Theorem 2sqlem10 27372

Description: Lemma for 2sq 27374. Every factor of a "proper" sum of two squares (where the summands are coprime) is a sum of two squares. (Contributed by Mario Carneiro, 19-Jun-2015.)

Hypotheses

Ref	Expression
2sq.1	⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))
2sqlem7.2	⊢ 𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))}

Assertion

Ref	Expression
2sqlem10	⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴) → 𝐵 ∈ 𝑆)

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦   𝑥,𝑆,𝑦,𝑧   𝑥,𝑌,𝑦

Allowed substitution hints:   𝐴(𝑤)   𝐵(𝑧,𝑤)   𝑆(𝑤)   𝑌(𝑧,𝑤)

Proof of Theorem 2sqlem10

Dummy variables 𝑎 𝑏 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5105	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑏 ∥ 𝑎 ↔ 𝐵 ∥ 𝑎))
2		eleq1 2816	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ 𝑆 ↔ 𝐵 ∈ 𝑆))
3	1, 2	imbi12d 344	. . . . 5 ⊢ (𝑏 = 𝐵 → ((𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ (𝐵 ∥ 𝑎 → 𝐵 ∈ 𝑆)))
4	3	ralbidv 3156	. . . 4 ⊢ (𝑏 = 𝐵 → (∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑎 ∈ 𝑌 (𝐵 ∥ 𝑎 → 𝐵 ∈ 𝑆)))
5		oveq2 7377	. . . . . 6 ⊢ (𝑚 = 1 → (1...𝑚) = (1...1))
6	5	raleqdv 3296	. . . . 5 ⊢ (𝑚 = 1 → (∀𝑏 ∈ (1...𝑚)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑏 ∈ (1...1)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
7		oveq2 7377	. . . . . 6 ⊢ (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
8	7	raleqdv 3296	. . . . 5 ⊢ (𝑚 = 𝑛 → (∀𝑏 ∈ (1...𝑚)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
9		oveq2 7377	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → (1...𝑚) = (1...(𝑛 + 1)))
10	9	raleqdv 3296	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (∀𝑏 ∈ (1...𝑚)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑏 ∈ (1...(𝑛 + 1))∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
11		oveq2 7377	. . . . . 6 ⊢ (𝑚 = 𝐵 → (1...𝑚) = (1...𝐵))
12	11	raleqdv 3296	. . . . 5 ⊢ (𝑚 = 𝐵 → (∀𝑏 ∈ (1...𝑚)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑏 ∈ (1...𝐵)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
13		elfz1eq 13472	. . . . . . . . 9 ⊢ (𝑏 ∈ (1...1) → 𝑏 = 1)
14		1z 12539	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
15		zgz 16880	. . . . . . . . . . . 12 ⊢ (1 ∈ ℤ → 1 ∈ ℤ[i])
16	14, 15	ax-mp 5	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ[i]
17		sq1 14136	. . . . . . . . . . . 12 ⊢ (1↑2) = 1
18	17	eqcomi 2738	. . . . . . . . . . 11 ⊢ 1 = (1↑2)
19		fveq2 6840	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 1 → (abs‘𝑥) = (abs‘1))
20		abs1 15239	. . . . . . . . . . . . . 14 ⊢ (abs‘1) = 1
21	19, 20	eqtrdi 2780	. . . . . . . . . . . . 13 ⊢ (𝑥 = 1 → (abs‘𝑥) = 1)
22	21	oveq1d 7384	. . . . . . . . . . . 12 ⊢ (𝑥 = 1 → ((abs‘𝑥)↑2) = (1↑2))
23	22	rspceeqv 3608	. . . . . . . . . . 11 ⊢ ((1 ∈ ℤ[i] ∧ 1 = (1↑2)) → ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
24	16, 18, 23	mp2an 692	. . . . . . . . . 10 ⊢ ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2)
25		2sq.1	. . . . . . . . . . 11 ⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))
26	25	2sqlem1 27361	. . . . . . . . . 10 ⊢ (1 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
27	24, 26	mpbir 231	. . . . . . . . 9 ⊢ 1 ∈ 𝑆
28	13, 27	eqeltrdi 2836	. . . . . . . 8 ⊢ (𝑏 ∈ (1...1) → 𝑏 ∈ 𝑆)
29	28	a1d 25	. . . . . . 7 ⊢ (𝑏 ∈ (1...1) → (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))
30	29	ralrimivw 3129	. . . . . 6 ⊢ (𝑏 ∈ (1...1) → ∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))
31	30	rgen 3046	. . . . 5 ⊢ ∀𝑏 ∈ (1...1)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)
32		2sqlem7.2	. . . . . . . . . . . . 13 ⊢ 𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))}
33		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))
34		nncn 12170	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
35	34	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → 𝑛 ∈ ℂ)
36		ax-1cn 11102	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℂ
37		pncan 11403	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
38	35, 36, 37	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → ((𝑛 + 1) − 1) = 𝑛)
39	38	oveq2d 7385	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (1...((𝑛 + 1) − 1)) = (1...𝑛))
40	33, 39	raleqtrrdv 3300	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → ∀𝑏 ∈ (1...((𝑛 + 1) − 1))∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))
41		simprr 772	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (𝑛 + 1) ∥ 𝑚)
42		peano2nn 12174	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
43	42	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (𝑛 + 1) ∈ ℕ)
44		simprl 770	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → 𝑚 ∈ 𝑌)
45	25, 32, 40, 41, 43, 44	2sqlem9 27371	. . . . . . . . . . . 12 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ (𝑚 ∈ 𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (𝑛 + 1) ∈ 𝑆)
46	45	expr 456	. . . . . . . . . . 11 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) ∧ 𝑚 ∈ 𝑌) → ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆))
47	46	ralrimiva 3125	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)) → ∀𝑚 ∈ 𝑌 ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆))
48	47	ex 412	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) → ∀𝑚 ∈ 𝑌 ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆)))
49		breq2 5106	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑚 → ((𝑛 + 1) ∥ 𝑎 ↔ (𝑛 + 1) ∥ 𝑚))
50	49	imbi1d 341	. . . . . . . . . 10 ⊢ (𝑎 = 𝑚 → (((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆) ↔ ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆)))
51	50	cbvralvw 3213	. . . . . . . . 9 ⊢ (∀𝑎 ∈ 𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆) ↔ ∀𝑚 ∈ 𝑌 ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆))
52	48, 51	imbitrrdi 252	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) → ∀𝑎 ∈ 𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
53		ovex 7402	. . . . . . . . 9 ⊢ (𝑛 + 1) ∈ V
54		breq1 5105	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑛 + 1) → (𝑏 ∥ 𝑎 ↔ (𝑛 + 1) ∥ 𝑎))
55		eleq1 2816	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑛 + 1) → (𝑏 ∈ 𝑆 ↔ (𝑛 + 1) ∈ 𝑆))
56	54, 55	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑏 = (𝑛 + 1) → ((𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
57	56	ralbidv 3156	. . . . . . . . 9 ⊢ (𝑏 = (𝑛 + 1) → (∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑎 ∈ 𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
58	53, 57	ralsn 4641	. . . . . . . 8 ⊢ (∀𝑏 ∈ {(𝑛 + 1)}∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑎 ∈ 𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆))
59	52, 58	imbitrrdi 252	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) → ∀𝑏 ∈ {(𝑛 + 1)}∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
60	59	ancld 550	. . . . . 6 ⊢ (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) → (∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ∧ ∀𝑏 ∈ {(𝑛 + 1)}∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))))
61		elnnuz 12813	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ≥‘1))
62		fzsuc 13508	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ≥‘1) → (1...(𝑛 + 1)) = ((1...𝑛) ∪ {(𝑛 + 1)}))
63	61, 62	sylbi 217	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ → (1...(𝑛 + 1)) = ((1...𝑛) ∪ {(𝑛 + 1)}))
64	63	raleqdv 3296	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...(𝑛 + 1))∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ ∀𝑏 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
65		ralunb 4156	. . . . . . 7 ⊢ (∀𝑏 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ (∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ∧ ∀𝑏 ∈ {(𝑛 + 1)}∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
66	64, 65	bitrdi 287	. . . . . 6 ⊢ (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...(𝑛 + 1))∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ↔ (∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) ∧ ∀𝑏 ∈ {(𝑛 + 1)}∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))))
67	60, 66	sylibrd 259	. . . . 5 ⊢ (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆) → ∀𝑏 ∈ (1...(𝑛 + 1))∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆)))
68	6, 8, 10, 12, 31, 67	nnind 12180	. . . 4 ⊢ (𝐵 ∈ ℕ → ∀𝑏 ∈ (1...𝐵)∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))
69		elfz1end 13491	. . . . 5 ⊢ (𝐵 ∈ ℕ ↔ 𝐵 ∈ (1...𝐵))
70	69	biimpi 216	. . . 4 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ (1...𝐵))
71	4, 68, 70	rspcdva 3586	. . 3 ⊢ (𝐵 ∈ ℕ → ∀𝑎 ∈ 𝑌 (𝐵 ∥ 𝑎 → 𝐵 ∈ 𝑆))
72		breq2 5106	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝐵 ∥ 𝑎 ↔ 𝐵 ∥ 𝐴))
73	72	imbi1d 341	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝐵 ∥ 𝑎 → 𝐵 ∈ 𝑆) ↔ (𝐵 ∥ 𝐴 → 𝐵 ∈ 𝑆)))
74	73	rspcv 3581	. . 3 ⊢ (𝐴 ∈ 𝑌 → (∀𝑎 ∈ 𝑌 (𝐵 ∥ 𝑎 → 𝐵 ∈ 𝑆) → (𝐵 ∥ 𝐴 → 𝐵 ∈ 𝑆)))
75	71, 74	syl5 34	. 2 ⊢ (𝐴 ∈ 𝑌 → (𝐵 ∈ ℕ → (𝐵 ∥ 𝐴 → 𝐵 ∈ 𝑆)))
76	75	3imp 1110	1 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ ℕ ∧ 𝐵 ∥ 𝐴) → 𝐵 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053   ∪ cun 3909  {csn 4585   class class class wbr 5102   ↦ cmpt 5183  ran crn 5632  ‘cfv 6499  (class class class)co 7369  ℂcc 11042  1c1 11045   + caddc 11047   − cmin 11381  ℕcn 12162  2c2 12217  ℤcz 12505  ℤ_≥cuz 12769  ...cfz 13444  ↑cexp 14002  abscabs 15176   ∥ cdvds 16198   gcd cgcd 16440  ℤ[i]cgz 16876

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 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-pre-sup 11122

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 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9369  df-inf 9370  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-div 11812  df-nn 12163  df-2 12225  df-3 12226  df-n0 12419  df-z 12506  df-uz 12770  df-rp 12928  df-fz 13445  df-fl 13730  df-mod 13808  df-seq 13943  df-exp 14003  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-dvds 16199  df-gcd 16441  df-prm 16618  df-gz 16877

This theorem is referenced by:  2sqlem11  27373