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Theorem acongeq 41293
Description: Two numbers in the fundamental domain are alternating-congruent iff they are equal. TODO: could be used to shorten jm2.26 41312. (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 12535 . . . . . . 7 2 ∈ ℤ
2 nnz 12520 . . . . . . . 8 (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
323ad2ant1 1133 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℤ)
4 zmulcl 12552 . . . . . . 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 13441 . . . . . . 7 (𝐵 ∈ (0...𝐴) → 𝐵 ∈ ℤ)
763ad2ant2 1134 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℤ)
8 congid 41281 . . . . . 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 7365 . . . . 5 (𝐵 = 𝐶 → (𝐵𝐵) = (𝐵𝐶))
1211adantl 482 . . . 4 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝐵 = 𝐶) → (𝐵𝐵) = (𝐵𝐶))
1310, 12breqtrd 5131 . . 3 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝐵 = 𝐶) → (2 · 𝐴) ∥ (𝐵𝐶))
1413orcd 871 . 2 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝐵 = 𝐶) → ((2 · 𝐴) ∥ (𝐵𝐶) ∨ (2 · 𝐴) ∥ (𝐵 − -𝐶)))
15 elfzelz 13441 . . . . . . . . . 10 (𝐶 ∈ (0...𝐴) → 𝐶 ∈ ℤ)
16153ad2ant3 1135 . . . . . . . . 9 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℤ)
177, 16zsubcld 12612 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵𝐶) ∈ ℤ)
1817zcnd 12608 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵𝐶) ∈ ℂ)
1918abscld 15321 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (abs‘(𝐵𝐶)) ∈ ℝ)
20 nnre 12160 . . . . . . . 8 (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
21203ad2ant1 1133 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℝ)
22 0re 11157 . . . . . . 7 0 ∈ ℝ
23 resubcl 11465 . . . . . . 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 12227 . . . . . . 7 2 ∈ ℝ
26 remulcl 11136 . . . . . . 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 1137 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ (0...𝐴))
29 simp3 1138 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ (0...𝐴))
3024leidd 11721 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 0) ≤ (𝐴 − 0))
31 fzmaxdif 41291 . . . . . . 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 1381 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (abs‘(𝐵𝐶)) ≤ (𝐴 − 0))
33 nnrp 12926 . . . . . . . . 9 (𝐴 ∈ ℕ → 𝐴 ∈ ℝ+)
34333ad2ant1 1133 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℝ+)
3521, 34ltaddrpd 12990 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 < (𝐴 + 𝐴))
3621recnd 11183 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℂ)
3736subid1d 11501 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 0) = 𝐴)
38362timesd 12396 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (2 · 𝐴) = (𝐴 + 𝐴))
3935, 37, 383brtr4d 5137 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 0) < (2 · 𝐴))
4019, 24, 27, 32, 39lelttrd 11313 . . . . 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 12226 . . . . . 6 2 ∈ ℕ
43 simpl1 1191 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵𝐶)) → 𝐴 ∈ ℕ)
44 nnmulcl 12177 . . . . . 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 1192 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵𝐶)) → 𝐵 ∈ (0...𝐴))
4746elfzelzd 13442 . . . . 5 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵𝐶)) → 𝐵 ∈ ℤ)
48 simpl3 1193 . . . . . 6 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵𝐶)) → 𝐶 ∈ (0...𝐴))
4948elfzelzd 13442 . . . . 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 41284 . . . . 5 ((((2 · 𝐴) ∈ ℕ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (2 · 𝐴) ∥ (𝐵𝐶)) → ((abs‘(𝐵𝐶)) < (2 · 𝐴) ↔ 𝐵 = 𝐶))
5245, 47, 49, 50, 51syl31anc 1373 . . . 4 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵𝐶)) → ((abs‘(𝐵𝐶)) < (2 · 𝐴) ↔ 𝐵 = 𝐶))
5341, 52mpbid 231 . . 3 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵𝐶)) → 𝐵 = 𝐶)
54 simpll2 1213 . . . . . . . . . . 11 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐵 ∈ (0...𝐴))
55 elfzle1 13444 . . . . . . . . . . 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 12607 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℝ)
5816zred 12607 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℝ)
5958renegcld 11582 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → -𝐶 ∈ ℝ)
6057, 59resubcld 11583 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − -𝐶) ∈ ℝ)
6160recnd 11183 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − -𝐶) ∈ ℂ)
6261abscld 15321 . . . . . . . . . . . . 13 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (abs‘(𝐵 − -𝐶)) ∈ ℝ)
6362ad2antrr 724 . . . . . . . . . . . 12 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (abs‘(𝐵 − -𝐶)) ∈ ℝ)
64 1re 11155 . . . . . . . . . . . . . . . 16 1 ∈ ℝ
65 resubcl 11465 . . . . . . . . . . . . . . . 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 11582 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → -(𝐴 − 1) ∈ ℝ)
6821, 67resubcld 11583 . . . . . . . . . . . . 13 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − -(𝐴 − 1)) ∈ ℝ)
6968ad2antrr 724 . . . . . . . . . . . 12 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (𝐴 − -(𝐴 − 1)) ∈ ℝ)
7027ad2antrr 724 . . . . . . . . . . . 12 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (2 · 𝐴) ∈ ℝ)
717ad2antrr 724 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐵 ∈ ℤ)
7271zcnd 12608 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐵 ∈ ℂ)
7316znegcld 12609 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → -𝐶 ∈ ℤ)
7473ad2antrr 724 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 ∈ ℤ)
7574zcnd 12608 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 ∈ ℂ)
7672, 75abssubd 15338 . . . . . . . . . . . . 13 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (abs‘(𝐵 − -𝐶)) = (abs‘(-𝐶𝐵)))
77 0zd 12511 . . . . . . . . . . . . . 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 12511 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 0 ∈ ℤ)
80 1z 12533 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℤ
81 zsubcl 12545 . . . . . . . . . . . . . . . . . . 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 41292 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ ℤ ∧ 0 ∈ ℤ ∧ (𝐴 − 1) ∈ ℤ) → (𝐶 ∈ (0...(𝐴 − 1)) ↔ -𝐶 ∈ (-(𝐴 − 1)...-0)))
8416, 79, 82, 83syl3anc 1371 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐶 ∈ (0...(𝐴 − 1)) ↔ -𝐶 ∈ (-(𝐴 − 1)...-0)))
8584ad2antrr 724 . . . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . . . 15 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 ∈ (-(𝐴 − 1)...-0))
87 neg0 11447 . . . . . . . . . . . . . . . . 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 7373 . . . . . . . . . . . . . . 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 2840 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 ∈ (-(𝐴 − 1)...0))
913ad2antrr 724 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐴 ∈ ℤ)
92 simp1 1136 . . . . . . . . . . . . . . . . . . 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 12454 . . . . . . . . . . . . . . . . . 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 12476 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 0 ≤ ((2 · 𝐴) − 1))
97 0m0e0 12273 . . . . . . . . . . . . . . . . 17 (0 − 0) = 0
9897a1i 11 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (0 − 0) = 0)
99 1cnd 11150 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 1 ∈ ℂ)
10036, 36, 99addsubassd 11532 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((𝐴 + 𝐴) − 1) = (𝐴 + (𝐴 − 1)))
10138oveq1d 7372 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((2 · 𝐴) − 1) = ((𝐴 + 𝐴) − 1))
102 ax-1cn 11109 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℂ
103 subcl 11400 . . . . . . . . . . . . . . . . . . 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 11519 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − -(𝐴 − 1)) = (𝐴 + (𝐴 − 1)))
106100, 101, 1053eqtr4rd 2787 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − -(𝐴 − 1)) = ((2 · 𝐴) − 1))
10796, 98, 1063brtr4d 5137 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (0 − 0) ≤ (𝐴 − -(𝐴 − 1)))
108107ad2antrr 724 . . . . . . . . . . . . . 14 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (0 − 0) ≤ (𝐴 − -(𝐴 − 1)))
109 fzmaxdif 41291 . . . . . . . . . . . . . 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 1381 . . . . . . . . . . . . 13 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (abs‘(-𝐶𝐵)) ≤ (𝐴 − -(𝐴 − 1)))
11176, 110eqbrtrd 5127 . . . . . . . . . . . 12 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (abs‘(𝐵 − -𝐶)) ≤ (𝐴 − -(𝐴 − 1)))
11227ltm1d 12087 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((2 · 𝐴) − 1) < (2 · 𝐴))
113106, 112eqbrtrd 5127 . . . . . . . . . . . . 13 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − -(𝐴 − 1)) < (2 · 𝐴))
114113ad2antrr 724 . . . . . . . . . . . 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 11313 . . . . . . . . . . 11 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (abs‘(𝐵 − -𝐶)) < (2 · 𝐴))
11693ad2antrr 724 . . . . . . . . . . . 12 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (2 · 𝐴) ∈ ℕ)
117 simplr 767 . . . . . . . . . . . 12 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (2 · 𝐴) ∥ (𝐵 − -𝐶))
118 congabseq 41284 . . . . . . . . . . . 12 ((((2 · 𝐴) ∈ ℕ ∧ 𝐵 ∈ ℤ ∧ -𝐶 ∈ ℤ) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) → ((abs‘(𝐵 − -𝐶)) < (2 · 𝐴) ↔ 𝐵 = -𝐶))
119116, 71, 74, 117, 118syl31anc 1373 . . . . . . . . . . 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 231 . . . . . . . . . 10 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐵 = -𝐶)
12156, 120breqtrd 5131 . . . . . . . . 9 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 0 ≤ -𝐶)
122 elfzelz 13441 . . . . . . . . . . . 12 (𝐶 ∈ (0...(𝐴 − 1)) → 𝐶 ∈ ℤ)
123122zred 12607 . . . . . . . . . . 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 11726 . . . . . . . . 9 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (𝐶 ≤ 0 ↔ 0 ≤ -𝐶))
126121, 125mpbird 256 . . . . . . . 8 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐶 ≤ 0)
127 elfzle1 13444 . . . . . . . . 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 11240 . . . . . . . . 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 711 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐶 = 0)
132131negeqd 11395 . . . . . 6 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 = -0)
133132, 88eqtrd 2776 . . . . 5 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 = 0)
134133, 120, 1313eqtr4d 2786 . . . 4 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐵 = 𝐶)
135 oveq2 7365 . . . . . . . . 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 6846 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (abs‘(𝐵𝐶)) = (abs‘(𝐵𝐴)))
13840ad2antrr 724 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (abs‘(𝐵𝐶)) < (2 · 𝐴))
139137, 138eqbrtrrd 5129 . . . . . 6 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (abs‘(𝐵𝐴)) < (2 · 𝐴))
14093ad2antrr 724 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (2 · 𝐴) ∈ ℕ)
1417ad2antrr 724 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → 𝐵 ∈ ℤ)
1423ad2antrr 724 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → 𝐴 ∈ ℤ)
143 simplr 767 . . . . . . . . 9 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (2 · 𝐴) ∥ (𝐵 − -𝐶))
1447zcnd 12608 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℂ)
14536, 36, 144ppncand 11552 . . . . . . . . . . . . 13 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((𝐴 + 𝐴) + (𝐵𝐴)) = (𝐴 + 𝐵))
14636, 144addcomd 11357 . . . . . . . . . . . . 13 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
147145, 146eqtrd 2776 . . . . . . . . . . . 12 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((𝐴 + 𝐴) + (𝐵𝐴)) = (𝐵 + 𝐴))
148147ad2antrr 724 . . . . . . . . . . 11 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → ((𝐴 + 𝐴) + (𝐵𝐴)) = (𝐵 + 𝐴))
149 oveq2 7365 . . . . . . . . . . . 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 2779 . . . . . . . . . 10 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → ((𝐴 + 𝐴) + (𝐵𝐴)) = (𝐵 + 𝐶))
15238oveq1d 7372 . . . . . . . . . . 11 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((2 · 𝐴) + (𝐵𝐴)) = ((𝐴 + 𝐴) + (𝐵𝐴)))
153152ad2antrr 724 . . . . . . . . . 10 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → ((2 · 𝐴) + (𝐵𝐴)) = ((𝐴 + 𝐴) + (𝐵𝐴)))
15416zcnd 12608 . . . . . . . . . . . 12 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℂ)
155144, 154subnegd 11519 . . . . . . . . . . 11 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − -𝐶) = (𝐵 + 𝐶))
156155ad2antrr 724 . . . . . . . . . 10 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (𝐵 − -𝐶) = (𝐵 + 𝐶))
157151, 153, 1563eqtr4d 2786 . . . . . . . . 9 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → ((2 · 𝐴) + (𝐵𝐴)) = (𝐵 − -𝐶))
158143, 157breqtrrd 5133 . . . . . . . 8 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (2 · 𝐴) ∥ ((2 · 𝐴) + (𝐵𝐴)))
1595ad2antrr 724 . . . . . . . . 9 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (2 · 𝐴) ∈ ℤ)
1607, 3zsubcld 12612 . . . . . . . . . 10 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵𝐴) ∈ ℤ)
161160ad2antrr 724 . . . . . . . . 9 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (𝐵𝐴) ∈ ℤ)
162 dvdsadd 16184 . . . . . . . . 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 256 . . . . . . 7 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (2 · 𝐴) ∥ (𝐵𝐴))
165 congabseq 41284 . . . . . . 7 ((((2 · 𝐴) ∈ ℕ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) ∧ (2 · 𝐴) ∥ (𝐵𝐴)) → ((abs‘(𝐵𝐴)) < (2 · 𝐴) ↔ 𝐵 = 𝐴))
166140, 141, 142, 164, 165syl31anc 1373 . . . . . 6 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → ((abs‘(𝐵𝐴)) < (2 · 𝐴) ↔ 𝐵 = 𝐴))
167139, 166mpbid 231 . . . . 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 2779 . . . 4 ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → 𝐵 = 𝐶)
170 nnnn0 12420 . . . . . . . 8 (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
1711703ad2ant1 1133 . . . . . . 7 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℕ0)
172 nn0uz 12805 . . . . . . 7 0 = (ℤ‘0)
173171, 172eleqtrdi 2848 . . . . . 6 ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ (ℤ‘0))
174 fzm1 13521 . . . . . . 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 957 . . 3 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) → 𝐵 = 𝐶)
17953, 178jaodan 956 . 2 (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ ((2 · 𝐴) ∥ (𝐵𝐶) ∨ (2 · 𝐴) ∥ (𝐵 − -𝐶))) → 𝐵 = 𝐶)
18014, 179impbida 799 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 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106   class class class wbr 5105  cfv 6496  (class class class)co 7357  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056   < clt 11189  cle 11190  cmin 11385  -cneg 11386  cn 12153  2c2 12208  0cn0 12413  cz 12499  cuz 12763  +crp 12915  ...cfz 13424  abscabs 15119  cdvds 16136
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-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  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-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-dvds 16137
This theorem is referenced by:  jm2.27a  41315
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