Theorem 2sqlem9 27411

Description: Lemma for 2sq 27414. (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 631	. . . . . . 7 ⊢ (𝑧 = 𝑁 → (((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2))) ↔ ((𝑥 gcd 𝑦) = 1 ∧ 𝑁 = ((𝑥↑2) + (𝑦↑2)))))
4	3	2rexbidv 3203	. . . . . 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 7377	. . . . . . . . 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 7377	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → (𝑥↑2) = (𝑢↑2))
8	7	oveq1d 7385	. . . . . . . . 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 633	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (((𝑥 gcd 𝑦) = 1 ∧ 𝑁 = ((𝑥↑2) + (𝑦↑2))) ↔ ((𝑢 gcd 𝑦) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑦↑2)))))
11		oveq2 7378	. . . . . . . . 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 7377	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → (𝑦↑2) = (𝑣↑2))
14	13	oveq2d 7386	. . . . . . . . 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 633	. . . . . . 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 3221	. . . . . 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 3637	. . . 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 12535	. . . . . . . . 9 ⊢ 1 ∈ ℤ
25		zgz 16875	. . . . . . . . 9 ⊢ (1 ∈ ℤ → 1 ∈ ℤ[i])
26	24, 25	ax-mp 5	. . . . . . . 8 ⊢ 1 ∈ ℤ[i]
27		sq1 14132	. . . . . . . . 9 ⊢ (1↑2) = 1
28	27	eqcomi 2746	. . . . . . . 8 ⊢ 1 = (1↑2)
29		fveq2 6844	. . . . . . . . . . 11 ⊢ (𝑥 = 1 → (abs‘𝑥) = (abs‘1))
30		abs1 15234	. . . . . . . . . . 11 ⊢ (abs‘1) = 1
31	29, 30	eqtrdi 2788	. . . . . . . . . 10 ⊢ (𝑥 = 1 → (abs‘𝑥) = 1)
32	31	oveq1d 7385	. . . . . . . . 9 ⊢ (𝑥 = 1 → ((abs‘𝑥)↑2) = (1↑2))
33	32	rspceeqv 3601	. . . . . . . 8 ⊢ ((1 ∈ ℤ[i] ∧ 1 = (1↑2)) → ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
34	26, 28, 33	mp2an 693	. . . . . . 7 ⊢ ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2)
35		2sq.1	. . . . . . . 8 ⊢ 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))
36	35	2sqlem1 27401	. . . . . . 7 ⊢ (1 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
37	34, 36	mpbir 231	. . . . . 6 ⊢ 1 ∈ 𝑆
38	23, 37	eqeltrdi 2845	. . . . 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 727	. . . . . . 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 727	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑢 ∈ ℤ ∧ 𝑣 ∈ ℤ)) ∧ (((𝑢 gcd 𝑣) = 1 ∧ 𝑁 = ((𝑢↑2) + (𝑣↑2))) ∧ 𝑀 ≠ 1)) → 𝑀 ∥ 𝑁)
43	35, 19	2sqlem7 27408	. . . . . . . . . 10 ⊢ 𝑌 ⊆ (𝑆 ∩ ℕ)
44		inss2 4192	. . . . . . . . . 10 ⊢ (𝑆 ∩ ℕ) ⊆ ℕ
45	43, 44	sstri 3945	. . . . . . . . 9 ⊢ 𝑌 ⊆ ℕ
46	45, 1	sselid 3933	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
47	46	ad2antrr 727	. . . . . . 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 727	. . . . . . . 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 12849	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ≥‘2) ↔ (𝑀 ∈ ℕ ∧ 𝑀 ≠ 1))
52	49, 50, 51	sylanbrc 584	. . . . . . 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 27410	. . . . . 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 3020	. . . 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 3195	. 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 1542   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∀wral 3052  ∃wrex 3062   ∩ cin 3902   class class class wbr 5100   ↦ cmpt 5181  ran crn 5635  ‘cfv 6502  (class class class)co 7370  1c1 11041   + caddc 11043   − cmin 11378   / cdiv 11808  ℕcn 12159  2c2 12214  ℤcz 12502  ℤ_≥cuz 12765  ...cfz 13437   mod cmo 13803  ↑cexp 13998  abscabs 15171   ∥ cdvds 16193   gcd cgcd 16435  ℤ[i]cgz 16871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117  ax-pre-sup 11118

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-2o 8410  df-er 8647  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-sup 9359  df-inf 9360  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-div 11809  df-nn 12160  df-2 12222  df-3 12223  df-n0 12416  df-z 12503  df-uz 12766  df-rp 12920  df-fz 13438  df-fl 13726  df-mod 13804  df-seq 13939  df-exp 13999  df-cj 15036  df-re 15037  df-im 15038  df-sqrt 15172  df-abs 15173  df-dvds 16194  df-gcd 16436  df-prm 16613  df-gz 16872

This theorem is referenced by:  2sqlem10  27412