acongeq - Mathbox for Stefan O'Rear

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Theorem acongeq 43225

Description: Two numbers in the fundamental domain are alternating-congruent iff they are equal. TODO: could be used to shorten jm2.26 43244. (Contributed by Stefan O'Rear, 4-Oct-2014.)

**Assertion**
Ref	Expression
acongeq	⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 = 𝐶 ↔ ((2 · 𝐴) ∥ (𝐵 − 𝐶) ∨ (2 · 𝐴) ∥ (𝐵 − -𝐶))))

**Proof of Theorem acongeq**
Step	Hyp	Ref	Expression
1		2z 12523	. . . . . . 7 ⊢ 2 ∈ ℤ
2		nnz 12509	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
3	2	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℤ)
4		zmulcl 12540	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (2 · 𝐴) ∈ ℤ)
5	1, 3, 4	sylancr 587	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (2 · 𝐴) ∈ ℤ)
6		elfzelz 13440	. . . . . . 7 ⊢ (𝐵 ∈ (0...𝐴) → 𝐵 ∈ ℤ)
7	6	3ad2ant2 1134	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℤ)
8		congid 43213	. . . . . 6 ⊢ (((2 · 𝐴) ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 · 𝐴) ∥ (𝐵 − 𝐵))
9	5, 7, 8	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (2 · 𝐴) ∥ (𝐵 − 𝐵))
10	9	adantr 480	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝐵 = 𝐶) → (2 · 𝐴) ∥ (𝐵 − 𝐵))
11		oveq2 7366	. . . . 5 ⊢ (𝐵 = 𝐶 → (𝐵 − 𝐵) = (𝐵 − 𝐶))
12	11	adantl 481	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝐵 = 𝐶) → (𝐵 − 𝐵) = (𝐵 − 𝐶))
13	10, 12	breqtrd 5124	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝐵 = 𝐶) → (2 · 𝐴) ∥ (𝐵 − 𝐶))
14	13	orcd 873	. 2 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝐵 = 𝐶) → ((2 · 𝐴) ∥ (𝐵 − 𝐶) ∨ (2 · 𝐴) ∥ (𝐵 − -𝐶)))
15		elfzelz 13440	. . . . . . . . . 10 ⊢ (𝐶 ∈ (0...𝐴) → 𝐶 ∈ ℤ)
16	15	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℤ)
17	7, 16	zsubcld 12601	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − 𝐶) ∈ ℤ)
18	17	zcnd 12597	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − 𝐶) ∈ ℂ)
19	18	abscld 15362	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (abs‘(𝐵 − 𝐶)) ∈ ℝ)
20		nnre 12152	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
21	20	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℝ)
22		0re 11134	. . . . . . 7 ⊢ 0 ∈ ℝ
23		resubcl 11445	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 − 0) ∈ ℝ)
24	21, 22, 23	sylancl 586	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 0) ∈ ℝ)
25		2re 12219	. . . . . . 7 ⊢ 2 ∈ ℝ
26		remulcl 11111	. . . . . . 7 ⊢ ((2 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (2 · 𝐴) ∈ ℝ)
27	25, 21, 26	sylancr 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))
30	24	leidd 11703	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 0) ≤ (𝐴 − 0))
31		fzmaxdif 43223	. . . . . . 7 ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ (0...𝐴)) ∧ (𝐴 ∈ ℤ ∧ 𝐶 ∈ (0...𝐴)) ∧ (𝐴 − 0) ≤ (𝐴 − 0)) → (abs‘(𝐵 − 𝐶)) ≤ (𝐴 − 0))
32	3, 28, 3, 29, 30, 31	syl221anc 1383	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (abs‘(𝐵 − 𝐶)) ≤ (𝐴 − 0))
33		nnrp 12917	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ⁺)
34	33	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℝ⁺)
35	21, 34	ltaddrpd 12982	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 < (𝐴 + 𝐴))
36	21	recnd 11160	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℂ)
37	36	subid1d 11481	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 0) = 𝐴)
38	36	2timesd 12384	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (2 · 𝐴) = (𝐴 + 𝐴))
39	35, 37, 38	3brtr4d 5130	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 0) < (2 · 𝐴))
40	19, 24, 27, 32, 39	lelttrd 11291	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (abs‘(𝐵 − 𝐶)) < (2 · 𝐴))
41	40	adantr 480	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − 𝐶)) → (abs‘(𝐵 − 𝐶)) < (2 · 𝐴))
42		2nn 12218	. . . . . 6 ⊢ 2 ∈ ℕ
43		simpl1 1192	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − 𝐶)) → 𝐴 ∈ ℕ)
44		nnmulcl 12169	. . . . . 6 ⊢ ((2 ∈ ℕ ∧ 𝐴 ∈ ℕ) → (2 · 𝐴) ∈ ℕ)
45	42, 43, 44	sylancr 587	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − 𝐶)) → (2 · 𝐴) ∈ ℕ)
46		simpl2 1193	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − 𝐶)) → 𝐵 ∈ (0...𝐴))
47	46	elfzelzd 13441	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − 𝐶)) → 𝐵 ∈ ℤ)
48		simpl3 1194	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − 𝐶)) → 𝐶 ∈ (0...𝐴))
49	48	elfzelzd 13441	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − 𝐶)) → 𝐶 ∈ ℤ)
50		simpr 484	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − 𝐶)) → (2 · 𝐴) ∥ (𝐵 − 𝐶))
51		congabseq 43216	. . . . 5 ⊢ ((((2 · 𝐴) ∈ ℕ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ (2 · 𝐴) ∥ (𝐵 − 𝐶)) → ((abs‘(𝐵 − 𝐶)) < (2 · 𝐴) ↔ 𝐵 = 𝐶))
52	45, 47, 49, 50, 51	syl31anc 1375	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − 𝐶)) → ((abs‘(𝐵 − 𝐶)) < (2 · 𝐴) ↔ 𝐵 = 𝐶))
53	41, 52	mpbid 232	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − 𝐶)) → 𝐵 = 𝐶)
54		simpll2 1214	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐵 ∈ (0...𝐴))
55		elfzle1 13443	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (0...𝐴) → 0 ≤ 𝐵)
56	54, 55	syl 17	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 0 ≤ 𝐵)
57	7	zred 12596	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℝ)
58	16	zred 12596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℝ)
59	58	renegcld 11564	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → -𝐶 ∈ ℝ)
60	57, 59	resubcld 11565	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − -𝐶) ∈ ℝ)
61	60	recnd 11160	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − -𝐶) ∈ ℂ)
62	61	abscld 15362	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (abs‘(𝐵 − -𝐶)) ∈ ℝ)
63	62	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (abs‘(𝐵 − -𝐶)) ∈ ℝ)
64		1re 11132	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
65		resubcl 11445	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐴 − 1) ∈ ℝ)
66	21, 64, 65	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 1) ∈ ℝ)
67	66	renegcld 11564	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → -(𝐴 − 1) ∈ ℝ)
68	21, 67	resubcld 11565	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − -(𝐴 − 1)) ∈ ℝ)
69	68	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (𝐴 − -(𝐴 − 1)) ∈ ℝ)
70	27	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (2 · 𝐴) ∈ ℝ)
71	7	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐵 ∈ ℤ)
72	71	zcnd 12597	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐵 ∈ ℂ)
73	16	znegcld 12598	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → -𝐶 ∈ ℤ)
74	73	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 ∈ ℤ)
75	74	zcnd 12597	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 ∈ ℂ)
76	72, 75	abssubd 15379	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (abs‘(𝐵 − -𝐶)) = (abs‘(-𝐶 − 𝐵)))
77		0zd 12500	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 0 ∈ ℤ)
78		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐶 ∈ (0...(𝐴 − 1)))
79		0zd 12500	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 0 ∈ ℤ)
80		1z 12521	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℤ
81		zsubcl 12533	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℤ ∧ 1 ∈ ℤ) → (𝐴 − 1) ∈ ℤ)
82	3, 80, 81	sylancl 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 1) ∈ ℤ)
83		fzneg 43224	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ ℤ ∧ 0 ∈ ℤ ∧ (𝐴 − 1) ∈ ℤ) → (𝐶 ∈ (0...(𝐴 − 1)) ↔ -𝐶 ∈ (-(𝐴 − 1)...-0)))
84	16, 79, 82, 83	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐶 ∈ (0...(𝐴 − 1)) ↔ -𝐶 ∈ (-(𝐴 − 1)...-0)))
85	84	ad2antrr 726	. . . . . . . . . . . . . . . 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)))
86	78, 85	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 ∈ (-(𝐴 − 1)...-0))
87		neg0 11427	. . . . . . . . . . . . . . . . 17 ⊢ -0 = 0
88	87	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -0 = 0)
89	88	oveq2d 7374	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (-(𝐴 − 1)...-0) = (-(𝐴 − 1)...0))
90	86, 89	eleqtrd 2838	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 ∈ (-(𝐴 − 1)...0))
91	3	ad2antrr 726	. . . . . . . . . . . . . 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)
93	42, 92, 44	sylancr 587	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (2 · 𝐴) ∈ ℕ)
94		nnm1nn0 12442	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 · 𝐴) ∈ ℕ → ((2 · 𝐴) − 1) ∈ ℕ₀)
95	93, 94	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((2 · 𝐴) − 1) ∈ ℕ₀)
96	95	nn0ge0d 12465	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 0 ≤ ((2 · 𝐴) − 1))
97		0m0e0 12260	. . . . . . . . . . . . . . . . 17 ⊢ (0 − 0) = 0
98	97	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (0 − 0) = 0)
99		1cnd 11127	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 1 ∈ ℂ)
100	36, 36, 99	addsubassd 11512	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((𝐴 + 𝐴) − 1) = (𝐴 + (𝐴 − 1)))
101	38	oveq1d 7373	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((2 · 𝐴) − 1) = ((𝐴 + 𝐴) − 1))
102		ax-1cn 11084	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℂ
103		subcl 11379	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ)
104	36, 102, 103	sylancl 586	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 1) ∈ ℂ)
105	36, 104	subnegd 11499	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − -(𝐴 − 1)) = (𝐴 + (𝐴 − 1)))
106	100, 101, 105	3eqtr4rd 2782	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − -(𝐴 − 1)) = ((2 · 𝐴) − 1))
107	96, 98, 106	3brtr4d 5130	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (0 − 0) ≤ (𝐴 − -(𝐴 − 1)))
108	107	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (0 − 0) ≤ (𝐴 − -(𝐴 − 1)))
109		fzmaxdif 43223	. . . . . . . . . . . . . 14 ⊢ (((0 ∈ ℤ ∧ -𝐶 ∈ (-(𝐴 − 1)...0)) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ (0...𝐴)) ∧ (0 − 0) ≤ (𝐴 − -(𝐴 − 1))) → (abs‘(-𝐶 − 𝐵)) ≤ (𝐴 − -(𝐴 − 1)))
110	77, 90, 91, 54, 108, 109	syl221anc 1383	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (abs‘(-𝐶 − 𝐵)) ≤ (𝐴 − -(𝐴 − 1)))
111	76, 110	eqbrtrd 5120	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (abs‘(𝐵 − -𝐶)) ≤ (𝐴 − -(𝐴 − 1)))
112	27	ltm1d 12074	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((2 · 𝐴) − 1) < (2 · 𝐴))
113	106, 112	eqbrtrd 5120	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − -(𝐴 − 1)) < (2 · 𝐴))
114	113	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (𝐴 − -(𝐴 − 1)) < (2 · 𝐴))
115	63, 69, 70, 111, 114	lelttrd 11291	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (abs‘(𝐵 − -𝐶)) < (2 · 𝐴))
116	93	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (2 · 𝐴) ∈ ℕ)
117		simplr 768	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (2 · 𝐴) ∥ (𝐵 − -𝐶))
118		congabseq 43216	. . . . . . . . . . . 12 ⊢ ((((2 · 𝐴) ∈ ℕ ∧ 𝐵 ∈ ℤ ∧ -𝐶 ∈ ℤ) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) → ((abs‘(𝐵 − -𝐶)) < (2 · 𝐴) ↔ 𝐵 = -𝐶))
119	116, 71, 74, 117, 118	syl31anc 1375	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → ((abs‘(𝐵 − -𝐶)) < (2 · 𝐴) ↔ 𝐵 = -𝐶))
120	115, 119	mpbid 232	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐵 = -𝐶)
121	56, 120	breqtrd 5124	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 0 ≤ -𝐶)
122		elfzelz 13440	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (0...(𝐴 − 1)) → 𝐶 ∈ ℤ)
123	122	zred 12596	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (0...(𝐴 − 1)) → 𝐶 ∈ ℝ)
124	123	adantl 481	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐶 ∈ ℝ)
125	124	le0neg1d 11708	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (𝐶 ≤ 0 ↔ 0 ≤ -𝐶))
126	121, 125	mpbird 257	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐶 ≤ 0)
127		elfzle1 13443	. . . . . . . . 9 ⊢ (𝐶 ∈ (0...(𝐴 − 1)) → 0 ≤ 𝐶)
128	127	adantl 481	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 0 ≤ 𝐶)
129		letri3 11218	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐶 = 0 ↔ (𝐶 ≤ 0 ∧ 0 ≤ 𝐶)))
130	124, 22, 129	sylancl 586	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (𝐶 = 0 ↔ (𝐶 ≤ 0 ∧ 0 ≤ 𝐶)))
131	126, 128, 130	mpbir2and 713	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐶 = 0)
132	131	negeqd 11374	. . . . . 6 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 = -0)
133	132, 88	eqtrd 2771	. . . . 5 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 = 0)
134	133, 120, 131	3eqtr4d 2781	. . . 4 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐵 = 𝐶)
135		oveq2 7366	. . . . . . . . 9 ⊢ (𝐶 = 𝐴 → (𝐵 − 𝐶) = (𝐵 − 𝐴))
136	135	adantl 481	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (𝐵 − 𝐶) = (𝐵 − 𝐴))
137	136	fveq2d 6838	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (abs‘(𝐵 − 𝐶)) = (abs‘(𝐵 − 𝐴)))
138	40	ad2antrr 726	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (abs‘(𝐵 − 𝐶)) < (2 · 𝐴))
139	137, 138	eqbrtrrd 5122	. . . . . 6 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (abs‘(𝐵 − 𝐴)) < (2 · 𝐴))
140	93	ad2antrr 726	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (2 · 𝐴) ∈ ℕ)
141	7	ad2antrr 726	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → 𝐵 ∈ ℤ)
142	3	ad2antrr 726	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → 𝐴 ∈ ℤ)
143		simplr 768	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (2 · 𝐴) ∥ (𝐵 − -𝐶))
144	7	zcnd 12597	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℂ)
145	36, 36, 144	ppncand 11532	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((𝐴 + 𝐴) + (𝐵 − 𝐴)) = (𝐴 + 𝐵))
146	36, 144	addcomd 11335	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
147	145, 146	eqtrd 2771	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((𝐴 + 𝐴) + (𝐵 − 𝐴)) = (𝐵 + 𝐴))
148	147	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → ((𝐴 + 𝐴) + (𝐵 − 𝐴)) = (𝐵 + 𝐴))
149		oveq2 7366	. . . . . . . . . . . 12 ⊢ (𝐶 = 𝐴 → (𝐵 + 𝐶) = (𝐵 + 𝐴))
150	149	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (𝐵 + 𝐶) = (𝐵 + 𝐴))
151	148, 150	eqtr4d 2774	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → ((𝐴 + 𝐴) + (𝐵 − 𝐴)) = (𝐵 + 𝐶))
152	38	oveq1d 7373	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((2 · 𝐴) + (𝐵 − 𝐴)) = ((𝐴 + 𝐴) + (𝐵 − 𝐴)))
153	152	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → ((2 · 𝐴) + (𝐵 − 𝐴)) = ((𝐴 + 𝐴) + (𝐵 − 𝐴)))
154	16	zcnd 12597	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℂ)
155	144, 154	subnegd 11499	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − -𝐶) = (𝐵 + 𝐶))
156	155	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (𝐵 − -𝐶) = (𝐵 + 𝐶))
157	151, 153, 156	3eqtr4d 2781	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → ((2 · 𝐴) + (𝐵 − 𝐴)) = (𝐵 − -𝐶))
158	143, 157	breqtrrd 5126	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (2 · 𝐴) ∥ ((2 · 𝐴) + (𝐵 − 𝐴)))
159	5	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (2 · 𝐴) ∈ ℤ)
160	7, 3	zsubcld 12601	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − 𝐴) ∈ ℤ)
161	160	ad2antrr 726	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (𝐵 − 𝐴) ∈ ℤ)
162		dvdsadd 16229	. . . . . . . . 9 ⊢ (((2 · 𝐴) ∈ ℤ ∧ (𝐵 − 𝐴) ∈ ℤ) → ((2 · 𝐴) ∥ (𝐵 − 𝐴) ↔ (2 · 𝐴) ∥ ((2 · 𝐴) + (𝐵 − 𝐴))))
163	159, 161, 162	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → ((2 · 𝐴) ∥ (𝐵 − 𝐴) ↔ (2 · 𝐴) ∥ ((2 · 𝐴) + (𝐵 − 𝐴))))
164	158, 163	mpbird 257	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (2 · 𝐴) ∥ (𝐵 − 𝐴))
165		congabseq 43216	. . . . . . 7 ⊢ ((((2 · 𝐴) ∈ ℕ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) ∧ (2 · 𝐴) ∥ (𝐵 − 𝐴)) → ((abs‘(𝐵 − 𝐴)) < (2 · 𝐴) ↔ 𝐵 = 𝐴))
166	140, 141, 142, 164, 165	syl31anc 1375	. . . . . 6 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → ((abs‘(𝐵 − 𝐴)) < (2 · 𝐴) ↔ 𝐵 = 𝐴))
167	139, 166	mpbid 232	. . . . 5 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → 𝐵 = 𝐴)
168		simpr 484	. . . . 5 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → 𝐶 = 𝐴)
169	167, 168	eqtr4d 2774	. . . 4 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → 𝐵 = 𝐶)
170		nnnn0 12408	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ₀)
171	170	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℕ₀)
172		nn0uz 12789	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
173	171, 172	eleqtrdi 2846	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ (ℤ_≥‘0))
174		fzm1 13523	. . . . . . 7 ⊢ (𝐴 ∈ (ℤ_≥‘0) → (𝐶 ∈ (0...𝐴) ↔ (𝐶 ∈ (0...(𝐴 − 1)) ∨ 𝐶 = 𝐴)))
175	174	biimpa 476	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘0) ∧ 𝐶 ∈ (0...𝐴)) → (𝐶 ∈ (0...(𝐴 − 1)) ∨ 𝐶 = 𝐴))
176	173, 29, 175	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐶 ∈ (0...(𝐴 − 1)) ∨ 𝐶 = 𝐴))
177	176	adantr 480	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) → (𝐶 ∈ (0...(𝐴 − 1)) ∨ 𝐶 = 𝐴))
178	134, 169, 177	mpjaodan 960	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) → 𝐵 = 𝐶)
179	53, 178	jaodan 959	. 2 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ ((2 · 𝐴) ∥ (𝐵 − 𝐶) ∨ (2 · 𝐴) ∥ (𝐵 − -𝐶))) → 𝐵 = 𝐶)
180	14, 179	impbida 800	1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 = 𝐶 ↔ ((2 · 𝐴) ∥ (𝐵 − 𝐶) ∨ (2 · 𝐴) ∥ (𝐵 − -𝐶))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 class class class wbr 5098 ‘cfv 6492 (class class class)co 7358 ℂcc 11024 ℝcr 11025 0cc0 11026 1c1 11027 + caddc 11029 · cmul 11031 < clt 11166 ≤ cle 11167 − cmin 11364 -cneg 11365 ℕcn 12145 2c2 12200 ℕ₀cn0 12401 ℤcz 12488 ℤ_≥cuz 12751 ℝ⁺crp 12905 ...cfz 13423 abscabs 15157 ∥ cdvds 16179

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9345 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-n0 12402 df-z 12489 df-uz 12752 df-rp 12906 df-fz 13424 df-seq 13925 df-exp 13985 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-dvds 16180

This theorem is referenced by: jm2.27a 43247

W3C validator