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Theorem pellfundglb 39366
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 39361 . . . . . . 7 (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) = inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ))
213ad2ant1 1127 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → (PellFund‘𝐷) = inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ))
3 simp3 1132 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → (PellFund‘𝐷) < 𝐴)
42, 3eqbrtrrd 5087 . . . . 5 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ) < 𝐴)
5 pellfundre 39362 . . . . . . . 8 (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ ℝ)
653ad2ant1 1127 . . . . . . 7 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → (PellFund‘𝐷) ∈ ℝ)
72, 6eqeltrrd 2919 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ) ∈ ℝ)
8 simp2 1131 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → 𝐴 ∈ ℝ)
97, 8ltnled 10781 . . . . 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 233 . . . 4 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ¬ 𝐴 ≤ inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ))
11 ssrab2 4060 . . . . . 6 {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ (Pell14QR‘𝐷)
12 pell14qrre 39338 . . . . . . . . 9 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑎 ∈ (Pell14QR‘𝐷)) → 𝑎 ∈ ℝ)
1312ex 413 . . . . . . . 8 (𝐷 ∈ (ℕ ∖ ◻NN) → (𝑎 ∈ (Pell14QR‘𝐷) → 𝑎 ∈ ℝ))
1413ssrdv 3977 . . . . . . 7 (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) ⊆ ℝ)
15143ad2ant1 1127 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → (Pell14QR‘𝐷) ⊆ ℝ)
1611, 15sstrid 3982 . . . . 5 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ ℝ)
17 pell1qrss14 39349 . . . . . . . 8 (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷))
18173ad2ant1 1127 . . . . . . 7 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → (Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷))
19 pellqrex 39360 . . . . . . . 8 (𝐷 ∈ (ℕ ∖ ◻NN) → ∃𝑎 ∈ (Pell1QR‘𝐷)1 < 𝑎)
20193ad2ant1 1127 . . . . . . 7 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ∃𝑎 ∈ (Pell1QR‘𝐷)1 < 𝑎)
21 ssrexv 4038 . . . . . . 7 ((Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷) → (∃𝑎 ∈ (Pell1QR‘𝐷)1 < 𝑎 → ∃𝑎 ∈ (Pell14QR‘𝐷)1 < 𝑎))
2218, 20, 21sylc 65 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ∃𝑎 ∈ (Pell14QR‘𝐷)1 < 𝑎)
23 rabn0 4343 . . . . . 6 ({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ≠ ∅ ↔ ∃𝑎 ∈ (Pell14QR‘𝐷)1 < 𝑎)
2422, 23sylibr 235 . . . . 5 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ≠ ∅)
25 infmrgelbi 39359 . . . . . 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 413 . . . . 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 1365 . . . 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 199 . . 3 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ¬ ∀𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}𝐴𝑥)
29 rexnal 3243 . . 3 (∃𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ¬ 𝐴𝑥 ↔ ¬ ∀𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}𝐴𝑥)
3028, 29sylibr 235 . 2 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ∃𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ¬ 𝐴𝑥)
31 breq2 5067 . . . . 5 (𝑎 = 𝑥 → (1 < 𝑎 ↔ 1 < 𝑥))
3231elrab 3684 . . . 4 (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ↔ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥))
33 simprl 767 . . . . . 6 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → 𝑥 ∈ (Pell14QR‘𝐷))
34 1red 10636 . . . . . . 7 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → 1 ∈ ℝ)
35 simpl1 1185 . . . . . . . 8 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → 𝐷 ∈ (ℕ ∖ ◻NN))
36 pell14qrre 39338 . . . . . . . 8 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ (Pell14QR‘𝐷)) → 𝑥 ∈ ℝ)
3735, 33, 36syl2anc 584 . . . . . . 7 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → 𝑥 ∈ ℝ)
38 simprr 769 . . . . . . 7 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → 1 < 𝑥)
3934, 37, 38ltled 10782 . . . . . 6 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → 1 ≤ 𝑥)
4033, 39jca 512 . . . . 5 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 ≤ 𝑥))
41 elpell1qr2 39353 . . . . . 6 (𝐷 ∈ (ℕ ∖ ◻NN) → (𝑥 ∈ (Pell1QR‘𝐷) ↔ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 ≤ 𝑥)))
4235, 41syl 17 . . . . 5 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → (𝑥 ∈ (Pell1QR‘𝐷) ↔ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 ≤ 𝑥)))
4340, 42mpbird 258 . . . 4 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → 𝑥 ∈ (Pell1QR‘𝐷))
4432, 43sylan2b 593 . . 3 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ 𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}) → 𝑥 ∈ (Pell1QR‘𝐷))
4544adantrr 713 . 2 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → 𝑥 ∈ (Pell1QR‘𝐷))
46 simpl1 1185 . . . 4 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → 𝐷 ∈ (ℕ ∖ ◻NN))
47 simprl 767 . . . . 5 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → 𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎})
4811, 47sseldi 3969 . . . 4 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → 𝑥 ∈ (Pell14QR‘𝐷))
49 simpr 485 . . . . . . . 8 ((𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥) → 1 < 𝑥)
5049a1i 11 . . . . . . 7 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ((𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥) → 1 < 𝑥))
5132, 50syl5bi 243 . . . . . 6 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} → 1 < 𝑥))
5251imp 407 . . . . 5 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ 𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}) → 1 < 𝑥)
5352adantrr 713 . . . 4 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → 1 < 𝑥)
54 pellfundlb 39365 . . . 4 ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥) → (PellFund‘𝐷) ≤ 𝑥)
5546, 48, 53, 54syl3anc 1365 . . 3 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → (PellFund‘𝐷) ≤ 𝑥)
56 simprr 769 . . . 4 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → ¬ 𝐴𝑥)
5715adantr 481 . . . . . 6 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → (Pell14QR‘𝐷) ⊆ ℝ)
5857, 48sseldd 3972 . . . . 5 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → 𝑥 ∈ ℝ)
59 simpl2 1186 . . . . 5 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → 𝐴 ∈ ℝ)
6058, 59ltnled 10781 . . . 4 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → (𝑥 < 𝐴 ↔ ¬ 𝐴𝑥))
6156, 60mpbird 258 . . 3 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → 𝑥 < 𝐴)
6255, 61jca 512 . 2 (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴𝑥)) → ((PellFund‘𝐷) ≤ 𝑥𝑥 < 𝐴))
6330, 45, 62reximssdv 3281 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 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021  wral 3143  wrex 3144  {crab 3147  cdif 3937  wss 3940  c0 4295   class class class wbr 5063  cfv 6354  infcinf 8899  cr 10530  1c1 10532   < clt 10669  cle 10670  cn 11632  NNcsquarenn 39317  Pell1QRcpell1qr 39318  Pell14QRcpell14qr 39320  PellFundcpellfund 39321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-inf2 9098  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-omul 8103  df-er 8284  df-map 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-sup 8900  df-inf 8901  df-oi 8968  df-card 9362  df-acn 9365  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-xnn0 11962  df-z 11976  df-uz 12238  df-q 12343  df-rp 12385  df-ico 12739  df-fz 12888  df-fl 13157  df-mod 13233  df-seq 13365  df-exp 13425  df-hash 13686  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-dvds 15603  df-gcd 15839  df-numer 16070  df-denom 16071  df-squarenn 39322  df-pell1qr 39323  df-pell14qr 39324  df-pell1234qr 39325  df-pellfund 39326
This theorem is referenced by:  pellfundex  39367
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