Theorem 2sqlem9 16123

Description: Lemma for 2sq . (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)))}
2sqlem9.5	⊢ (𝜑 → ∀𝑏 ∈ (1...(𝑀 − 1))∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))
2sqlem9.7	⊢ (𝜑 → 𝑀 ∥ 𝑁)
2sqlem9.6	⊢ (𝜑 → 𝑀 ∈ ℕ)
2sqlem9.4	⊢ (𝜑 → 𝑁 ∈ 𝑌)

Assertion

Ref	Expression
2sqlem9	⊢ (𝜑 → 𝑀 ∈ 𝑆)

Distinct variable groups:   𝑎,𝑏,𝑤,𝑥,𝑦,𝑧   𝜑,𝑥,𝑦   𝑀,𝑎,𝑏,𝑥,𝑦,𝑧   𝑆,𝑎,𝑏,𝑥,𝑦,𝑧   𝑥,𝑁,𝑦,𝑧   𝑌,𝑎,𝑏,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑧,𝑤,𝑎,𝑏)   𝑆(𝑤)   𝑀(𝑤)   𝑁(𝑤,𝑎,𝑏)   𝑌(𝑧,𝑤)

Proof of Theorem 2sqlem9

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2sqlem9.4	. . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑌)
2		eqeq1 2241	. . . . . . . 8 ⊢ (𝑧 = 𝑁 → (𝑧 = ((𝑥↑2) + (𝑦↑2)) ↔ 𝑁 = ((𝑥↑2) + (𝑦↑2))))
3	2	anbi2d 464	. . . . . . 7 ⊢ (𝑧 = 𝑁 → (((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) ↔ ((𝑥 gcd 𝑦) = 1 ∧ 𝑁 = ((𝑥↑2) + (𝑦↑2)))))
4	3	2rexbidv 2569	. . . . . 6 ⊢ (𝑧 = 𝑁 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑁 = ((𝑥↑2) + (𝑦↑2)))))
5		oveq1 6065	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (𝑥 gcd 𝑦) = (𝑢 gcd 𝑦))
6	5	eqeq1d 2243	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ((𝑥 gcd 𝑦) = 1 ↔ (𝑢 gcd 𝑦) = 1))
7		oveq1 6065	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → (𝑥↑2) = (𝑢↑2))
8	7	oveq1d 6073	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → ((𝑥↑2) + (𝑦↑2)) = ((𝑢↑2) + (𝑦↑2)))
9	8	eqeq2d 2246	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (𝑁 = ((𝑥↑2) + (𝑦↑2)) ↔ 𝑁 = ((𝑢↑2) + (𝑦↑2))))
10	6, 9	anbi12d 473	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (((𝑥 gcd 𝑦) = 1 ∧ 𝑁 = ((𝑥↑2) + (𝑦↑2))) ↔ ((𝑢 gcd 𝑦) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑦↑2)))))
11		oveq2 6066	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑢 gcd 𝑦) = (𝑢 gcd 𝑣))
12	11	eqeq1d 2243	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝑢 gcd 𝑦) = 1 ↔ (𝑢 gcd 𝑣) = 1))
13		oveq1 6065	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (𝑦↑2) = (𝑣↑2))
14	13	oveq2d 6074	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → ((𝑢↑2) + (𝑦↑2)) = ((𝑢↑2) + (𝑣↑2)))
15	14	eqeq2d 2246	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → (𝑁 = ((𝑢↑2) + (𝑦↑2)) ↔ 𝑁 = ((𝑢↑2) + (𝑣↑2))))
16	12, 15	anbi12d 473	. . . . . . 7 ⊢ (𝑦 = 𝑣 → (((𝑢 gcd 𝑦) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑦↑2))) ↔ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2)))))
17	10, 16	cbvrex2vw 2792	. . . . . 6 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑁 = ((𝑥↑2) + (𝑦↑2))) ↔ ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))))
18	4, 17	bitrdi 196	. . . . 5 ⊢ (𝑧 = 𝑁 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) ↔ ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2)))))
19		2sqlem7.2	. . . . 5 ⊢ 𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))}
20	18, 19	elab2g 2967	. . . 4 ⊢ (𝑁 ∈ 𝑌 → (𝑁 ∈ 𝑌 ↔ ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2)))))
21	20	ibi 176	. . 3 ⊢ (𝑁 ∈ 𝑌 → ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))))
22	1, 21	syl 14	. 2 ⊢ (𝜑 → ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))))
23		simpr 110	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2)))) ∧ 𝑀 = 1) → 𝑀 = 1)
24		1z 9620	. . . . . . . . 9 ⊢ 1 ∈ ℤ
25		zgz 13096	. . . . . . . . 9 ⊢ (1 ∈ ℤ → 1 ∈ ℤ[i])
26	24, 25	ax-mp 5	. . . . . . . 8 ⊢ 1 ∈ ℤ[i]
27		sq1 11019	. . . . . . . . 9 ⊢ (1↑2) = 1
28	27	eqcomi 2238	. . . . . . . 8 ⊢ 1 = (1↑2)
29		fveq2 5675	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (abs‘𝑥) = (abs‘1))
30		abs1 11782	. . . . . . . . . . 11 ⊢ (abs‘1) = 1
31	29, 30	eqtrdi 2283	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (abs‘𝑥) = 1)
32	31	oveq1d 6073	. . . . . . . . 9 ⊢ (𝑥 = 1 → ((abs‘𝑥)↑2) = (1↑2))
33	32	rspceeqv 2942	. . . . . . . 8 ⊢ ((1 ∈ ℤ[i] ∧ 1 = (1↑2)) → ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
34	26, 28, 33	mp2an 426	. . . . . . 7 ⊢ ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2)
35		2sq.1	. . . . . . . 8 ⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))
36	35	2sqlem1 16113	. . . . . . 7 ⊢ (1 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
37	34, 36	mpbir 146	. . . . . 6 ⊢ 1 ∈ 𝑆
38	23, 37	eqeltrdi 2325	. . . . 5 ⊢ ((((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2)))) ∧ 𝑀 = 1) → 𝑀 ∈ 𝑆)
39		2sqlem9.5	. . . . . . . 8 ⊢ (𝜑 → ∀𝑏 ∈ (1...(𝑀 − 1))∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))
40	39	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → ∀𝑏 ∈ (1...(𝑀 − 1))∀𝑎 ∈ 𝑌 (𝑏 ∥ 𝑎 → 𝑏 ∈ 𝑆))
41		2sqlem9.7	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∥ 𝑁)
42	41	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑀 ∥ 𝑁)
43	35, 19	2sqlem7 16120	. . . . . . . . . 10 ⊢ 𝑌 ⊆ (𝑆 ∩ ℕ)
44		inss2 3446	. . . . . . . . . 10 ⊢ (𝑆 ∩ ℕ) ⊆ ℕ
45	43, 44	sstri 3251	. . . . . . . . 9 ⊢ 𝑌 ⊆ ℕ
46	45, 1	sselid 3240	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
47	46	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑁 ∈ ℕ)
48		2sqlem9.6	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ)
49	48	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑀 ∈ ℕ)
50		simprr 533	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑀 ≠ 1)
51		eluz2b3 9954	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ≥‘2) ↔ (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1))
52	49, 50, 51	sylanbrc 417	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑀 ∈ (ℤ≥‘2))
53		simplrl 537	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑢 ∈ ℤ)
54		simplrr 538	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑣 ∈ ℤ)
55		simprll 539	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → (𝑢 gcd 𝑣) = 1)
56		simprlr 540	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑁 = ((𝑢↑2) + (𝑣↑2)))
57		eqid 2234	. . . . . . 7 ⊢ (((𝑢 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (((𝑢 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
58		eqid 2234	. . . . . . 7 ⊢ (((𝑣 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (((𝑣 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
59		eqid 2234	. . . . . . 7 ⊢ ((((𝑢 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) / ((((𝑢 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) gcd (((𝑣 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)))) = ((((𝑢 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) / ((((𝑢 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) gcd (((𝑣 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))))
60		eqid 2234	. . . . . . 7 ⊢ ((((𝑣 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) / ((((𝑢 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) gcd (((𝑣 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)))) = ((((𝑣 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) / ((((𝑢 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) gcd (((𝑣 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))))
61	35, 19, 40, 42, 47, 52, 53, 54, 55, 56, 57, 58, 59, 60	2sqlem8 16122	. . . . . 6 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑀 ∈ 𝑆)
62	61	anassrs 400	. . . . 5 ⊢ ((((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2)))) ∧ 𝑀 ≠ 1) → 𝑀 ∈ 𝑆)
63	48	nnzd 9717	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
64	63	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2)))) → 𝑀 ∈ ℤ)
65		zdceq 9670	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 1 ∈ ℤ) → DECID 𝑀 = 1)
66	64, 24, 65	sylancl 413	. . . . . 6 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2)))) → DECID 𝑀 = 1)
67		dcne 2425	. . . . . 6 ⊢ (DECID 𝑀 = 1 ↔ (𝑀 = 1 ∨ 𝑀 ≠ 1))
68	66, 67	sylib 122	. . . . 5 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2)))) → (𝑀 = 1 ∨ 𝑀 ≠ 1))
69	38, 62, 68	mpjaodan 806	. . . 4 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2)))) → 𝑀 ∈ 𝑆)
70	69	ex 115	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) → 𝑀 ∈ 𝑆))
71	70	rexlimdvva 2670	. 2 ⊢ (𝜑 → (∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) → 𝑀 ∈ 𝑆))
72	22, 71	mpd 13	1 ⊢ (𝜑 → 𝑀 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 716  DECID wdc 842   = wceq 1398   ∈ wcel 2205  {cab 2220   ≠ wne 2414  ∀wral 2522  ∃wrex 2523   ∩ cin 3213   class class class wbr 4114   ↦ cmpt 4176  ran crn 4755  ‘cfv 5357  (class class class)co 6058  1c1 8144   + caddc 8146   − cmin 8460   / cdiv 8963  ℕcn 9254  2c2 9305  ℤcz 9594  ℤ_≥cuz 9871  ...cfz 10361   mod cmo 10708  ↑cexp 10924  abscabs 11707   ∥ cdvds 12498   gcd cgcd 12674  ℤ[i]cgz 13092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-2o 6661  df-er 6780  df-en 6989  df-sup 7288  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-fl 10654  df-mod 10709  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-dvds 12499  df-gcd 12675  df-prm 12830  df-gz 13093

This theorem is referenced by:  2sqlem10  16124