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Theorem 2sqlem10 25443
Description: Lemma for 2sq 25445. 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 4811 . . . . . 6 (𝑏 = 𝐵 → (𝑏𝑎𝐵𝑎))
2 eleq1 2831 . . . . . 6 (𝑏 = 𝐵 → (𝑏𝑆𝐵𝑆))
31, 2imbi12d 335 . . . . 5 (𝑏 = 𝐵 → ((𝑏𝑎𝑏𝑆) ↔ (𝐵𝑎𝐵𝑆)))
43ralbidv 3132 . . . 4 (𝑏 = 𝐵 → (∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑎𝑌 (𝐵𝑎𝐵𝑆)))
5 oveq2 6849 . . . . . 6 (𝑚 = 1 → (1...𝑚) = (1...1))
65raleqdv 3291 . . . . 5 (𝑚 = 1 → (∀𝑏 ∈ (1...𝑚)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑏 ∈ (1...1)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
7 oveq2 6849 . . . . . 6 (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
87raleqdv 3291 . . . . 5 (𝑚 = 𝑛 → (∀𝑏 ∈ (1...𝑚)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
9 oveq2 6849 . . . . . 6 (𝑚 = (𝑛 + 1) → (1...𝑚) = (1...(𝑛 + 1)))
109raleqdv 3291 . . . . 5 (𝑚 = (𝑛 + 1) → (∀𝑏 ∈ (1...𝑚)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑏 ∈ (1...(𝑛 + 1))∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
11 oveq2 6849 . . . . . 6 (𝑚 = 𝐵 → (1...𝑚) = (1...𝐵))
1211raleqdv 3291 . . . . 5 (𝑚 = 𝐵 → (∀𝑏 ∈ (1...𝑚)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑏 ∈ (1...𝐵)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
13 elfz1eq 12558 . . . . . . . . 9 (𝑏 ∈ (1...1) → 𝑏 = 1)
14 1z 11653 . . . . . . . . . . . 12 1 ∈ ℤ
15 zgz 15917 . . . . . . . . . . . 12 (1 ∈ ℤ → 1 ∈ ℤ[i])
1614, 15ax-mp 5 . . . . . . . . . . 11 1 ∈ ℤ[i]
17 sq1 13164 . . . . . . . . . . . 12 (1↑2) = 1
1817eqcomi 2773 . . . . . . . . . . 11 1 = (1↑2)
19 fveq2 6374 . . . . . . . . . . . . . 14 (𝑥 = 1 → (abs‘𝑥) = (abs‘1))
20 abs1 14323 . . . . . . . . . . . . . 14 (abs‘1) = 1
2119, 20syl6eq 2814 . . . . . . . . . . . . 13 (𝑥 = 1 → (abs‘𝑥) = 1)
2221oveq1d 6856 . . . . . . . . . . . 12 (𝑥 = 1 → ((abs‘𝑥)↑2) = (1↑2))
2322rspceeqv 3478 . . . . . . . . . . 11 ((1 ∈ ℤ[i] ∧ 1 = (1↑2)) → ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
2416, 18, 23mp2an 683 . . . . . . . . . 10 𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2)
25 2sq.1 . . . . . . . . . . 11 𝑆 = ran (𝑤 ∈ ℤ[i] ↦ ((abs‘𝑤)↑2))
26252sqlem1 25432 . . . . . . . . . 10 (1 ∈ 𝑆 ↔ ∃𝑥 ∈ ℤ[i] 1 = ((abs‘𝑥)↑2))
2724, 26mpbir 222 . . . . . . . . 9 1 ∈ 𝑆
2813, 27syl6eqel 2851 . . . . . . . 8 (𝑏 ∈ (1...1) → 𝑏𝑆)
2928a1d 25 . . . . . . 7 (𝑏 ∈ (1...1) → (𝑏𝑎𝑏𝑆))
3029ralrimivw 3113 . . . . . 6 (𝑏 ∈ (1...1) → ∀𝑎𝑌 (𝑏𝑎𝑏𝑆))
3130rgen 3068 . . . . 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 785 . . . . . . . . . . . . . 14 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆))
34 nncn 11282 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
3534ad2antrr 717 . . . . . . . . . . . . . . . . 17 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → 𝑛 ∈ ℂ)
36 ax-1cn 10246 . . . . . . . . . . . . . . . . 17 1 ∈ ℂ
37 pncan 10540 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
3835, 36, 37sylancl 580 . . . . . . . . . . . . . . . 16 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → ((𝑛 + 1) − 1) = 𝑛)
3938oveq2d 6857 . . . . . . . . . . . . . . 15 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (1...((𝑛 + 1) − 1)) = (1...𝑛))
4039raleqdv 3291 . . . . . . . . . . . . . 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 248 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → ∀𝑏 ∈ (1...((𝑛 + 1) − 1))∀𝑎𝑌 (𝑏𝑎𝑏𝑆))
42 simprr 789 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (𝑛 + 1) ∥ 𝑚)
43 peano2nn 11287 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → (𝑛 + 1) ∈ ℕ)
4443ad2antrr 717 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (𝑛 + 1) ∈ ℕ)
45 simprl 787 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → 𝑚𝑌)
4625, 32, 41, 42, 44, 452sqlem9 25442 . . . . . . . . . . . 12 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ (𝑚𝑌 ∧ (𝑛 + 1) ∥ 𝑚)) → (𝑛 + 1) ∈ 𝑆)
4746expr 448 . . . . . . . . . . 11 (((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) ∧ 𝑚𝑌) → ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆))
4847ralrimiva 3112 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ ∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆)) → ∀𝑚𝑌 ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆))
4948ex 401 . . . . . . . . 9 (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) → ∀𝑚𝑌 ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆)))
50 breq2 4812 . . . . . . . . . . 11 (𝑎 = 𝑚 → ((𝑛 + 1) ∥ 𝑎 ↔ (𝑛 + 1) ∥ 𝑚))
5150imbi1d 332 . . . . . . . . . 10 (𝑎 = 𝑚 → (((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆) ↔ ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆)))
5251cbvralv 3318 . . . . . . . . 9 (∀𝑎𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆) ↔ ∀𝑚𝑌 ((𝑛 + 1) ∥ 𝑚 → (𝑛 + 1) ∈ 𝑆))
5349, 52syl6ibr 243 . . . . . . . 8 (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) → ∀𝑎𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
54 ovex 6873 . . . . . . . . 9 (𝑛 + 1) ∈ V
55 breq1 4811 . . . . . . . . . . 11 (𝑏 = (𝑛 + 1) → (𝑏𝑎 ↔ (𝑛 + 1) ∥ 𝑎))
56 eleq1 2831 . . . . . . . . . . 11 (𝑏 = (𝑛 + 1) → (𝑏𝑆 ↔ (𝑛 + 1) ∈ 𝑆))
5755, 56imbi12d 335 . . . . . . . . . 10 (𝑏 = (𝑛 + 1) → ((𝑏𝑎𝑏𝑆) ↔ ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
5857ralbidv 3132 . . . . . . . . 9 (𝑏 = (𝑛 + 1) → (∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑎𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆)))
5954, 58ralsn 4378 . . . . . . . 8 (∀𝑏 ∈ {(𝑛 + 1)}∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑎𝑌 ((𝑛 + 1) ∥ 𝑎 → (𝑛 + 1) ∈ 𝑆))
6053, 59syl6ibr 243 . . . . . . 7 (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) → ∀𝑏 ∈ {(𝑛 + 1)}∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
6160ancld 546 . . . . . 6 (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) → (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ∧ ∀𝑏 ∈ {(𝑛 + 1)}∀𝑎𝑌 (𝑏𝑎𝑏𝑆))))
62 elnnuz 11923 . . . . . . . . 9 (𝑛 ∈ ℕ ↔ 𝑛 ∈ (ℤ‘1))
63 fzsuc 12594 . . . . . . . . 9 (𝑛 ∈ (ℤ‘1) → (1...(𝑛 + 1)) = ((1...𝑛) ∪ {(𝑛 + 1)}))
6462, 63sylbi 208 . . . . . . . 8 (𝑛 ∈ ℕ → (1...(𝑛 + 1)) = ((1...𝑛) ∪ {(𝑛 + 1)}))
6564raleqdv 3291 . . . . . . 7 (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...(𝑛 + 1))∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ ∀𝑏 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
66 ralunb 3955 . . . . . . 7 (∀𝑏 ∈ ((1...𝑛) ∪ {(𝑛 + 1)})∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ∧ ∀𝑏 ∈ {(𝑛 + 1)}∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
6765, 66syl6bb 278 . . . . . 6 (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...(𝑛 + 1))∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ↔ (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) ∧ ∀𝑏 ∈ {(𝑛 + 1)}∀𝑎𝑌 (𝑏𝑎𝑏𝑆))))
6861, 67sylibrd 250 . . . . 5 (𝑛 ∈ ℕ → (∀𝑏 ∈ (1...𝑛)∀𝑎𝑌 (𝑏𝑎𝑏𝑆) → ∀𝑏 ∈ (1...(𝑛 + 1))∀𝑎𝑌 (𝑏𝑎𝑏𝑆)))
696, 8, 10, 12, 31, 68nnind 11293 . . . 4 (𝐵 ∈ ℕ → ∀𝑏 ∈ (1...𝐵)∀𝑎𝑌 (𝑏𝑎𝑏𝑆))
70 elfz1end 12577 . . . . 5 (𝐵 ∈ ℕ ↔ 𝐵 ∈ (1...𝐵))
7170biimpi 207 . . . 4 (𝐵 ∈ ℕ → 𝐵 ∈ (1...𝐵))
724, 69, 71rspcdva 3466 . . 3 (𝐵 ∈ ℕ → ∀𝑎𝑌 (𝐵𝑎𝐵𝑆))
73 breq2 4812 . . . . 5 (𝑎 = 𝐴 → (𝐵𝑎𝐵𝐴))
7473imbi1d 332 . . . 4 (𝑎 = 𝐴 → ((𝐵𝑎𝐵𝑆) ↔ (𝐵𝐴𝐵𝑆)))
7574rspcv 3456 . . 3 (𝐴𝑌 → (∀𝑎𝑌 (𝐵𝑎𝐵𝑆) → (𝐵𝐴𝐵𝑆)))
7672, 75syl5 34 . 2 (𝐴𝑌 → (𝐵 ∈ ℕ → (𝐵𝐴𝐵𝑆)))
77763imp 1137 1 ((𝐴𝑌𝐵 ∈ ℕ ∧ 𝐵𝐴) → 𝐵𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107   = wceq 1652  wcel 2155  {cab 2750  wral 3054  wrex 3055  cun 3729  {csn 4333   class class class wbr 4808  cmpt 4887  ran crn 5277  cfv 6067  (class class class)co 6841  cc 10186  1c1 10189   + caddc 10191  cmin 10519  cn 11273  2c2 11326  cz 11623  cuz 11885  ...cfz 12532  cexp 13066  abscabs 14260  cdvds 15266   gcd cgcd 15498  ℤ[i]cgz 15913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146  ax-cnex 10244  ax-resscn 10245  ax-1cn 10246  ax-icn 10247  ax-addcl 10248  ax-addrcl 10249  ax-mulcl 10250  ax-mulrcl 10251  ax-mulcom 10252  ax-addass 10253  ax-mulass 10254  ax-distr 10255  ax-i2m1 10256  ax-1ne0 10257  ax-1rid 10258  ax-rnegex 10259  ax-rrecex 10260  ax-cnre 10261  ax-pre-lttri 10262  ax-pre-lttrn 10263  ax-pre-ltadd 10264  ax-pre-mulgt0 10265  ax-pre-sup 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-iun 4677  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-om 7263  df-1st 7365  df-2nd 7366  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-1o 7763  df-2o 7764  df-er 7946  df-en 8160  df-dom 8161  df-sdom 8162  df-fin 8163  df-sup 8554  df-inf 8555  df-pnf 10329  df-mnf 10330  df-xr 10331  df-ltxr 10332  df-le 10333  df-sub 10521  df-neg 10522  df-div 10938  df-nn 11274  df-2 11334  df-3 11335  df-n0 11538  df-z 11624  df-uz 11886  df-rp 12028  df-fz 12533  df-fl 12800  df-mod 12876  df-seq 13008  df-exp 13067  df-cj 14125  df-re 14126  df-im 14127  df-sqrt 14261  df-abs 14262  df-dvds 15267  df-gcd 15499  df-prm 15667  df-gz 15914
This theorem is referenced by:  2sqlem11  25444
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