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Theorem acongeq 39458
Description: Two numbers in the fundamental domain are alternating-congruent iff they are equal. TODO: could be used to shorten jm2.26 39477. (Contributed by Stefan O'Rear, 4-Oct-2014.)
Assertion
Ref Expression
acongeq ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 = 𝐶 ↔ ((2 · 𝐴) ∥ (𝐵𝐶) ∨ (2 · 𝐴) ∥ (𝐵 − -𝐶))))

Proof of Theorem acongeq
StepHypRef Expression
1 2z 12002 . . . . . . 7 2 ∈ ℤ
2 nnz 11992 . . . . . . . 8 (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
323ad2ant1 1125 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℤ)
4 zmulcl 12019 . . . . . . 7 ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (2 · 𝐴) ∈ ℤ)
51, 3, 4sylancr 587 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (2 · 𝐴) ∈ ℤ)
6 elfzelz 12896 . . . . . . 7 (𝐵 ∈ (0...𝐴) → 𝐵 ∈ ℤ)
763ad2ant2 1126 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℤ)
8 congid 39446 . . . . . 6 (((2 · 𝐴) ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 · 𝐴) ∥ (𝐵𝐵))
95, 7, 8syl2anc 584 . . . . 5 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (2 · 𝐴) ∥ (𝐵𝐵))
109adantr 481 . . . 4 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝐵 = 𝐶) → (2 · 𝐴) ∥ (𝐵𝐵))
11 oveq2 7153 . . . . 5 (𝐵 = 𝐶 → (𝐵𝐵) = (𝐵𝐶))
1211adantl 482 . . . 4 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝐵 = 𝐶) → (𝐵𝐵) = (𝐵𝐶))
1310, 12breqtrd 5083 . . 3 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝐵 = 𝐶) → (2 · 𝐴) ∥ (𝐵𝐶))
1413orcd 869 . 2 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝐵 = 𝐶) → ((2 · 𝐴) ∥ (𝐵𝐶) ∨ (2 · 𝐴) ∥ (𝐵 − -𝐶)))
15 elfzelz 12896 . . . . . . . . . 10 (𝐶 ∈ (0...𝐴) → 𝐶 ∈ ℤ)
16153ad2ant3 1127 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℤ)
177, 16zsubcld 12080 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵𝐶) ∈ ℤ)
1817zcnd 12076 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵𝐶) ∈ ℂ)
1918abscld 14784 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (abs‘(𝐵𝐶)) ∈ ℝ)
20 nnre 11633 . . . . . . . 8 (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
21203ad2ant1 1125 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℝ)
22 0re 10631 . . . . . . 7 0 ∈ ℝ
23 resubcl 10938 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 − 0) ∈ ℝ)
2421, 22, 23sylancl 586 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 0) ∈ ℝ)
25 2re 11699 . . . . . . 7 2 ∈ ℝ
26 remulcl 10610 . . . . . . 7 ((2 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (2 · 𝐴) ∈ ℝ)
2725, 21, 26sylancr 587 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (2 · 𝐴) ∈ ℝ)
28 simp2 1129 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ (0...𝐴))
29 simp3 1130 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ (0...𝐴))
3024leidd 11194 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 0) ≤ (𝐴 − 0))
31 fzmaxdif 39456 . . . . . . 7 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ (0...𝐴)) ∧ (𝐴 ∈ ℤ ∧ 𝐶 ∈ (0...𝐴)) ∧ (𝐴 − 0) ≤ (𝐴 − 0)) → (abs‘(𝐵𝐶)) ≤ (𝐴 − 0))
323, 28, 3, 29, 30, 31syl221anc 1373 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (abs‘(𝐵𝐶)) ≤ (𝐴 − 0))
33 nnrp 12388 . . . . . . . . 9 (𝐴 ∈ ℕ → 𝐴 ∈ ℝ+)
34333ad2ant1 1125 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℝ+)
3521, 34ltaddrpd 12452 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 < (𝐴 + 𝐴))
3621recnd 10657 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℂ)
3736subid1d 10974 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 0) = 𝐴)
38362timesd 11868 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (2 · 𝐴) = (𝐴 + 𝐴))
3935, 37, 383brtr4d 5089 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 0) < (2 · 𝐴))
4019, 24, 27, 32, 39lelttrd 10786 . . . . 5 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (abs‘(𝐵𝐶)) < (2 · 𝐴))
4140adantr 481 . . . 4 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵𝐶)) → (abs‘(𝐵𝐶)) < (2 · 𝐴))
42 2nn 11698 . . . . . 6 2 ∈ ℕ
43 simpl1 1183 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵𝐶)) → 𝐴 ∈ ℕ)
44 nnmulcl 11649 . . . . . 6 ((2 ∈ ℕ ∧ 𝐴 ∈ ℕ) → (2 · 𝐴) ∈ ℕ)
4542, 43, 44sylancr 587 . . . . 5 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵𝐶)) → (2 · 𝐴) ∈ ℕ)
46 simpl2 1184 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵𝐶)) → 𝐵 ∈ (0...𝐴))
4746, 6syl 17 . . . . 5 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵𝐶)) → 𝐵 ∈ ℤ)
48 simpl3 1185 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵𝐶)) → 𝐶 ∈ (0...𝐴))
4948, 15syl 17 . . . . 5 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵𝐶)) → 𝐶 ∈ ℤ)
50 simpr 485 . . . . 5 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵𝐶)) → (2 · 𝐴) ∥ (𝐵𝐶))
51 congabseq 39449 . . . . 5 ((((2 · 𝐴) ∈ ℕ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (2 · 𝐴) ∥ (𝐵𝐶)) → ((abs‘(𝐵𝐶)) < (2 · 𝐴) ↔ 𝐵 = 𝐶))
5245, 47, 49, 50, 51syl31anc 1365 . . . 4 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵𝐶)) → ((abs‘(𝐵𝐶)) < (2 · 𝐴) ↔ 𝐵 = 𝐶))
5341, 52mpbid 233 . . 3 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵𝐶)) → 𝐵 = 𝐶)
54 simpll2 1205 . . . . . . . . . . 11 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐵 ∈ (0...𝐴))
55 elfzle1 12898 . . . . . . . . . . 11 (𝐵 ∈ (0...𝐴) → 0 ≤ 𝐵)
5654, 55syl 17 . . . . . . . . . 10 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 0 ≤ 𝐵)
577zred 12075 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℝ)
5816zred 12075 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℝ)
5958renegcld 11055 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → -𝐶 ∈ ℝ)
6057, 59resubcld 11056 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − -𝐶) ∈ ℝ)
6160recnd 10657 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − -𝐶) ∈ ℂ)
6261abscld 14784 . . . . . . . . . . . . 13 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (abs‘(𝐵 − -𝐶)) ∈ ℝ)
6362ad2antrr 722 . . . . . . . . . . . 12 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (abs‘(𝐵 − -𝐶)) ∈ ℝ)
64 1re 10629 . . . . . . . . . . . . . . . 16 1 ∈ ℝ
65 resubcl 10938 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐴 − 1) ∈ ℝ)
6621, 64, 65sylancl 586 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 1) ∈ ℝ)
6766renegcld 11055 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → -(𝐴 − 1) ∈ ℝ)
6821, 67resubcld 11056 . . . . . . . . . . . . 13 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − -(𝐴 − 1)) ∈ ℝ)
6968ad2antrr 722 . . . . . . . . . . . 12 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (𝐴 − -(𝐴 − 1)) ∈ ℝ)
7027ad2antrr 722 . . . . . . . . . . . 12 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (2 · 𝐴) ∈ ℝ)
717ad2antrr 722 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐵 ∈ ℤ)
7271zcnd 12076 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐵 ∈ ℂ)
7316znegcld 12077 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → -𝐶 ∈ ℤ)
7473ad2antrr 722 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 ∈ ℤ)
7574zcnd 12076 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 ∈ ℂ)
7672, 75abssubd 14801 . . . . . . . . . . . . 13 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (abs‘(𝐵 − -𝐶)) = (abs‘(-𝐶𝐵)))
77 0zd 11981 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 0 ∈ ℤ)
78 simpr 485 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐶 ∈ (0...(𝐴 − 1)))
79 0zd 11981 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 0 ∈ ℤ)
80 1z 12000 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℤ
81 zsubcl 12012 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℤ ∧ 1 ∈ ℤ) → (𝐴 − 1) ∈ ℤ)
823, 80, 81sylancl 586 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 1) ∈ ℤ)
83 fzneg 39457 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ ℤ ∧ 0 ∈ ℤ ∧ (𝐴 − 1) ∈ ℤ) → (𝐶 ∈ (0...(𝐴 − 1)) ↔ -𝐶 ∈ (-(𝐴 − 1)...-0)))
8416, 79, 82, 83syl3anc 1363 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐶 ∈ (0...(𝐴 − 1)) ↔ -𝐶 ∈ (-(𝐴 − 1)...-0)))
8584ad2antrr 722 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (𝐶 ∈ (0...(𝐴 − 1)) ↔ -𝐶 ∈ (-(𝐴 − 1)...-0)))
8678, 85mpbid 233 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 ∈ (-(𝐴 − 1)...-0))
87 neg0 10920 . . . . . . . . . . . . . . . . 17 -0 = 0
8887a1i 11 . . . . . . . . . . . . . . . 16 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -0 = 0)
8988oveq2d 7161 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (-(𝐴 − 1)...-0) = (-(𝐴 − 1)...0))
9086, 89eleqtrd 2912 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 ∈ (-(𝐴 − 1)...0))
913ad2antrr 722 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐴 ∈ ℤ)
92 simp1 1128 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℕ)
9342, 92, 44sylancr 587 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (2 · 𝐴) ∈ ℕ)
94 nnm1nn0 11926 . . . . . . . . . . . . . . . . . 18 ((2 · 𝐴) ∈ ℕ → ((2 · 𝐴) − 1) ∈ ℕ0)
9593, 94syl 17 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((2 · 𝐴) − 1) ∈ ℕ0)
9695nn0ge0d 11946 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 0 ≤ ((2 · 𝐴) − 1))
97 0m0e0 11745 . . . . . . . . . . . . . . . . 17 (0 − 0) = 0
9897a1i 11 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (0 − 0) = 0)
99 1cnd 10624 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 1 ∈ ℂ)
10036, 36, 99addsubassd 11005 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((𝐴 + 𝐴) − 1) = (𝐴 + (𝐴 − 1)))
10138oveq1d 7160 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((2 · 𝐴) − 1) = ((𝐴 + 𝐴) − 1))
102 ax-1cn 10583 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℂ
103 subcl 10873 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ)
10436, 102, 103sylancl 586 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 1) ∈ ℂ)
10536, 104subnegd 10992 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − -(𝐴 − 1)) = (𝐴 + (𝐴 − 1)))
106100, 101, 1053eqtr4rd 2864 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − -(𝐴 − 1)) = ((2 · 𝐴) − 1))
10796, 98, 1063brtr4d 5089 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (0 − 0) ≤ (𝐴 − -(𝐴 − 1)))
108107ad2antrr 722 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (0 − 0) ≤ (𝐴 − -(𝐴 − 1)))
109 fzmaxdif 39456 . . . . . . . . . . . . . 14 (((0 ∈ ℤ ∧ -𝐶 ∈ (-(𝐴 − 1)...0)) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ (0...𝐴)) ∧ (0 − 0) ≤ (𝐴 − -(𝐴 − 1))) → (abs‘(-𝐶𝐵)) ≤ (𝐴 − -(𝐴 − 1)))
11077, 90, 91, 54, 108, 109syl221anc 1373 . . . . . . . . . . . . 13 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (abs‘(-𝐶𝐵)) ≤ (𝐴 − -(𝐴 − 1)))
11176, 110eqbrtrd 5079 . . . . . . . . . . . 12 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (abs‘(𝐵 − -𝐶)) ≤ (𝐴 − -(𝐴 − 1)))
11227ltm1d 11560 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((2 · 𝐴) − 1) < (2 · 𝐴))
113106, 112eqbrtrd 5079 . . . . . . . . . . . . 13 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − -(𝐴 − 1)) < (2 · 𝐴))
114113ad2antrr 722 . . . . . . . . . . . 12 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (𝐴 − -(𝐴 − 1)) < (2 · 𝐴))
11563, 69, 70, 111, 114lelttrd 10786 . . . . . . . . . . 11 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (abs‘(𝐵 − -𝐶)) < (2 · 𝐴))
11693ad2antrr 722 . . . . . . . . . . . 12 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (2 · 𝐴) ∈ ℕ)
117 simplr 765 . . . . . . . . . . . 12 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (2 · 𝐴) ∥ (𝐵 − -𝐶))
118 congabseq 39449 . . . . . . . . . . . 12 ((((2 · 𝐴) ∈ ℕ ∧ 𝐵 ∈ ℤ ∧ -𝐶 ∈ ℤ) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) → ((abs‘(𝐵 − -𝐶)) < (2 · 𝐴) ↔ 𝐵 = -𝐶))
119116, 71, 74, 117, 118syl31anc 1365 . . . . . . . . . . 11 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → ((abs‘(𝐵 − -𝐶)) < (2 · 𝐴) ↔ 𝐵 = -𝐶))
120115, 119mpbid 233 . . . . . . . . . 10 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐵 = -𝐶)
12156, 120breqtrd 5083 . . . . . . . . 9 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 0 ≤ -𝐶)
122 elfzelz 12896 . . . . . . . . . . . 12 (𝐶 ∈ (0...(𝐴 − 1)) → 𝐶 ∈ ℤ)
123122zred 12075 . . . . . . . . . . 11 (𝐶 ∈ (0...(𝐴 − 1)) → 𝐶 ∈ ℝ)
124123adantl 482 . . . . . . . . . 10 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐶 ∈ ℝ)
125124le0neg1d 11199 . . . . . . . . 9 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (𝐶 ≤ 0 ↔ 0 ≤ -𝐶))
126121, 125mpbird 258 . . . . . . . 8 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐶 ≤ 0)
127 elfzle1 12898 . . . . . . . . 9 (𝐶 ∈ (0...(𝐴 − 1)) → 0 ≤ 𝐶)
128127adantl 482 . . . . . . . 8 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 0 ≤ 𝐶)
129 letri3 10714 . . . . . . . . 9 ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐶 = 0 ↔ (𝐶 ≤ 0 ∧ 0 ≤ 𝐶)))
130124, 22, 129sylancl 586 . . . . . . . 8 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (𝐶 = 0 ↔ (𝐶 ≤ 0 ∧ 0 ≤ 𝐶)))
131126, 128, 130mpbir2and 709 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐶 = 0)
132131negeqd 10868 . . . . . 6 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 = -0)
133132, 88eqtrd 2853 . . . . 5 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 = 0)
134133, 120, 1313eqtr4d 2863 . . . 4 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐵 = 𝐶)
135 oveq2 7153 . . . . . . . . 9 (𝐶 = 𝐴 → (𝐵𝐶) = (𝐵𝐴))
136135adantl 482 . . . . . . . 8 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (𝐵𝐶) = (𝐵𝐴))
137136fveq2d 6667 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (abs‘(𝐵𝐶)) = (abs‘(𝐵𝐴)))
13840ad2antrr 722 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (abs‘(𝐵𝐶)) < (2 · 𝐴))
139137, 138eqbrtrrd 5081 . . . . . 6 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (abs‘(𝐵𝐴)) < (2 · 𝐴))
14093ad2antrr 722 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (2 · 𝐴) ∈ ℕ)
1417ad2antrr 722 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → 𝐵 ∈ ℤ)
1423ad2antrr 722 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → 𝐴 ∈ ℤ)
143 simplr 765 . . . . . . . . 9 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (2 · 𝐴) ∥ (𝐵 − -𝐶))
1447zcnd 12076 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℂ)
14536, 36, 144ppncand 11025 . . . . . . . . . . . . 13 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((𝐴 + 𝐴) + (𝐵𝐴)) = (𝐴 + 𝐵))
14636, 144addcomd 10830 . . . . . . . . . . . . 13 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
147145, 146eqtrd 2853 . . . . . . . . . . . 12 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((𝐴 + 𝐴) + (𝐵𝐴)) = (𝐵 + 𝐴))
148147ad2antrr 722 . . . . . . . . . . 11 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → ((𝐴 + 𝐴) + (𝐵𝐴)) = (𝐵 + 𝐴))
149 oveq2 7153 . . . . . . . . . . . 12 (𝐶 = 𝐴 → (𝐵 + 𝐶) = (𝐵 + 𝐴))
150149adantl 482 . . . . . . . . . . 11 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (𝐵 + 𝐶) = (𝐵 + 𝐴))
151148, 150eqtr4d 2856 . . . . . . . . . 10 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → ((𝐴 + 𝐴) + (𝐵𝐴)) = (𝐵 + 𝐶))
15238oveq1d 7160 . . . . . . . . . . 11 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((2 · 𝐴) + (𝐵𝐴)) = ((𝐴 + 𝐴) + (𝐵𝐴)))
153152ad2antrr 722 . . . . . . . . . 10 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → ((2 · 𝐴) + (𝐵𝐴)) = ((𝐴 + 𝐴) + (𝐵𝐴)))
15416zcnd 12076 . . . . . . . . . . . 12 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℂ)
155144, 154subnegd 10992 . . . . . . . . . . 11 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − -𝐶) = (𝐵 + 𝐶))
156155ad2antrr 722 . . . . . . . . . 10 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (𝐵 − -𝐶) = (𝐵 + 𝐶))
157151, 153, 1563eqtr4d 2863 . . . . . . . . 9 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → ((2 · 𝐴) + (𝐵𝐴)) = (𝐵 − -𝐶))
158143, 157breqtrrd 5085 . . . . . . . 8 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (2 · 𝐴) ∥ ((2 · 𝐴) + (𝐵𝐴)))
1595ad2antrr 722 . . . . . . . . 9 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (2 · 𝐴) ∈ ℤ)
1607, 3zsubcld 12080 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵𝐴) ∈ ℤ)
161160ad2antrr 722 . . . . . . . . 9 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (𝐵𝐴) ∈ ℤ)
162 dvdsadd 15640 . . . . . . . . 9 (((2 · 𝐴) ∈ ℤ ∧ (𝐵𝐴) ∈ ℤ) → ((2 · 𝐴) ∥ (𝐵𝐴) ↔ (2 · 𝐴) ∥ ((2 · 𝐴) + (𝐵𝐴))))
163159, 161, 162syl2anc 584 . . . . . . . 8 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → ((2 · 𝐴) ∥ (𝐵𝐴) ↔ (2 · 𝐴) ∥ ((2 · 𝐴) + (𝐵𝐴))))
164158, 163mpbird 258 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (2 · 𝐴) ∥ (𝐵𝐴))
165 congabseq 39449 . . . . . . 7 ((((2 · 𝐴) ∈ ℕ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) ∧ (2 · 𝐴) ∥ (𝐵𝐴)) → ((abs‘(𝐵𝐴)) < (2 · 𝐴) ↔ 𝐵 = 𝐴))
166140, 141, 142, 164, 165syl31anc 1365 . . . . . 6 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → ((abs‘(𝐵𝐴)) < (2 · 𝐴) ↔ 𝐵 = 𝐴))
167139, 166mpbid 233 . . . . 5 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → 𝐵 = 𝐴)
168 simpr 485 . . . . 5 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → 𝐶 = 𝐴)
169167, 168eqtr4d 2856 . . . 4 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → 𝐵 = 𝐶)
170 nnnn0 11892 . . . . . . . 8 (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
1711703ad2ant1 1125 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℕ0)
172 nn0uz 12268 . . . . . . 7 0 = (ℤ‘0)
173171, 172eleqtrdi 2920 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ (ℤ‘0))
174 fzm1 12975 . . . . . . 7 (𝐴 ∈ (ℤ‘0) → (𝐶 ∈ (0...𝐴) ↔ (𝐶 ∈ (0...(𝐴 − 1)) ∨ 𝐶 = 𝐴)))
175174biimpa 477 . . . . . 6 ((𝐴 ∈ (ℤ‘0) ∧ 𝐶 ∈ (0...𝐴)) → (𝐶 ∈ (0...(𝐴 − 1)) ∨ 𝐶 = 𝐴))
176173, 29, 175syl2anc 584 . . . . 5 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐶 ∈ (0...(𝐴 − 1)) ∨ 𝐶 = 𝐴))
177176adantr 481 . . . 4 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) → (𝐶 ∈ (0...(𝐴 − 1)) ∨ 𝐶 = 𝐴))
178134, 169, 177mpjaodan 952 . . 3 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) → 𝐵 = 𝐶)
17953, 178jaodan 951 . 2 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ ((2 · 𝐴) ∥ (𝐵𝐶) ∨ (2 · 𝐴) ∥ (𝐵 − -𝐶))) → 𝐵 = 𝐶)
18014, 179impbida 797 1 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 = 𝐶 ↔ ((2 · 𝐴) ∥ (𝐵𝐶) ∨ (2 · 𝐴) ∥ (𝐵 − -𝐶))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105   class class class wbr 5057  cfv 6348  (class class class)co 7145  cc 10523  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   · cmul 10530   < clt 10663  cle 10664  cmin 10858  -cneg 10859  cn 11626  2c2 11680  0cn0 11885  cz 11969  cuz 12231  +crp 12377  ...cfz 12880  abscabs 14581  cdvds 15595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12881  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-dvds 15596
This theorem is referenced by:  jm2.27a  39480
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