Theorem 2sqlem9 27471

Description: Lemma for 2sq 27474. (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 2741	. . . . . . . 8 ⊢ (𝑧 = 𝑁 → (𝑧 = ((𝑥↑2) + (𝑦↑2)) ↔ 𝑁 = ((𝑥↑2) + (𝑦↑2))))
3	2	anbi2d 630	. . . . . . 7 ⊢ (𝑧 = 𝑁 → (((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) ↔ ((𝑥 gcd 𝑦) = 1 ∧ 𝑁 = ((𝑥↑2) + (𝑦↑2)))))
4	3	2rexbidv 3222	. . . . . 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 7438	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (𝑥 gcd 𝑦) = (𝑢 gcd 𝑦))
6	5	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ((𝑥 gcd 𝑦) = 1 ↔ (𝑢 gcd 𝑦) = 1))
7		oveq1 7438	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → (𝑥↑2) = (𝑢↑2))
8	7	oveq1d 7446	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → ((𝑥↑2) + (𝑦↑2)) = ((𝑢↑2) + (𝑦↑2)))
9	8	eqeq2d 2748	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (𝑁 = ((𝑥↑2) + (𝑦↑2)) ↔ 𝑁 = ((𝑢↑2) + (𝑦↑2))))
10	6, 9	anbi12d 632	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (((𝑥 gcd 𝑦) = 1 ∧ 𝑁 = ((𝑥↑2) + (𝑦↑2))) ↔ ((𝑢 gcd 𝑦) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑦↑2)))))
11		oveq2 7439	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑢 gcd 𝑦) = (𝑢 gcd 𝑣))
12	11	eqeq1d 2739	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝑢 gcd 𝑦) = 1 ↔ (𝑢 gcd 𝑣) = 1))
13		oveq1 7438	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (𝑦↑2) = (𝑣↑2))
14	13	oveq2d 7447	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → ((𝑢↑2) + (𝑦↑2)) = ((𝑢↑2) + (𝑣↑2)))
15	14	eqeq2d 2748	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → (𝑁 = ((𝑢↑2) + (𝑦↑2)) ↔ 𝑁 = ((𝑢↑2) + (𝑣↑2))))
16	12, 15	anbi12d 632	. . . . . . 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 3242	. . . . . 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 287	. . . . 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 3680	. . . 4 ⊢ (𝑁 ∈ 𝑌 → (𝑁 ∈ 𝑌 ↔ ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2)))))
21	20	ibi 267	. . 3 ⊢ (𝑁 ∈ 𝑌 → ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))))
22	1, 21	syl 17	. 2 ⊢ (𝜑 → ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))))
23		simpr 484	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2)))) ∧ 𝑀 = 1) → 𝑀 = 1)
24		1z 12647	. . . . . . . . 9 ⊢ 1 ∈ ℤ
25		zgz 16971	. . . . . . . . 9 ⊢ (1 ∈ ℤ → 1 ∈ ℤ[i])
26	24, 25	ax-mp 5	. . . . . . . 8 ⊢ 1 ∈ ℤ[i]
27		sq1 14234	. . . . . . . . 9 ⊢ (1↑2) = 1
28	27	eqcomi 2746	. . . . . . . 8 ⊢ 1 = (1↑2)
29		fveq2 6906	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (abs‘𝑥) = (abs‘1))
30		abs1 15336	. . . . . . . . . . 11 ⊢ (abs‘1) = 1
31	29, 30	eqtrdi 2793	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (abs‘𝑥) = 1)
32	31	oveq1d 7446	. . . . . . . . 9 ⊢ (𝑥 = 1 → ((abs‘𝑥)↑2) = (1↑2))
33	32	rspceeqv 3645	. . . . . . . 8 ⊢ ((1 ∈ ℤ[i] ∧ 1 = (1↑2)) → ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
34	26, 28, 33	mp2an 692	. . . . . . 7 ⊢ ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2)
35		2sq.1	. . . . . . . 8 ⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))
36	35	2sqlem1 27461	. . . . . . 7 ⊢ (1 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
37	34, 36	mpbir 231	. . . . . 6 ⊢ 1 ∈ 𝑆
38	23, 37	eqeltrdi 2849	. . . . 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 726	. . . . . . 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 726	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑀 ∥ 𝑁)
43	35, 19	2sqlem7 27468	. . . . . . . . . 10 ⊢ 𝑌 ⊆ (𝑆 ∩ ℕ)
44		inss2 4238	. . . . . . . . . 10 ⊢ (𝑆 ∩ ℕ) ⊆ ℕ
45	43, 44	sstri 3993	. . . . . . . . 9 ⊢ 𝑌 ⊆ ℕ
46	45, 1	sselid 3981	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
47	46	ad2antrr 726	. . . . . . 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 726	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑀 ∈ ℕ)
50		simprr 773	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑀 ≠ 1)
51		eluz2b3 12964	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ≥‘2) ↔ (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1))
52	49, 50, 51	sylanbrc 583	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑀 ∈ (ℤ≥‘2))
53		simplrl 777	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑢 ∈ ℤ)
54		simplrr 778	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑣 ∈ ℤ)
55		simprll 779	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → (𝑢 gcd 𝑣) = 1)
56		simprlr 780	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑁 = ((𝑢↑2) + (𝑣↑2)))
57		eqid 2737	. . . . . . 7 ⊢ (((𝑢 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (((𝑢 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
58		eqid 2737	. . . . . . 7 ⊢ (((𝑣 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (((𝑣 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
59		eqid 2737	. . . . . . 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 2737	. . . . . . 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 27470	. . . . . 6 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑀 ∈ 𝑆)
62	61	anassrs 467	. . . . 5 ⊢ ((((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2)))) ∧ 𝑀 ≠ 1) → 𝑀 ∈ 𝑆)
63	38, 62	pm2.61dane 3029	. . . 4 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2)))) → 𝑀 ∈ 𝑆)
64	63	ex 412	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) → 𝑀 ∈ 𝑆))
65	64	rexlimdvva 3213	. 2 ⊢ (𝜑 → (∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) → 𝑀 ∈ 𝑆))
66	22, 65	mpd 15	1 ⊢ (𝜑 → 𝑀 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ∩ cin 3950   class class class wbr 5143   ↦ cmpt 5225  ran crn 5686  ‘cfv 6561  (class class class)co 7431  1c1 11156   + caddc 11158   − cmin 11492   / cdiv 11920  ℕcn 12266  2c2 12321  ℤcz 12613  ℤ_≥cuz 12878  ...cfz 13547   mod cmo 13909  ↑cexp 14102  abscabs 15273   ∥ cdvds 16290   gcd cgcd 16531  ℤ[i]cgz 16967

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-dvds 16291  df-gcd 16532  df-prm 16709  df-gz 16968

This theorem is referenced by:  2sqlem10  27472