 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ∃𝑥 ∈ (Pell1QR‘𝐷)((PellFund‘𝐷) ≤ 𝑥𝑥 < 𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583  ‘cfv 5804  infcinf 8230  ℝcr 9814  1c1 9816   < clt 9953   ≤ cle 9954  ℕcn 10897  ◻NNcsquarenn 36418  Pell1QRcpell1qr 36419  Pell14QRcpell14qr 36421  PellFundcpellfund 36422 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-ico 12052  df-fz 12198  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-dvds 14822  df-gcd 15055  df-numer 15281  df-denom 15282  df-squarenn 36423  df-pell1qr 36424  df-pell14qr 36425  df-pell1234qr 36426  df-pellfund 36427 This theorem is referenced by:  pellfundex  36468
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