Theorem 2sqlem9 27754

Description: Lemma for 2sq 27757. (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 2764	. . . . . . . 8 ⊢ (𝑧 = 𝑁 → (𝑧 = ((𝑥↑2) + (𝑦↑2)) ↔ 𝑁 = ((𝑥↑2) + (𝑦↑2))))
3	2	anbi2d 642	. . . . . . 7 ⊢ (𝑧 = 𝑁 → (((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) ↔ ((𝑥 gcd 𝑦) = 1 ∧ 𝑁 = ((𝑥↑2) + (𝑦↑2)))))
4	3	2rexbidv 3227	. . . . . 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 7427	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → (𝑥 gcd 𝑦) = (𝑢 gcd 𝑦))
6	5	eqeq1d 2762	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → ((𝑥 gcd 𝑦) = 1 ↔ (𝑢 gcd 𝑦) = 1))
7		oveq1 7427	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → (𝑥↑2) = (𝑢↑2))
8	7	oveq1d 7435	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → ((𝑥↑2) + (𝑦↑2)) = ((𝑢↑2) + (𝑦↑2)))
9	8	eqeq2d 2771	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (𝑁 = ((𝑥↑2) + (𝑦↑2)) ↔ 𝑁 = ((𝑢↑2) + (𝑦↑2))))
10	6, 9	anbi12d 644	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (((𝑥 gcd 𝑦) = 1 ∧ 𝑁 = ((𝑥↑2) + (𝑦↑2))) ↔ ((𝑢 gcd 𝑦) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑦↑2)))))
11		oveq2 7428	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → (𝑢 gcd 𝑦) = (𝑢 gcd 𝑣))
12	11	eqeq1d 2762	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝑢 gcd 𝑦) = 1 ↔ (𝑢 gcd 𝑣) = 1))
13		oveq1 7427	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (𝑦↑2) = (𝑣↑2))
14	13	oveq2d 7436	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → ((𝑢↑2) + (𝑦↑2)) = ((𝑢↑2) + (𝑣↑2)))
15	14	eqeq2d 2771	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → (𝑁 = ((𝑢↑2) + (𝑦↑2)) ↔ 𝑁 = ((𝑢↑2) + (𝑣↑2))))
16	12, 15	anbi12d 644	. . . . . . 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 3245	. . . . . 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 290	. . . . 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 3633	. . . 4 ⊢ (𝑁 ∈ 𝑌 → (𝑁 ∈ 𝑌 ↔ ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2)))))
21	20	ibi 270	. . 3 ⊢ (𝑁 ∈ 𝑌 → ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))))
22	1, 21	syl 18	. 2 ⊢ (𝜑 → ∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))))
23		simpr 490	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2)))) ∧ 𝑀 = 1) → 𝑀 = 1)
24		1z 12726	. . . . . . . . 9 ⊢ 1 ∈ ℤ
25		zgz 17111	. . . . . . . . 9 ⊢ (1 ∈ ℤ → 1 ∈ ℤ[i])
26	24, 25	ax-mp 5	. . . . . . . 8 ⊢ 1 ∈ ℤ[i]
27		sq1 14338	. . . . . . . . 9 ⊢ (1↑2) = 1
28	27	eqcomi 2769	. . . . . . . 8 ⊢ 1 = (1↑2)
29		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (abs‘𝑥) = (abs‘1))
30		abs1 15464	. . . . . . . . . . 11 ⊢ (abs‘1) = 1
31	29, 30	eqtrdi 2811	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (abs‘𝑥) = 1)
32	31	oveq1d 7435	. . . . . . . . 9 ⊢ (𝑥 = 1 → ((abs‘𝑥)↑2) = (1↑2))
33	32	rspceeqv 3598	. . . . . . . 8 ⊢ ((1 ∈ ℤ[i] ∧ 1 = (1↑2)) → ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
34	26, 28, 33	mp2an 705	. . . . . . 7 ⊢ ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2)
35		2sq.1	. . . . . . . 8 ⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))
36	35	2sqlem1 27744	. . . . . . 7 ⊢ (1 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
37	34, 36	mpbir 234	. . . . . 6 ⊢ 1 ∈ 𝑆
38	23, 37	eqeltrdi 2868	. . . . 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 739	. . . . . . 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 739	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑀 ∥ 𝑁)
43	35, 19	2sqlem7 27751	. . . . . . . . . 10 ⊢ 𝑌 ⊆ (𝑆 ∩ ℕ)
44		inss2 4134	. . . . . . . . . 10 ⊢ (𝑆 ∩ ℕ) ⊆ ℕ
45	43, 44	sstri 3917	. . . . . . . . 9 ⊢ 𝑌 ⊆ ℕ
46	45, 1	sselid 3906	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
47	46	ad2antrr 739	. . . . . . 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 739	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑀 ∈ ℕ)
50		simprr 785	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑀 ≠ 1)
51		eluz2b3 13049	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ≥‘2) ↔ (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1))
52	49, 50, 51	sylanbrc 595	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑀 ∈ (ℤ≥‘2))
53		simplrl 789	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑢 ∈ ℤ)
54		simplrr 790	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑣 ∈ ℤ)
55		simprll 791	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → (𝑢 gcd 𝑣) = 1)
56		simprlr 792	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑁 = ((𝑢↑2) + (𝑣↑2)))
57		eqid 2760	. . . . . . 7 ⊢ (((𝑢 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (((𝑢 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
58		eqid 2760	. . . . . . 7 ⊢ (((𝑣 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (((𝑣 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
59		eqid 2760	. . . . . . 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 2760	. . . . . . 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 27753	. . . . . 6 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑀 ∈ 𝑆)
62	61	anassrs 473	. . . . 5 ⊢ ((((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2)))) ∧ 𝑀 ≠ 1) → 𝑀 ∈ 𝑆)
63	38, 62	pm2.61dane 3042	. . . 4 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2)))) → 𝑀 ∈ 𝑆)
64	63	ex 418	. . 3 ⊢ ((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) → (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) → 𝑀 ∈ 𝑆))
65	64	rexlimdvva 3219	. 2 ⊢ (𝜑 → (∃𝑢 ∈ ℤ ∃𝑣 ∈ ℤ ((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) → 𝑀 ∈ 𝑆))
66	22, 65	mpd 16	1 ⊢ (𝜑 → 𝑀 ∈ 𝑆)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086   ∩ cin 4089   class class class wbr 5102   ↦ cmpt 5185  ran crn 5651  ‘cfv 6538  (class class class)co 7420  1c1 11201   + caddc 11203   − cmin 11541   / cdiv 11973  ℕcn 12335  2c2 12397  ℤcz 12693  ℤ_≥cuz 12965  ...cfz 13639   mod cmo 14009  ↑cexp 14204  abscabs 15401   ∥ cdvds 16422   gcd cgcd 16664  ℤ[i]cgz 17107

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-sup 9434  df-inf 9435  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-3 12406  df-n0 12607  df-z 12694  df-uz 12966  df-rp 13121  df-fz 13640  df-fl 13932  df-mod 14010  df-seq 14145  df-exp 14205  df-cj 15266  df-re 15267  df-im 15268  df-sqrt 15402  df-abs 15403  df-dvds 16423  df-gcd 16665  df-prm 16847  df-gz 17108

This theorem is used by:  2sqlem10  27755