Theorem pythagtriplem11 16791

Description: Lemma for pythagtrip 16800. Show that 𝑀 (which will eventually be closely related to the 𝑚 in the final statement) is a natural. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Hypothesis

Ref	Expression
pythagtriplem11.1	⊢ 𝑀 = (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2)

Assertion

Ref	Expression
pythagtriplem11	⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝑀 ∈ ℕ)

Proof of Theorem pythagtriplem11

Step	Hyp	Ref	Expression
1		pythagtriplem11.1	. 2 ⊢ 𝑀 = (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2)
2		pythagtriplem9 16790	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 + 𝐵)) ∈ ℕ)
3	2	nnzd 12545	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 + 𝐵)) ∈ ℤ)
4		simp3r 1204	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ¬ 2 ∥ 𝐴)
5		2z 12554	. . . . . . . . . 10 ⊢ 2 ∈ ℤ
6		nnz 12540	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ ℕ → 𝐶 ∈ ℤ)
7	6	3ad2ant3 1136	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐶 ∈ ℤ)
8		nnz 12540	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℤ)
9	8	3ad2ant2 1135	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐵 ∈ ℤ)
10	7, 9	zaddcld 12632	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 + 𝐵) ∈ ℤ)
11	10	3ad2ant1 1134	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 + 𝐵) ∈ ℤ)
12		nnz 12540	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
13	12	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → 𝐴 ∈ ℤ)
14	13	3ad2ant1 1134	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝐴 ∈ ℤ)
15		dvdsgcdb 16509	. . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ (𝐶 + 𝐵) ∈ ℤ ∧ 𝐴 ∈ ℤ) → ((2 ∥ (𝐶 + 𝐵) ∧ 2 ∥ 𝐴) ↔ 2 ∥ ((𝐶 + 𝐵) gcd 𝐴)))
16	5, 11, 14, 15	mp3an2i 1469	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((2 ∥ (𝐶 + 𝐵) ∧ 2 ∥ 𝐴) ↔ 2 ∥ ((𝐶 + 𝐵) gcd 𝐴)))
17	16	biimpar 477	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 2 ∥ ((𝐶 + 𝐵) gcd 𝐴)) → (2 ∥ (𝐶 + 𝐵) ∧ 2 ∥ 𝐴))
18	17	simprd 495	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 2 ∥ ((𝐶 + 𝐵) gcd 𝐴)) → 2 ∥ 𝐴)
19	4, 18	mtand 816	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ¬ 2 ∥ ((𝐶 + 𝐵) gcd 𝐴))
20		pythagtriplem7 16788	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 + 𝐵)) = ((𝐶 + 𝐵) gcd 𝐴))
21	20	breq2d 5098	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 ∥ (√‘(𝐶 + 𝐵)) ↔ 2 ∥ ((𝐶 + 𝐵) gcd 𝐴)))
22	19, 21	mtbird 325	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ¬ 2 ∥ (√‘(𝐶 + 𝐵)))
23		pythagtriplem8 16789	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 − 𝐵)) ∈ ℕ)
24	23	nnzd 12545	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 − 𝐵)) ∈ ℤ)
25	7, 9	zsubcld 12633	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (𝐶 − 𝐵) ∈ ℤ)
26	25	3ad2ant1 1134	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (𝐶 − 𝐵) ∈ ℤ)
27		dvdsgcdb 16509	. . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ (𝐶 − 𝐵) ∈ ℤ ∧ 𝐴 ∈ ℤ) → ((2 ∥ (𝐶 − 𝐵) ∧ 2 ∥ 𝐴) ↔ 2 ∥ ((𝐶 − 𝐵) gcd 𝐴)))
28	5, 26, 14, 27	mp3an2i 1469	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((2 ∥ (𝐶 − 𝐵) ∧ 2 ∥ 𝐴) ↔ 2 ∥ ((𝐶 − 𝐵) gcd 𝐴)))
29	28	biimpar 477	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 2 ∥ ((𝐶 − 𝐵) gcd 𝐴)) → (2 ∥ (𝐶 − 𝐵) ∧ 2 ∥ 𝐴))
30	29	simprd 495	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) ∧ 2 ∥ ((𝐶 − 𝐵) gcd 𝐴)) → 2 ∥ 𝐴)
31	4, 30	mtand 816	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ¬ 2 ∥ ((𝐶 − 𝐵) gcd 𝐴))
32		pythagtriplem6 16787	. . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 − 𝐵)) = ((𝐶 − 𝐵) gcd 𝐴))
33	32	breq2d 5098	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 ∥ (√‘(𝐶 − 𝐵)) ↔ 2 ∥ ((𝐶 − 𝐵) gcd 𝐴)))
34	31, 33	mtbird 325	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ¬ 2 ∥ (√‘(𝐶 − 𝐵)))
35		opoe 16327	. . . . 5 ⊢ ((((√‘(𝐶 + 𝐵)) ∈ ℤ ∧ ¬ 2 ∥ (√‘(𝐶 + 𝐵))) ∧ ((√‘(𝐶 − 𝐵)) ∈ ℤ ∧ ¬ 2 ∥ (√‘(𝐶 − 𝐵)))) → 2 ∥ ((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))))
36	3, 22, 24, 34, 35	syl22anc 839	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 2 ∥ ((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))))
37	2, 23	nnaddcld 12224	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) ∈ ℕ)
38	37	nnzd 12545	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) ∈ ℤ)
39		evend2 16321	. . . . 5 ⊢ (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) ∈ ℤ → (2 ∥ ((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) ↔ (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2) ∈ ℤ))
40	38, 39	syl 17	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (2 ∥ ((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) ↔ (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2) ∈ ℤ))
41	36, 40	mpbid 232	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2) ∈ ℤ)
42	2	nnred 12184	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 + 𝐵)) ∈ ℝ)
43	23	nnred 12184	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (√‘(𝐶 − 𝐵)) ∈ ℝ)
44	2	nngt0d 12221	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 < (√‘(𝐶 + 𝐵)))
45	23	nngt0d 12221	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 < (√‘(𝐶 − 𝐵)))
46	42, 43, 44, 45	addgt0d 11720	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 < ((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))))
47	37	nnred 12184	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → ((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) ∈ ℝ)
48		halfpos2 12401	. . . . 5 ⊢ (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) ∈ ℝ → (0 < ((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) ↔ 0 < (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2)))
49	47, 48	syl 17	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (0 < ((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) ↔ 0 < (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2)))
50	46, 49	mpbid 232	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 0 < (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2))
51		elnnz 12529	. . 3 ⊢ ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2) ∈ ℕ ↔ ((((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2) ∈ ℤ ∧ 0 < (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2)))
52	41, 50, 51	sylanbrc 584	. 2 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → (((√‘(𝐶 + 𝐵)) + (√‘(𝐶 − 𝐵))) / 2) ∈ ℕ)
53	1, 52	eqeltrid 2841	1 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) → 𝑀 ∈ ℕ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   class class class wbr 5086  ‘cfv 6494  (class class class)co 7362  ℝcr 11032  0cc0 11033  1c1 11034   + caddc 11036   < clt 11174   − cmin 11372   / cdiv 11802  ℕcn 12169  2c2 12231  ℤcz 12519  ↑cexp 14018  √csqrt 15190   ∥ cdvds 16216   gcd cgcd 16458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110  ax-pre-sup 11111

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-2o 8401  df-er 8638  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-sup 9350  df-inf 9351  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-div 11803  df-nn 12170  df-2 12239  df-3 12240  df-n0 12433  df-z 12520  df-uz 12784  df-rp 12938  df-fz 13457  df-fl 13746  df-mod 13824  df-seq 13959  df-exp 14019  df-cj 15056  df-re 15057  df-im 15058  df-sqrt 15192  df-abs 15193  df-dvds 16217  df-gcd 16459  df-prm 16636

This theorem is referenced by:  pythagtriplem18  16798  flt4lem5  43101  flt4lem5a  43103  flt4lem5c  43105  flt4lem5d  43106  flt4lem5e  43107