Theorem pellfundglb 42872

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) ≤ 𝑥 ∧ 𝑥 < 𝐴))

Distinct variable groups:   𝑥,𝐷   𝑥,𝐴

Proof of Theorem pellfundglb

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pellfundval 42867	. . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) = inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ))
2	1	3ad2ant1 1132	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → (PellFund‘𝐷) = inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ))
3		simp3 1137	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → (PellFund‘𝐷) < 𝐴)
4	2, 3	eqbrtrrd 5171	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ) < 𝐴)
5		pellfundre 42868	. . . . . . . 8 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ ℝ)
6	5	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → (PellFund‘𝐷) ∈ ℝ)
7	2, 6	eqeltrrd 2839	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ) ∈ ℝ)
8		simp2 1136	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → 𝐴 ∈ ℝ)
9	7, 8	ltnled 11405	. . . . 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, < )))
10	4, 9	mpbid 232	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ¬ 𝐴 ≤ inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ))
11		ssrab2 4089	. . . . . 6 ⊢ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ (Pell14QR‘𝐷)
12		pell14qrre 42844	. . . . . . . . 9 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑎 ∈ (Pell14QR‘𝐷)) → 𝑎 ∈ ℝ)
13	12	ex 412	. . . . . . . 8 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝑎 ∈ (Pell14QR‘𝐷) → 𝑎 ∈ ℝ))
14	13	ssrdv 4000	. . . . . . 7 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) ⊆ ℝ)
15	14	3ad2ant1 1132	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → (Pell14QR‘𝐷) ⊆ ℝ)
16	11, 15	sstrid 4006	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ ℝ)
17		pell1qrss14 42855	. . . . . . . 8 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷))
18	17	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → (Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷))
19		pellqrex 42866	. . . . . . . 8 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → ∃𝑎 ∈ (Pell1QR‘𝐷)1 < 𝑎)
20	19	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ∃𝑎 ∈ (Pell1QR‘𝐷)1 < 𝑎)
21		ssrexv 4064	. . . . . . 7 ⊢ ((Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷) → (∃𝑎 ∈ (Pell1QR‘𝐷)1 < 𝑎 → ∃𝑎 ∈ (Pell14QR‘𝐷)1 < 𝑎))
22	18, 20, 21	sylc 65	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ∃𝑎 ∈ (Pell14QR‘𝐷)1 < 𝑎)
23		rabn0 4394	. . . . . 6 ⊢ ({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ≠ ∅ ↔ ∃𝑎 ∈ (Pell14QR‘𝐷)1 < 𝑎)
24	22, 23	sylibr 234	. . . . 5 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ≠ ∅)
25		infmrgelbi 42865	. . . . . 6 ⊢ ((({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ ℝ ∧ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ≠ ∅ ∧ 𝐴 ∈ ℝ) ∧ ∀𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}𝐴 ≤ 𝑥) → 𝐴 ≤ inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < ))
26	25	ex 412	. . . . 5 ⊢ (({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ⊆ ℝ ∧ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ≠ ∅ ∧ 𝐴 ∈ ℝ) → (∀𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}𝐴 ≤ 𝑥 → 𝐴 ≤ inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < )))
27	16, 24, 8, 26	syl3anc 1370	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → (∀𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}𝐴 ≤ 𝑥 → 𝐴 ≤ inf({𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}, ℝ, < )))
28	10, 27	mtod 198	. . 3 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ¬ ∀𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}𝐴 ≤ 𝑥)
29		rexnal 3097	. . 3 ⊢ (∃𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ¬ 𝐴 ≤ 𝑥 ↔ ¬ ∀𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}𝐴 ≤ 𝑥)
30	28, 29	sylibr 234	. 2 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ∃𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ¬ 𝐴 ≤ 𝑥)
31		breq2 5151	. . . . 5 ⊢ (𝑎 = 𝑥 → (1 < 𝑎 ↔ 1 < 𝑥))
32	31	elrab 3694	. . . 4 ⊢ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ↔ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥))
33		simprl 771	. . . . . 6 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → 𝑥 ∈ (Pell14QR‘𝐷))
34		1red 11259	. . . . . . 7 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → 1 ∈ ℝ)
35		simpl1 1190	. . . . . . . 8 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → 𝐷 ∈ (ℕ ∖ ◻NN))
36		pell14qrre 42844	. . . . . . . 8 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ (Pell14QR‘𝐷)) → 𝑥 ∈ ℝ)
37	35, 33, 36	syl2anc 584	. . . . . . 7 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → 𝑥 ∈ ℝ)
38		simprr 773	. . . . . . 7 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → 1 < 𝑥)
39	34, 37, 38	ltled 11406	. . . . . 6 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → 1 ≤ 𝑥)
40	33, 39	jca 511	. . . . 5 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 ≤ 𝑥))
41		elpell1qr2 42859	. . . . . 6 ⊢ (𝐷 ∈ (ℕ ∖ ◻NN) → (𝑥 ∈ (Pell1QR‘𝐷) ↔ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 ≤ 𝑥)))
42	35, 41	syl 17	. . . . 5 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → (𝑥 ∈ (Pell1QR‘𝐷) ↔ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 ≤ 𝑥)))
43	40, 42	mpbird 257	. . . 4 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥)) → 𝑥 ∈ (Pell1QR‘𝐷))
44	32, 43	sylan2b 594	. . 3 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ 𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}) → 𝑥 ∈ (Pell1QR‘𝐷))
45	44	adantrr 717	. 2 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴 ≤ 𝑥)) → 𝑥 ∈ (Pell1QR‘𝐷))
46		simpl1 1190	. . . 4 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴 ≤ 𝑥)) → 𝐷 ∈ (ℕ ∖ ◻NN))
47		simprl 771	. . . . 5 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴 ≤ 𝑥)) → 𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎})
48	11, 47	sselid 3992	. . . 4 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴 ≤ 𝑥)) → 𝑥 ∈ (Pell14QR‘𝐷))
49		simpr 484	. . . . . . . 8 ⊢ ((𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥) → 1 < 𝑥)
50	49	a1i 11	. . . . . . 7 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ((𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥) → 1 < 𝑥))
51	32, 50	biimtrid 242	. . . . . 6 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} → 1 < 𝑥))
52	51	imp 406	. . . . 5 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ 𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎}) → 1 < 𝑥)
53	52	adantrr 717	. . . 4 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴 ≤ 𝑥)) → 1 < 𝑥)
54		pellfundlb 42871	. . . 4 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝑥 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝑥) → (PellFund‘𝐷) ≤ 𝑥)
55	46, 48, 53, 54	syl3anc 1370	. . 3 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴 ≤ 𝑥)) → (PellFund‘𝐷) ≤ 𝑥)
56		simprr 773	. . . 4 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴 ≤ 𝑥)) → ¬ 𝐴 ≤ 𝑥)
57	15	adantr 480	. . . . . 6 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴 ≤ 𝑥)) → (Pell14QR‘𝐷) ⊆ ℝ)
58	57, 48	sseldd 3995	. . . . 5 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴 ≤ 𝑥)) → 𝑥 ∈ ℝ)
59		simpl2 1191	. . . . 5 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴 ≤ 𝑥)) → 𝐴 ∈ ℝ)
60	58, 59	ltnled 11405	. . . 4 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴 ≤ 𝑥)) → (𝑥 < 𝐴 ↔ ¬ 𝐴 ≤ 𝑥))
61	56, 60	mpbird 257	. . 3 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴 ≤ 𝑥)) → 𝑥 < 𝐴)
62	55, 61	jca 511	. 2 ⊢ (((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) ∧ (𝑥 ∈ {𝑎 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑎} ∧ ¬ 𝐴 ≤ 𝑥)) → ((PellFund‘𝐷) ≤ 𝑥 ∧ 𝑥 < 𝐴))
63	30, 45, 62	reximssdv 3170	1 ⊢ ((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ∃𝑥 ∈ (Pell1QR‘𝐷)((PellFund‘𝐷) ≤ 𝑥 ∧ 𝑥 < 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1536   ∈ wcel 2105   ≠ wne 2937  ∀wral 3058  ∃wrex 3067  {crab 3432   ∖ cdif 3959   ⊆ wss 3962  ∅c0 4338   class class class wbr 5147  ‘cfv 6562  infcinf 9478  ℝcr 11151  1c1 11153   < clt 11292   ≤ cle 11293  ℕcn 12263  ◻_NNcsquarenn 42823  Pell1QRcpell1qr 42824  Pell14QRcpell14qr 42826  PellFundcpellfund 42827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-omul 8509  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-oi 9547  df-card 9976  df-acn 9979  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-q 12988  df-rp 13032  df-ico 13389  df-fz 13544  df-fl 13828  df-mod 13906  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-dvds 16287  df-gcd 16528  df-numer 16768  df-denom 16769  df-squarenn 42828  df-pell1qr 42829  df-pell14qr 42830  df-pell1234qr 42831  df-pellfund 42832

This theorem is referenced by:  pellfundex  42873