Theorem 2sqlem9 27412

Description: Lemma for 2sq 27415. (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 2745	. . . . . . . 8 ⊢ (𝑧 = 𝑁 → (𝑧 = ((𝑥↑2) + (𝑦↑2)) ↔ 𝑁 = ((𝑥↑2) + (𝑦↑2))))
3	2	anbi2d 637	. . . . . . 7 ⊢ (𝑧 = 𝑁 → (((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) ↔ ((𝑥 gcd 𝑦) = 1 ∧ 𝑁 = ((𝑥↑2) + (𝑦↑2)))))
4	3	2rexbidv 3206	. . . . . 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 7367	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (𝑥 gcd 𝑦) = (𝑢 gcd 𝑦))
6	5	eqeq1d 2743	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ((𝑥 gcd 𝑦) = 1 ↔ (𝑢 gcd 𝑦) = 1))
7		oveq1 7367	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → (𝑥↑2) = (𝑢↑2))
8	7	oveq1d 7375	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → ((𝑥↑2) + (𝑦↑2)) = ((𝑢↑2) + (𝑦↑2)))
9	8	eqeq2d 2752	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (𝑁 = ((𝑥↑2) + (𝑦↑2)) ↔ 𝑁 = ((𝑢↑2) + (𝑦↑2))))
10	6, 9	anbi12d 639	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (((𝑥 gcd 𝑦) = 1 ∧ 𝑁 = ((𝑥↑2) + (𝑦↑2))) ↔ ((𝑢 gcd 𝑦) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑦↑2)))))
11		oveq2 7368	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑢 gcd 𝑦) = (𝑢 gcd 𝑣))
12	11	eqeq1d 2743	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝑢 gcd 𝑦) = 1 ↔ (𝑢 gcd 𝑣) = 1))
13		oveq1 7367	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (𝑦↑2) = (𝑣↑2))
14	13	oveq2d 7376	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → ((𝑢↑2) + (𝑦↑2)) = ((𝑢↑2) + (𝑣↑2)))
15	14	eqeq2d 2752	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → (𝑁 = ((𝑢↑2) + (𝑦↑2)) ↔ 𝑁 = ((𝑢↑2) + (𝑣↑2))))
16	12, 15	anbi12d 639	. . . . . . 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 3224	. . . . . 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 289	. . . . 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 3620	. . . 4 ⊢ (𝑁 ∈ 𝑌 → (𝑁 ∈ 𝑌 ↔ ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2)))))
21	20	ibi 269	. . 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 486	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2)))) ∧ 𝑀 = 1) → 𝑀 = 1)
24		1z 12552	. . . . . . . . 9 ⊢ 1 ∈ ℤ
25		zgz 16899	. . . . . . . . 9 ⊢ (1 ∈ ℤ → 1 ∈ ℤ[i])
26	24, 25	ax-mp 5	. . . . . . . 8 ⊢ 1 ∈ ℤ[i]
27		sq1 14152	. . . . . . . . 9 ⊢ (1↑2) = 1
28	27	eqcomi 2750	. . . . . . . 8 ⊢ 1 = (1↑2)
29		fveq2 6831	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (abs‘𝑥) = (abs‘1))
30		abs1 15254	. . . . . . . . . . 11 ⊢ (abs‘1) = 1
31	29, 30	eqtrdi 2792	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (abs‘𝑥) = 1)
32	31	oveq1d 7375	. . . . . . . . 9 ⊢ (𝑥 = 1 → ((abs‘𝑥)↑2) = (1↑2))
33	32	rspceeqv 3585	. . . . . . . 8 ⊢ ((1 ∈ ℤ[i] ∧ 1 = (1↑2)) → ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
34	26, 28, 33	mp2an 699	. . . . . . 7 ⊢ ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2)
35		2sq.1	. . . . . . . 8 ⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))
36	35	2sqlem1 27402	. . . . . . 7 ⊢ (1 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
37	34, 36	mpbir 233	. . . . . 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 733	. . . . . . 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 733	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑀 ∥ 𝑁)
43	35, 19	2sqlem7 27409	. . . . . . . . . 10 ⊢ 𝑌 ⊆ (𝑆 ∩ ℕ)
44		inss2 4169	. . . . . . . . . 10 ⊢ (𝑆 ∩ ℕ) ⊆ ℕ
45	43, 44	sstri 3926	. . . . . . . . 9 ⊢ 𝑌 ⊆ ℕ
46	45, 1	sselid 3915	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
47	46	ad2antrr 733	. . . . . . 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 733	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑀 ∈ ℕ)
50		simprr 779	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑀 ≠ 1)
51		eluz2b3 12867	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ≥‘2) ↔ (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1))
52	49, 50, 51	sylanbrc 590	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑀 ∈ (ℤ≥‘2))
53		simplrl 783	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑢 ∈ ℤ)
54		simplrr 784	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑣 ∈ ℤ)
55		simprll 785	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → (𝑢 gcd 𝑣) = 1)
56		simprlr 786	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑁 = ((𝑢↑2) + (𝑣↑2)))
57		eqid 2741	. . . . . . 7 ⊢ (((𝑢 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (((𝑢 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
58		eqid 2741	. . . . . . 7 ⊢ (((𝑣 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (((𝑣 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
59		eqid 2741	. . . . . . 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 2741	. . . . . . 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 27411	. . . . . 6 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑀 ∈ 𝑆)
62	61	anassrs 469	. . . . 5 ⊢ ((((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2)))) ∧ 𝑀 ≠ 1) → 𝑀 ∈ 𝑆)
63	38, 62	pm2.61dane 3023	. . . 4 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2)))) → 𝑀 ∈ 𝑆)
64	63	ex 414	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) → 𝑀 ∈ 𝑆))
65	64	rexlimdvva 3198	. 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 397   = wceq 1548   ∈ wcel 2121  {cab 2719   ≠ wne 2936  ∀wral 3055  ∃wrex 3065   ∩ cin 3884   class class class wbr 5075   ↦ cmpt 5156  ran crn 5622  ‘cfv 6489  (class class class)co 7360  1c1 11034   + caddc 11036   − cmin 11372   / cdiv 11802  ℕcn 12169  2c2 12231  ℤcz 12519  ℤ_≥cuz 12783  ...cfz 13456   mod cmo 13823  ↑cexp 14018  abscabs 15191   ∥ cdvds 16216   gcd cgcd 16458  ℤ[i]cgz 16895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  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 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-nel 3041  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-sup 9349  df-inf 9350  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  df-gz 16896

This theorem is referenced by:  2sqlem10  27413