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Theorem 2sqlem10 14012
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 4001 . . . . . 6 (𝑏 = 𝐵 → (𝑏𝑎𝐵𝑎))
2 eleq1 2238 . . . . . 6 (𝑏 = 𝐵 → (𝑏𝑆𝐵𝑆))
31, 2imbi12d 234 . . . . 5 (𝑏 = 𝐵 → ((𝑏𝑎𝑏𝑆) ↔ (𝐵𝑎𝐵𝑆)))
43ralbidv 2475 . . . 4 (𝑏 = 𝐵 → (∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑎𝑌 (𝐵𝑎𝐵𝑆)))
5 oveq2 5873 . . . . . 6 (𝑚 = 1 → (1...𝑚) = (1...1))
65raleqdv 2676 . . . . 5 (𝑚 = 1 → (∀𝑏 ∈ (1...𝑚)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑏 ∈ (1...1)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
7 oveq2 5873 . . . . . 6 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
87raleqdv 2676 . . . . 5 (𝑚 = 𝑛 → (∀𝑏 ∈ (1...𝑚)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
9 oveq2 5873 . . . . . 6 (𝑚 = (𝑛 + 1) → (1...𝑚) = (1...(𝑛 + 1)))
109raleqdv 2676 . . . . 5 (𝑚 = (𝑛 + 1) → (∀𝑏 ∈ (1...𝑚)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑏 ∈ (1...(𝑛 + 1))∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
11 oveq2 5873 . . . . . 6 (𝑚 = 𝐵 → (1...𝑚) = (1...𝐵))
1211raleqdv 2676 . . . . 5 (𝑚 = 𝐵 → (∀𝑏 ∈ (1...𝑚)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑏 ∈ (1...𝐵)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
13 elfz1eq 10003 . . . . . . . . 9 (𝑏 ∈ (1...1) → 𝑏 = 1)
14 1z 9250 . . . . . . . . . . . 12 1 ∈ ℤ
15 zgz 12336 . . . . . . . . . . . 12 (1 ∈ ℤ → 1 ∈ ℤ[i])
1614, 15ax-mp 5 . . . . . . . . . . 11 1 ∈ ℤ[i]
17 sq1 10581 . . . . . . . . . . . 12 (1↑2) = 1
1817eqcomi 2179 . . . . . . . . . . 11 1 = (1↑2)
19 fveq2 5507 . . . . . . . . . . . . . 14 (𝑥 = 1 → (abs‘𝑥) = (abs‘1))
20 abs1 11047 . . . . . . . . . . . . . 14 (abs‘1) = 1
2119, 20eqtrdi 2224 . . . . . . . . . . . . 13 (𝑥 = 1 → (abs‘𝑥) = 1)
2221oveq1d 5880 . . . . . . . . . . . 12 (𝑥 = 1 → ((abs‘𝑥)↑2) = (1↑2))
2322rspceeqv 2857 . . . . . . . . . . 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 14001 . . . . . . . . . 10 (1 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
2724, 26mpbir 146 . . . . . . . . 9 1 ∈ 𝑆
2813, 27eqeltrdi 2266 . . . . . . . 8 (𝑏 ∈ (1...1) → 𝑏𝑆)
2928a1d 22 . . . . . . 7 (𝑏 ∈ (1...1) → (𝑏𝑎𝑏𝑆))
3029ralrimivw 2549 . . . . . 6 (𝑏 ∈ (1...1) → ∀𝑎𝑌 (𝑏𝑎𝑏𝑆))
3130rgen 2528 . . . . 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 8898 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
3534ad2antrr 488 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → 𝑛 ∈ ℂ)
36 ax-1cn 7879 . . . . . . . . . . . . . . . . 17 1 ∈ ℂ
37 pncan 8137 . . . . . . . . . . . . . . . . 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 5881 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (1...((𝑛 + 1) − 1)) = (1...𝑛))
4039raleqdv 2676 . . . . . . . . . . . . . 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 8902 . . . . . . . . . . . . . 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 14011 . . . . . . . . . . . 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 2548 . . . . . . . . . 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 4002 . . . . . . . . . . 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 2705 . . . . . . . . 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 4001 . . . . . . . . . . . 12 (𝑏 = (𝑛 + 1) → (𝑏𝑎 ↔ (𝑛 + 1) ∥ 𝑎))
55 eleq1 2238 . . . . . . . . . . . 12 (𝑏 = (𝑛 + 1) → (𝑏𝑆 ↔ (𝑛 + 1) ∈ 𝑆))
5654, 55imbi12d 234 . . . . . . . . . . 11 (𝑏 = (𝑛 + 1) → ((𝑏𝑎𝑏𝑆) ↔ ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
5756ralbidv 2475 . . . . . . . . . 10 (𝑏 = (𝑛 + 1) → (∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑎𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
5857ralsng 3629 . . . . . . . . 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 9535 . . . . . . . . 9 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
63 fzsuc 10037 . . . . . . . . 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 2676 . . . . . . 7 (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...(𝑛 + 1))∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑏 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
66 ralunb 3314 . . . . . . 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 8906 . . . 4 (𝐵 ∈ ℕ → ∀𝑏 ∈ (1...𝐵)∀𝑎𝑌 (𝑏𝑎𝑏𝑆))
70 elfz1end 10023 . . . . 5 (𝐵 ∈ ℕ ↔ 𝐵 ∈ (1...𝐵))
7170biimpi 120 . . . 4 (𝐵 ∈ ℕ → 𝐵 ∈ (1...𝐵))
724, 69, 71rspcdva 2844 . . 3 (𝐵 ∈ ℕ → ∀𝑎𝑌 (𝐵𝑎𝐵𝑆))
73 breq2 4002 . . . . 5 (𝑎 = 𝐴 → (𝐵𝑎𝐵𝐴))
7473imbi1d 231 . . . 4 (𝑎 = 𝐴 → ((𝐵𝑎𝐵𝑆) ↔ (𝐵𝐴𝐵𝑆)))
7574rspcv 2835 . . 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 2146  {cab 2161  wral 2453  wrex 2454  cun 3125  {csn 3589   class class class wbr 3998  cmpt 4059  ran crn 4621  cfv 5208  (class class class)co 5865  cc 7784  1c1 7787   + caddc 7789  cmin 8102  cn 8890  2c2 8941  cz 9224  cuz 9499  ...cfz 9977  cexp 10487  abscabs 10972  cdvds 11760   gcd cgcd 11908  ℤ[i]cgz 12332
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906
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 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-1o 6407  df-2o 6408  df-er 6525  df-en 6731  df-sup 6973  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-n0 9148  df-z 9225  df-uz 9500  df-q 9591  df-rp 9623  df-fz 9978  df-fzo 10111  df-fl 10238  df-mod 10291  df-seqfrec 10414  df-exp 10488  df-cj 10817  df-re 10818  df-im 10819  df-rsqrt 10973  df-abs 10974  df-dvds 11761  df-gcd 11909  df-prm 12073  df-gz 12333
This theorem is referenced by: (None)
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