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Theorem 2sqlem10 14128
Description: Lemma for 2sq . Every factor of a "proper" sum of two squares (where the summands are coprime) is a sum of two squares. (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)))}
Assertion
Ref Expression
2sqlem10 ((𝐴𝑌𝐵 ∈ ℕ ∧ 𝐵𝐴) → 𝐵𝑆)
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦   𝑥,𝑆,𝑦,𝑧   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐴(𝑤)   𝐵(𝑧,𝑤)   𝑆(𝑤)   𝑌(𝑧,𝑤)

Proof of Theorem 2sqlem10
Dummy variables 𝑎 𝑏 𝑛 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 4003 . . . . . 6 (𝑏 = 𝐵 → (𝑏𝑎𝐵𝑎))
2 eleq1 2240 . . . . . 6 (𝑏 = 𝐵 → (𝑏𝑆𝐵𝑆))
31, 2imbi12d 234 . . . . 5 (𝑏 = 𝐵 → ((𝑏𝑎𝑏𝑆) ↔ (𝐵𝑎𝐵𝑆)))
43ralbidv 2477 . . . 4 (𝑏 = 𝐵 → (∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑎𝑌 (𝐵𝑎𝐵𝑆)))
5 oveq2 5877 . . . . . 6 (𝑚 = 1 → (1...𝑚) = (1...1))
65raleqdv 2678 . . . . 5 (𝑚 = 1 → (∀𝑏 ∈ (1...𝑚)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑏 ∈ (1...1)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
7 oveq2 5877 . . . . . 6 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
87raleqdv 2678 . . . . 5 (𝑚 = 𝑛 → (∀𝑏 ∈ (1...𝑚)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
9 oveq2 5877 . . . . . 6 (𝑚 = (𝑛 + 1) → (1...𝑚) = (1...(𝑛 + 1)))
109raleqdv 2678 . . . . 5 (𝑚 = (𝑛 + 1) → (∀𝑏 ∈ (1...𝑚)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑏 ∈ (1...(𝑛 + 1))∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
11 oveq2 5877 . . . . . 6 (𝑚 = 𝐵 → (1...𝑚) = (1...𝐵))
1211raleqdv 2678 . . . . 5 (𝑚 = 𝐵 → (∀𝑏 ∈ (1...𝑚)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑏 ∈ (1...𝐵)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
13 elfz1eq 10021 . . . . . . . . 9 (𝑏 ∈ (1...1) → 𝑏 = 1)
14 1z 9268 . . . . . . . . . . . 12 1 ∈ ℤ
15 zgz 12354 . . . . . . . . . . . 12 (1 ∈ ℤ → 1 ∈ ℤ[i])
1614, 15ax-mp 5 . . . . . . . . . . 11 1 ∈ ℤ[i]
17 sq1 10599 . . . . . . . . . . . 12 (1↑2) = 1
1817eqcomi 2181 . . . . . . . . . . 11 1 = (1↑2)
19 fveq2 5511 . . . . . . . . . . . . . 14 (𝑥 = 1 → (abs‘𝑥) = (abs‘1))
20 abs1 11065 . . . . . . . . . . . . . 14 (abs‘1) = 1
2119, 20eqtrdi 2226 . . . . . . . . . . . . 13 (𝑥 = 1 → (abs‘𝑥) = 1)
2221oveq1d 5884 . . . . . . . . . . . 12 (𝑥 = 1 → ((abs‘𝑥)↑2) = (1↑2))
2322rspceeqv 2859 . . . . . . . . . . 11 ((1 ∈ ℤ[i] ∧ 1 = (1↑2)) → ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
2416, 18, 23mp2an 426 . . . . . . . . . 10 𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2)
25 2sq.1 . . . . . . . . . . 11 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))
26252sqlem1 14117 . . . . . . . . . 10 (1 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
2724, 26mpbir 146 . . . . . . . . 9 1 ∈ 𝑆
2813, 27eqeltrdi 2268 . . . . . . . 8 (𝑏 ∈ (1...1) → 𝑏𝑆)
2928a1d 22 . . . . . . 7 (𝑏 ∈ (1...1) → (𝑏𝑎𝑏𝑆))
3029ralrimivw 2551 . . . . . 6 (𝑏 ∈ (1...1) → ∀𝑎𝑌 (𝑏𝑎𝑏𝑆))
3130rgen 2530 . . . . 5 𝑏 ∈ (1...1)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)
32 2sqlem7.2 . . . . . . . . . . . . 13 𝑌 = {𝑧 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ((𝑥 gcd 𝑦) = 1 ∧ 𝑧 = ((𝑥↑2) + (𝑦↑2)))}
33 simplr 528 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆))
34 nncn 8916 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
3534ad2antrr 488 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → 𝑛 ∈ ℂ)
36 ax-1cn 7895 . . . . . . . . . . . . . . . . 17 1 ∈ ℂ
37 pncan 8153 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
3835, 36, 37sylancl 413 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → ((𝑛 + 1) − 1) = 𝑛)
3938oveq2d 5885 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (1...((𝑛 + 1) − 1)) = (1...𝑛))
4039raleqdv 2678 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (∀𝑏 ∈ (1...((𝑛 + 1) − 1))∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
4133, 40mpbird 167 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → ∀𝑏 ∈ (1...((𝑛 + 1) − 1))∀𝑎𝑌 (𝑏𝑎𝑏𝑆))
42 simprr 531 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (𝑛 + 1) ∥ 𝑚)
43 peano2nn 8920 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
4443ad2antrr 488 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (𝑛 + 1) ∈ ℕ)
45 simprl 529 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → 𝑚𝑌)
4625, 32, 41, 42, 44, 452sqlem9 14127 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (𝑛 + 1) ∈ 𝑆)
4746expr 375 . . . . . . . . . . 11 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ 𝑚𝑌) → ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆))
4847ralrimiva 2550 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) → ∀𝑚𝑌 ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆))
4948ex 115 . . . . . . . . 9 (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) → ∀𝑚𝑌 ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆)))
50 breq2 4004 . . . . . . . . . . 11 (𝑎 = 𝑚 → ((𝑛 + 1) ∥ 𝑎 ↔ (𝑛 + 1) ∥ 𝑚))
5150imbi1d 231 . . . . . . . . . 10 (𝑎 = 𝑚 → (((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆) ↔ ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆)))
5251cbvralvw 2707 . . . . . . . . 9 (∀𝑎𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆) ↔ ∀𝑚𝑌 ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆))
5349, 52syl6ibr 162 . . . . . . . 8 (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) → ∀𝑎𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
54 breq1 4003 . . . . . . . . . . . 12 (𝑏 = (𝑛 + 1) → (𝑏𝑎 ↔ (𝑛 + 1) ∥ 𝑎))
55 eleq1 2240 . . . . . . . . . . . 12 (𝑏 = (𝑛 + 1) → (𝑏𝑆 ↔ (𝑛 + 1) ∈ 𝑆))
5654, 55imbi12d 234 . . . . . . . . . . 11 (𝑏 = (𝑛 + 1) → ((𝑏𝑎𝑏𝑆) ↔ ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
5756ralbidv 2477 . . . . . . . . . 10 (𝑏 = (𝑛 + 1) → (∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑎𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
5857ralsng 3631 . . . . . . . . 9 ((𝑛 + 1) ∈ ℕ → (∀𝑏 ∈ {(𝑛 + 1)}∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑎𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
5943, 58syl 14 . . . . . . . 8 (𝑛 ∈ ℕ → (∀𝑏 ∈ {(𝑛 + 1)}∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑎𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
6053, 59sylibrd 169 . . . . . . 7 (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) → ∀𝑏 ∈ {(𝑛 + 1)}∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
6160ancld 325 . . . . . 6 (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) → (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ∧ ∀𝑏 ∈ {(𝑛 + 1)}∀𝑎𝑌 (𝑏𝑎𝑏𝑆))))
62 elnnuz 9553 . . . . . . . . 9 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
63 fzsuc 10055 . . . . . . . . 9 (𝑛 ∈ (ℤ‘1) → (1...(𝑛 + 1)) = ((1...𝑛) ∪ {(𝑛 + 1)}))
6462, 63sylbi 121 . . . . . . . 8 (𝑛 ∈ ℕ → (1...(𝑛 + 1)) = ((1...𝑛) ∪ {(𝑛 + 1)}))
6564raleqdv 2678 . . . . . . 7 (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...(𝑛 + 1))∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑏 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
66 ralunb 3316 . . . . . . 7 (∀𝑏 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ∧ ∀𝑏 ∈ {(𝑛 + 1)}∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
6765, 66bitrdi 196 . . . . . 6 (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...(𝑛 + 1))∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ∧ ∀𝑏 ∈ {(𝑛 + 1)}∀𝑎𝑌 (𝑏𝑎𝑏𝑆))))
6861, 67sylibrd 169 . . . . 5 (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) → ∀𝑏 ∈ (1...(𝑛 + 1))∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
696, 8, 10, 12, 31, 68nnind 8924 . . . 4 (𝐵 ∈ ℕ → ∀𝑏 ∈ (1...𝐵)∀𝑎𝑌 (𝑏𝑎𝑏𝑆))
70 elfz1end 10041 . . . . 5 (𝐵 ∈ ℕ ↔ 𝐵 ∈ (1...𝐵))
7170biimpi 120 . . . 4 (𝐵 ∈ ℕ → 𝐵 ∈ (1...𝐵))
724, 69, 71rspcdva 2846 . . 3 (𝐵 ∈ ℕ → ∀𝑎𝑌 (𝐵𝑎𝐵𝑆))
73 breq2 4004 . . . . 5 (𝑎 = 𝐴 → (𝐵𝑎𝐵𝐴))
7473imbi1d 231 . . . 4 (𝑎 = 𝐴 → ((𝐵𝑎𝐵𝑆) ↔ (𝐵𝐴𝐵𝑆)))
7574rspcv 2837 . . 3 (𝐴𝑌 → (∀𝑎𝑌 (𝐵𝑎𝐵𝑆) → (𝐵𝐴𝐵𝑆)))
7672, 75syl5 32 . 2 (𝐴𝑌 → (𝐵 ∈ ℕ → (𝐵𝐴𝐵𝑆)))
77763imp 1193 1 ((𝐴𝑌𝐵 ∈ ℕ ∧ 𝐵𝐴) → 𝐵𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wcel 2148  {cab 2163  wral 2455  wrex 2456  cun 3127  {csn 3591   class class class wbr 4000  cmpt 4061  ran crn 4624  cfv 5212  (class class class)co 5869  cc 7800  1c1 7803   + caddc 7805  cmin 8118  cn 8908  2c2 8959  cz 9242  cuz 9517  ...cfz 9995  cexp 10505  abscabs 10990  cdvds 11778   gcd cgcd 11926  ℤ[i]cgz 12350
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-1o 6411  df-2o 6412  df-er 6529  df-en 6735  df-sup 6977  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-dvds 11779  df-gcd 11927  df-prm 12091  df-gz 12351
This theorem is referenced by: (None)
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