acongeq - Mathbox for Stefan O'Rear

	Mathbox for Stefan O'Rear		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > acongeq		Structured version Visualization version GIF version

Theorem acongeq 44256

Description: Two numbers in the fundamental domain are alternating-congruent iff they are equal. TODO: could be used to shorten jm2.26 44275. (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 12768	. . . . . . 7 ⊢ 2 ∈ ℤ
2		nnz 12754	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
3	2	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℤ)
4		zmulcl 12785	. . . . . . 7 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (2 · 𝐴) ∈ ℤ)
5	1, 3, 4	sylancr 599	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (2 · 𝐴) ∈ ℤ)
6		elfzelz 13696	. . . . . . 7 ⊢ (𝐵 ∈ (0...𝐴) → 𝐵 ∈ ℤ)
7	6	3ad2ant2 1152	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℤ)
8		congid 44244	. . . . . 6 ⊢ (((2 · 𝐴) ∈ ℤ ∧ 𝐵 ∈ ℤ) → (2 · 𝐴) ∥ (𝐵 − 𝐵))
9	5, 7, 8	syl2anc 596	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (2 · 𝐴) ∥ (𝐵 − 𝐵))
10	9	adantr 486	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝐵 = 𝐶) → (2 · 𝐴) ∥ (𝐵 − 𝐵))
11		oveq2 7459	. . . . 5 ⊢ (𝐵 = 𝐶 → (𝐵 − 𝐵) = (𝐵 − 𝐶))
12	11	adantl 487	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝐵 = 𝐶) → (𝐵 − 𝐵) = (𝐵 − 𝐶))
13	10, 12	breqtrd 5137	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝐵 = 𝐶) → (2 · 𝐴) ∥ (𝐵 − 𝐶))
14	13	orcd 887	. 2 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ 𝐵 = 𝐶) → ((2 · 𝐴) ∥ (𝐵 − 𝐶) ∨ (2 · 𝐴) ∥ (𝐵 − -𝐶)))
15		elfzelz 13696	. . . . . . . . . 10 ⊢ (𝐶 ∈ (0...𝐴) → 𝐶 ∈ ℤ)
16	15	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℤ)
17	7, 16	zsubcld 12848	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − 𝐶) ∈ ℤ)
18	17	zcnd 12844	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − 𝐶) ∈ ℂ)
19	18	abscld 15650	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (abs‘(𝐵 − 𝐶)) ∈ ℝ)
20		nnre 12382	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
21	20	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℝ)
22		0re 11349	. . . . . . 7 ⊢ 0 ∈ ℝ
23		resubcl 11661	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 − 0) ∈ ℝ)
24	21, 22, 23	sylancl 598	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 0) ∈ ℝ)
25		2re 12457	. . . . . . 7 ⊢ 2 ∈ ℝ
26		remulcl 11324	. . . . . . 7 ⊢ ((2 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (2 · 𝐴) ∈ ℝ)
27	25, 21, 26	sylancr 599	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (2 · 𝐴) ∈ ℝ)
28		simp2 1155	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ (0...𝐴))
29		simp3 1156	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ (0...𝐴))
30	24	leidd 11922	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 0) ≤ (𝐴 − 0))
31		fzmaxdif 44254	. . . . . . 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 1409	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (abs‘(𝐵 − 𝐶)) ≤ (𝐴 − 0))
33		nnrp 13172	. . . . . . . . 9 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℝ⁺)
34	33	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℝ⁺)
35	21, 34	ltaddrpd 13237	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 < (𝐴 + 𝐴))
36	21	recnd 11376	. . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℂ)
37	36	subid1d 11697	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 0) = 𝐴)
38	36	2timesd 12629	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (2 · 𝐴) = (𝐴 + 𝐴))
39	35, 37, 38	3brtr4d 5143	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 0) < (2 · 𝐴))
40	19, 24, 27, 32, 39	lelttrd 11507	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (abs‘(𝐵 − 𝐶)) < (2 · 𝐴))
41	40	adantr 486	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − 𝐶)) → (abs‘(𝐵 − 𝐶)) < (2 · 𝐴))
42		2nn 12456	. . . . . 6 ⊢ 2 ∈ ℕ
43		simpl1 1210	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − 𝐶)) → 𝐴 ∈ ℕ)
44		nnmulcl 12399	. . . . . 6 ⊢ ((2 ∈ ℕ ∧ 𝐴 ∈ ℕ) → (2 · 𝐴) ∈ ℕ)
45	42, 43, 44	sylancr 599	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − 𝐶)) → (2 · 𝐴) ∈ ℕ)
46		simpl2 1211	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − 𝐶)) → 𝐵 ∈ (0...𝐴))
47	46	elfzelzd 13697	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − 𝐶)) → 𝐵 ∈ ℤ)
48		simpl3 1212	. . . . . 6 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − 𝐶)) → 𝐶 ∈ (0...𝐴))
49	48	elfzelzd 13697	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − 𝐶)) → 𝐶 ∈ ℤ)
50		simpr 490	. . . . 5 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − 𝐶)) → (2 · 𝐴) ∥ (𝐵 − 𝐶))
51		congabseq 44247	. . . . 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 1401	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − 𝐶)) → ((abs‘(𝐵 − 𝐶)) < (2 · 𝐴) ↔ 𝐵 = 𝐶))
53	41, 52	mpbid 235	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − 𝐶)) → 𝐵 = 𝐶)
54		simpll2 1232	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐵 ∈ (0...𝐴))
55		elfzle1 13700	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (0...𝐴) → 0 ≤ 𝐵)
56	54, 55	syl 18	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 0 ≤ 𝐵)
57	7	zred 12843	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℝ)
58	16	zred 12843	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℝ)
59	58	renegcld 11782	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → -𝐶 ∈ ℝ)
60	57, 59	resubcld 11783	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − -𝐶) ∈ ℝ)
61	60	recnd 11376	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − -𝐶) ∈ ℂ)
62	61	abscld 15650	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (abs‘(𝐵 − -𝐶)) ∈ ℝ)
63	62	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (abs‘(𝐵 − -𝐶)) ∈ ℝ)
64		1re 11347	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
65		resubcl 11661	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐴 − 1) ∈ ℝ)
66	21, 64, 65	sylancl 598	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 1) ∈ ℝ)
67	66	renegcld 11782	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → -(𝐴 − 1) ∈ ℝ)
68	21, 67	resubcld 11783	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − -(𝐴 − 1)) ∈ ℝ)
69	68	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (𝐴 − -(𝐴 − 1)) ∈ ℝ)
70	27	ad2antrr 739	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (2 · 𝐴) ∈ ℝ)
71	7	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐵 ∈ ℤ)
72	71	zcnd 12844	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐵 ∈ ℂ)
73	16	znegcld 12845	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → -𝐶 ∈ ℤ)
74	73	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 ∈ ℤ)
75	74	zcnd 12844	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 ∈ ℂ)
76	72, 75	abssubd 15667	. . . . . . . . . . . . 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 12745	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 0 ∈ ℤ)
78		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐶 ∈ (0...(𝐴 − 1)))
79		0zd 12745	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 0 ∈ ℤ)
80		1z 12766	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℤ
81		zsubcl 12778	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℤ ∧ 1 ∈ ℤ) → (𝐴 − 1) ∈ ℤ)
82	3, 80, 81	sylancl 598	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 1) ∈ ℤ)
83		fzneg 44255	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ ℤ ∧ 0 ∈ ℤ ∧ (𝐴 − 1) ∈ ℤ) → (𝐶 ∈ (0...(𝐴 − 1)) ↔ -𝐶 ∈ ( -(𝐴 − 1)... -0)))
84	16, 79, 82, 83	syl3anc 1399	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐶 ∈ (0...(𝐴 − 1)) ↔ -𝐶 ∈ ( -(𝐴 − 1)... -0)))
85	84	ad2antrr 739	. . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 ∈ ( -(𝐴 − 1)... -0))
87		neg0 11643	. . . . . . . . . . . . . . . . 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 7467	. . . . . . . . . . . . . . 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 2866	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 ∈ ( -(𝐴 − 1)...0))
91	3	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐴 ∈ ℤ)
92		simp1 1154	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℕ)
93	42, 92, 44	sylancr 599	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (2 · 𝐴) ∈ ℕ)
94		nnm1nn0 12687	. . . . . . . . . . . . . . . . . 18 ⊢ ((2 · 𝐴) ∈ ℕ → ((2 · 𝐴) − 1) ∈ ℕ₀)
95	93, 94	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((2 · 𝐴) − 1) ∈ ℕ₀)
96	95	nn0ge0d 12710	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 0 ≤ ((2 · 𝐴) − 1))
97		0m0e0 12501	. . . . . . . . . . . . . . . . 17 ⊢ (0 − 0) = 0
98	97	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (0 − 0) = 0)
99		1cnd 11341	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 1 ∈ ℂ)
100	36, 36, 99	addsubassd 11728	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((𝐴 + 𝐴) − 1) = (𝐴 + (𝐴 − 1)))
101	38	oveq1d 7466	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((2 · 𝐴) − 1) = ((𝐴 + 𝐴) − 1))
102		ax-1cn 11297	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℂ
103		subcl 11595	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 − 1) ∈ ℂ)
104	36, 102, 103	sylancl 598	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − 1) ∈ ℂ)
105	36, 104	subnegd 11715	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − -(𝐴 − 1)) = (𝐴 + (𝐴 − 1)))
106	100, 101, 105	3eqtr4rd 2809	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − -(𝐴 − 1)) = ((2 · 𝐴) − 1))
107	96, 98, 106	3brtr4d 5143	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (0 − 0) ≤ (𝐴 − -(𝐴 − 1)))
108	107	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (0 − 0) ≤ (𝐴 − -(𝐴 − 1)))
109		fzmaxdif 44254	. . . . . . . . . . . . . 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 1409	. . . . . . . . . . . . 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 5133	. . . . . . . . . . . 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 12289	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((2 · 𝐴) − 1) < (2 · 𝐴))
113	106, 112	eqbrtrd 5133	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 − -(𝐴 − 1)) < (2 · 𝐴))
114	113	ad2antrr 739	. . . . . . . . . . . 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 11507	. . . . . . . . . . 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 739	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (2 · 𝐴) ∈ ℕ)
117		simplr 781	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → (2 · 𝐴) ∥ (𝐵 − -𝐶))
118		congabseq 44247	. . . . . . . . . . . 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 1401	. . . . . . . . . . 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 235	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐵 = -𝐶)
121	56, 120	breqtrd 5137	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 0 ≤ -𝐶)
122		elfzelz 13696	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (0...(𝐴 − 1)) → 𝐶 ∈ ℤ)
123	122	zred 12843	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (0...(𝐴 − 1)) → 𝐶 ∈ ℝ)
124	123	adantl 487	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐶 ∈ ℝ)
125	124	le0neg1d 11927	. . . . . . . . 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 260	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐶 ≤ 0)
127		elfzle1 13700	. . . . . . . . 9 ⊢ (𝐶 ∈ (0...(𝐴 − 1)) → 0 ≤ 𝐶)
128	127	adantl 487	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 0 ≤ 𝐶)
129		letri3 11434	. . . . . . . . 9 ⊢ ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐶 = 0 ↔ (𝐶 ≤ 0 ∧ 0 ≤ 𝐶)))
130	124, 22, 129	sylancl 598	. . . . . . . 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 726	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐶 = 0)
132	131	negeqd 11590	. . . . . 6 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 = -0)
133	132, 88	eqtrd 2798	. . . . 5 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → -𝐶 = 0)
134	133, 120, 131	3eqtr4d 2808	. . . 4 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 ∈ (0...(𝐴 − 1))) → 𝐵 = 𝐶)
135		oveq2 7459	. . . . . . . . 9 ⊢ (𝐶 = 𝐴 → (𝐵 − 𝐶) = (𝐵 − 𝐴))
136	135	adantl 487	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (𝐵 − 𝐶) = (𝐵 − 𝐴))
137	136	fveq2d 6920	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (abs‘(𝐵 − 𝐶)) = (abs‘(𝐵 − 𝐴)))
138	40	ad2antrr 739	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (abs‘(𝐵 − 𝐶)) < (2 · 𝐴))
139	137, 138	eqbrtrrd 5135	. . . . . 6 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (abs‘(𝐵 − 𝐴)) < (2 · 𝐴))
140	93	ad2antrr 739	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (2 · 𝐴) ∈ ℕ)
141	7	ad2antrr 739	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → 𝐵 ∈ ℤ)
142	3	ad2antrr 739	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → 𝐴 ∈ ℤ)
143		simplr 781	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (2 · 𝐴) ∥ (𝐵 − -𝐶))
144	7	zcnd 12844	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐵 ∈ ℂ)
145	36, 36, 144	ppncand 11748	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((𝐴 + 𝐴) + (𝐵 − 𝐴)) = (𝐴 + 𝐵))
146	36, 144	addcomd 11551	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
147	145, 146	eqtrd 2798	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((𝐴 + 𝐴) + (𝐵 − 𝐴)) = (𝐵 + 𝐴))
148	147	ad2antrr 739	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → ((𝐴 + 𝐴) + (𝐵 − 𝐴)) = (𝐵 + 𝐴))
149		oveq2 7459	. . . . . . . . . . . 12 ⊢ (𝐶 = 𝐴 → (𝐵 + 𝐶) = (𝐵 + 𝐴))
150	149	adantl 487	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (𝐵 + 𝐶) = (𝐵 + 𝐴))
151	148, 150	eqtr4d 2801	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → ((𝐴 + 𝐴) + (𝐵 − 𝐴)) = (𝐵 + 𝐶))
152	38	oveq1d 7466	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → ((2 · 𝐴) + (𝐵 − 𝐴)) = ((𝐴 + 𝐴) + (𝐵 − 𝐴)))
153	152	ad2antrr 739	. . . . . . . . . 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 12844	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐶 ∈ ℂ)
155	144, 154	subnegd 11715	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − -𝐶) = (𝐵 + 𝐶))
156	155	ad2antrr 739	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (𝐵 − -𝐶) = (𝐵 + 𝐶))
157	151, 153, 156	3eqtr4d 2808	. . . . . . . . 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 5139	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (2 · 𝐴) ∥ ((2 · 𝐴) + (𝐵 − 𝐴)))
159	5	ad2antrr 739	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (2 · 𝐴) ∈ ℤ)
160	7, 3	zsubcld 12848	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐵 − 𝐴) ∈ ℤ)
161	160	ad2antrr 739	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (𝐵 − 𝐴) ∈ ℤ)
162		dvdsadd 16516	. . . . . . . . 9 ⊢ (((2 · 𝐴) ∈ ℤ ∧ (𝐵 − 𝐴) ∈ ℤ) → ((2 · 𝐴) ∥ (𝐵 − 𝐴) ↔ (2 · 𝐴) ∥ ((2 · 𝐴) + (𝐵 − 𝐴))))
163	159, 161, 162	syl2anc 596	. . . . . . . 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 260	. . . . . . 7 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → (2 · 𝐴) ∥ (𝐵 − 𝐴))
165		congabseq 44247	. . . . . . 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 1401	. . . . . 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 235	. . . . 5 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → 𝐵 = 𝐴)
168		simpr 490	. . . . 5 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → 𝐶 = 𝐴)
169	167, 168	eqtr4d 2801	. . . 4 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) ∧ 𝐶 = 𝐴) → 𝐵 = 𝐶)
170		nnnn0 12653	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℕ₀)
171	170	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ ℕ₀)
172		nn0uz 13043	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
173	171, 172	eleqtrdi 2874	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → 𝐴 ∈ (ℤ_≥‘0))
174		fzm1 13781	. . . . . . 7 ⊢ (𝐴 ∈ (ℤ_≥‘0) → (𝐶 ∈ (0...𝐴) ↔ (𝐶 ∈ (0...(𝐴 − 1)) ∨ 𝐶 = 𝐴)))
175	174	biimpa 482	. . . . . 6 ⊢ ((𝐴 ∈ (ℤ_≥‘0) ∧ 𝐶 ∈ (0...𝐴)) → (𝐶 ∈ (0...(𝐴 − 1)) ∨ 𝐶 = 𝐴))
176	173, 29, 175	syl2anc 596	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) → (𝐶 ∈ (0...(𝐴 − 1)) ∨ 𝐶 = 𝐴))
177	176	adantr 486	. . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) → (𝐶 ∈ (0...(𝐴 − 1)) ∨ 𝐶 = 𝐴))
178	134, 169, 177	mpjaodan 973	. . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ (2 · 𝐴) ∥ (𝐵 − -𝐶)) → 𝐵 = 𝐶)
179	53, 178	jaodan 972	. 2 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ (0...𝐴) ∧ 𝐶 ∈ (0...𝐴)) ∧ ((2 · 𝐴) ∥ (𝐵 − 𝐶) ∨ (2 · 𝐴) ∥ (𝐵 − -𝐶))) → 𝐵 = 𝐶)
180	14, 179	impbida 813	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 class class class wbr 5108 ‘cfv 6566 (class class class)co 7451 ℂcc 11237 ℝcr 11238 0cc0 11239 1c1 11240 + caddc 11242 · cmul 11244 < clt 11382 ≤ cle 11383 − cmin 11580 -cneg 11581 ℕcn 12375 2c2 12437 ℕ₀cn0 12646 ℤcz 12733 ℤ_≥cuz 13005 ℝ⁺crp 13160 ...cfz 13679 abscabs 15444 ∥ cdvds 16466

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-sup 9471 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-n0 12647 df-z 12734 df-uz 13006 df-rp 13161 df-fz 13680 df-seq 14185 df-exp 14245 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-dvds 16467

This theorem is used by: jm2.27a 44278

W3C validator