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Theorem 2sqlem10 26776
Description: Lemma for 2sq 26778. 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 5108 . . . . . 6 (𝑏 = 𝐵 → (𝑏𝑎𝐵𝑎))
2 eleq1 2825 . . . . . 6 (𝑏 = 𝐵 → (𝑏𝑆𝐵𝑆))
31, 2imbi12d 344 . . . . 5 (𝑏 = 𝐵 → ((𝑏𝑎𝑏𝑆) ↔ (𝐵𝑎𝐵𝑆)))
43ralbidv 3174 . . . 4 (𝑏 = 𝐵 → (∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑎𝑌 (𝐵𝑎𝐵𝑆)))
5 oveq2 7365 . . . . . 6 (𝑚 = 1 → (1...𝑚) = (1...1))
65raleqdv 3313 . . . . 5 (𝑚 = 1 → (∀𝑏 ∈ (1...𝑚)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑏 ∈ (1...1)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
7 oveq2 7365 . . . . . 6 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
87raleqdv 3313 . . . . 5 (𝑚 = 𝑛 → (∀𝑏 ∈ (1...𝑚)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
9 oveq2 7365 . . . . . 6 (𝑚 = (𝑛 + 1) → (1...𝑚) = (1...(𝑛 + 1)))
109raleqdv 3313 . . . . 5 (𝑚 = (𝑛 + 1) → (∀𝑏 ∈ (1...𝑚)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑏 ∈ (1...(𝑛 + 1))∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
11 oveq2 7365 . . . . . 6 (𝑚 = 𝐵 → (1...𝑚) = (1...𝐵))
1211raleqdv 3313 . . . . 5 (𝑚 = 𝐵 → (∀𝑏 ∈ (1...𝑚)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑏 ∈ (1...𝐵)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
13 elfz1eq 13452 . . . . . . . . 9 (𝑏 ∈ (1...1) → 𝑏 = 1)
14 1z 12533 . . . . . . . . . . . 12 1 ∈ ℤ
15 zgz 16805 . . . . . . . . . . . 12 (1 ∈ ℤ → 1 ∈ ℤ[i])
1614, 15ax-mp 5 . . . . . . . . . . 11 1 ∈ ℤ[i]
17 sq1 14099 . . . . . . . . . . . 12 (1↑2) = 1
1817eqcomi 2745 . . . . . . . . . . 11 1 = (1↑2)
19 fveq2 6842 . . . . . . . . . . . . . 14 (𝑥 = 1 → (abs‘𝑥) = (abs‘1))
20 abs1 15182 . . . . . . . . . . . . . 14 (abs‘1) = 1
2119, 20eqtrdi 2792 . . . . . . . . . . . . 13 (𝑥 = 1 → (abs‘𝑥) = 1)
2221oveq1d 7372 . . . . . . . . . . . 12 (𝑥 = 1 → ((abs‘𝑥)↑2) = (1↑2))
2322rspceeqv 3595 . . . . . . . . . . 11 ((1 ∈ ℤ[i] ∧ 1 = (1↑2)) → ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
2416, 18, 23mp2an 690 . . . . . . . . . 10 𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2)
25 2sq.1 . . . . . . . . . . 11 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))
26252sqlem1 26765 . . . . . . . . . 10 (1 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
2724, 26mpbir 230 . . . . . . . . 9 1 ∈ 𝑆
2813, 27eqeltrdi 2846 . . . . . . . 8 (𝑏 ∈ (1...1) → 𝑏𝑆)
2928a1d 25 . . . . . . 7 (𝑏 ∈ (1...1) → (𝑏𝑎𝑏𝑆))
3029ralrimivw 3147 . . . . . 6 (𝑏 ∈ (1...1) → ∀𝑎𝑌 (𝑏𝑎𝑏𝑆))
3130rgen 3066 . . . . 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 767 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆))
34 nncn 12161 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
3534ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → 𝑛 ∈ ℂ)
36 ax-1cn 11109 . . . . . . . . . . . . . . . . 17 1 ∈ ℂ
37 pncan 11407 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
3835, 36, 37sylancl 586 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → ((𝑛 + 1) − 1) = 𝑛)
3938oveq2d 7373 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (1...((𝑛 + 1) − 1)) = (1...𝑛))
4039raleqdv 3313 . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → ∀𝑏 ∈ (1...((𝑛 + 1) − 1))∀𝑎𝑌 (𝑏𝑎𝑏𝑆))
42 simprr 771 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (𝑛 + 1) ∥ 𝑚)
43 peano2nn 12165 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
4443ad2antrr 724 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (𝑛 + 1) ∈ ℕ)
45 simprl 769 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → 𝑚𝑌)
4625, 32, 41, 42, 44, 452sqlem9 26775 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (𝑛 + 1) ∈ 𝑆)
4746expr 457 . . . . . . . . . . 11 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ 𝑚𝑌) → ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆))
4847ralrimiva 3143 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) → ∀𝑚𝑌 ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆))
4948ex 413 . . . . . . . . 9 (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) → ∀𝑚𝑌 ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆)))
50 breq2 5109 . . . . . . . . . . 11 (𝑎 = 𝑚 → ((𝑛 + 1) ∥ 𝑎 ↔ (𝑛 + 1) ∥ 𝑚))
5150imbi1d 341 . . . . . . . . . 10 (𝑎 = 𝑚 → (((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆) ↔ ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆)))
5251cbvralvw 3225 . . . . . . . . 9 (∀𝑎𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆) ↔ ∀𝑚𝑌 ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆))
5349, 52syl6ibr 251 . . . . . . . 8 (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) → ∀𝑎𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
54 ovex 7390 . . . . . . . . 9 (𝑛 + 1) ∈ V
55 breq1 5108 . . . . . . . . . . 11 (𝑏 = (𝑛 + 1) → (𝑏𝑎 ↔ (𝑛 + 1) ∥ 𝑎))
56 eleq1 2825 . . . . . . . . . . 11 (𝑏 = (𝑛 + 1) → (𝑏𝑆 ↔ (𝑛 + 1) ∈ 𝑆))
5755, 56imbi12d 344 . . . . . . . . . 10 (𝑏 = (𝑛 + 1) → ((𝑏𝑎𝑏𝑆) ↔ ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
5857ralbidv 3174 . . . . . . . . 9 (𝑏 = (𝑛 + 1) → (∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑎𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
5954, 58ralsn 4642 . . . . . . . 8 (∀𝑏 ∈ {(𝑛 + 1)}∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑎𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆))
6053, 59syl6ibr 251 . . . . . . 7 (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) → ∀𝑏 ∈ {(𝑛 + 1)}∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
6160ancld 551 . . . . . 6 (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) → (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ∧ ∀𝑏 ∈ {(𝑛 + 1)}∀𝑎𝑌 (𝑏𝑎𝑏𝑆))))
62 elnnuz 12807 . . . . . . . . 9 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
63 fzsuc 13488 . . . . . . . . 9 (𝑛 ∈ (ℤ‘1) → (1...(𝑛 + 1)) = ((1...𝑛) ∪ {(𝑛 + 1)}))
6462, 63sylbi 216 . . . . . . . 8 (𝑛 ∈ ℕ → (1...(𝑛 + 1)) = ((1...𝑛) ∪ {(𝑛 + 1)}))
6564raleqdv 3313 . . . . . . 7 (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...(𝑛 + 1))∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑏 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
66 ralunb 4151 . . . . . . 7 (∀𝑏 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ∧ ∀𝑏 ∈ {(𝑛 + 1)}∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
6765, 66bitrdi 286 . . . . . 6 (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...(𝑛 + 1))∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ∧ ∀𝑏 ∈ {(𝑛 + 1)}∀𝑎𝑌 (𝑏𝑎𝑏𝑆))))
6861, 67sylibrd 258 . . . . 5 (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) → ∀𝑏 ∈ (1...(𝑛 + 1))∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
696, 8, 10, 12, 31, 68nnind 12171 . . . 4 (𝐵 ∈ ℕ → ∀𝑏 ∈ (1...𝐵)∀𝑎𝑌 (𝑏𝑎𝑏𝑆))
70 elfz1end 13471 . . . . 5 (𝐵 ∈ ℕ ↔ 𝐵 ∈ (1...𝐵))
7170biimpi 215 . . . 4 (𝐵 ∈ ℕ → 𝐵 ∈ (1...𝐵))
724, 69, 71rspcdva 3582 . . 3 (𝐵 ∈ ℕ → ∀𝑎𝑌 (𝐵𝑎𝐵𝑆))
73 breq2 5109 . . . . 5 (𝑎 = 𝐴 → (𝐵𝑎𝐵𝐴))
7473imbi1d 341 . . . 4 (𝑎 = 𝐴 → ((𝐵𝑎𝐵𝑆) ↔ (𝐵𝐴𝐵𝑆)))
7574rspcv 3577 . . 3 (𝐴𝑌 → (∀𝑎𝑌 (𝐵𝑎𝐵𝑆) → (𝐵𝐴𝐵𝑆)))
7672, 75syl5 34 . 2 (𝐴𝑌 → (𝐵 ∈ ℕ → (𝐵𝐴𝐵𝑆)))
77763imp 1111 1 ((𝐴𝑌𝐵 ∈ ℕ ∧ 𝐵𝐴) → 𝐵𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  {cab 2713  wral 3064  wrex 3073  cun 3908  {csn 4586   class class class wbr 5105  cmpt 5188  ran crn 5634  cfv 6496  (class class class)co 7357  cc 11049  1c1 11052   + caddc 11054  cmin 11385  cn 12153  2c2 12208  cz 12499  cuz 12763  ...cfz 13424  cexp 13967  abscabs 15119  cdvds 16136   gcd cgcd 16374  ℤ[i]cgz 16801
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-dvds 16137  df-gcd 16375  df-prm 16548  df-gz 16802
This theorem is referenced by:  2sqlem11  26777
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