4sqlem10 - Metamath Proof Explorer

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Theorem 4sqlem10 15489

Description: Lemma for 4sq 15506. (Contributed by Mario Carneiro, 16-Jul-2014.)

**Hypotheses**
Ref	Expression
4sqlem5.2	⊢ (𝜑 → 𝐴 ∈ ℤ)
4sqlem5.3	⊢ (𝜑 → 𝑀 ∈ ℕ)
4sqlem5.4	⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
4sqlem10.5	⊢ ((𝜑 ∧ 𝜓) → ((((𝑀↑2) / 2) / 2) − (𝐵↑2)) = 0)

**Assertion**
Ref	Expression
4sqlem10	⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))

**Proof of Theorem 4sqlem10**
Step	Hyp	Ref	Expression
1		4sqlem5.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℤ)
2	1	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℤ)
3		4sqlem5.3	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℕ)
4	3	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℕ)
5	4	nnred 10912	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℝ)
6	5	rehalfcld 11156	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℝ)
7	6	recnd 9947	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℂ)
8	7	negnegd 10262	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → --(𝑀 / 2) = (𝑀 / 2))
9		4sqlem5.4	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
10	1, 3, 9	4sqlem5 15484	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ))
11	10	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ))
12	11	simpld 474	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℤ)
13	12	zred 11358	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℝ)
14	1, 3, 9	4sqlem6 15485	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2)))
15	14	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2)))
16	15	simprd 478	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 < (𝑀 / 2))
17	13, 16	ltned 10052	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ≠ (𝑀 / 2))
18	17	neneqd 2787	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝐵 = (𝑀 / 2))
19		2cnd 10970	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝜓) → 2 ∈ ℂ)
20	19	sqvald 12867	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜓) → (2↑2) = (2 · 2))
21	20	oveq2d 6565	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀↑2) / (2↑2)) = ((𝑀↑2) / (2 · 2)))
22	4	nncnd 10913	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℂ)
23		2ne0 10990	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ≠ 0
24	23	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜓) → 2 ≠ 0)
25	22, 19, 24	sqdivd 12883	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2)↑2) = ((𝑀↑2) / (2↑2)))
26	22	sqcld 12868	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∈ ℂ)
27	26, 19, 19, 24, 24	divdiv1d 10711	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) = ((𝑀↑2) / (2 · 2)))
28	21, 25, 27	3eqtr4d 2654	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2)↑2) = (((𝑀↑2) / 2) / 2))
29	26	halfcld 11154	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀↑2) / 2) ∈ ℂ)
30	29	halfcld 11154	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) ∈ ℂ)
31	12	zcnd 11359	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℂ)
32	31	sqcld 12868	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → (𝐵↑2) ∈ ℂ)
33		4sqlem10.5	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → ((((𝑀↑2) / 2) / 2) − (𝐵↑2)) = 0)
34	30, 32, 33	subeq0d 10279	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) = (𝐵↑2))
35	28, 34	eqtr2d 2645	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → (𝐵↑2) = ((𝑀 / 2)↑2))
36		sqeqor 12840	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ ℂ ∧ (𝑀 / 2) ∈ ℂ) → ((𝐵↑2) = ((𝑀 / 2)↑2) ↔ (𝐵 = (𝑀 / 2) ∨ 𝐵 = -(𝑀 / 2))))
37	31, 7, 36	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → ((𝐵↑2) = ((𝑀 / 2)↑2) ↔ (𝐵 = (𝑀 / 2) ∨ 𝐵 = -(𝑀 / 2))))
38	35, 37	mpbid 221	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → (𝐵 = (𝑀 / 2) ∨ 𝐵 = -(𝑀 / 2)))
39	38	ord 391	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → (¬ 𝐵 = (𝑀 / 2) → 𝐵 = -(𝑀 / 2)))
40	18, 39	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 = -(𝑀 / 2))
41	40, 12	eqeltrrd 2689	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → -(𝑀 / 2) ∈ ℤ)
42	41	znegcld 11360	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → --(𝑀 / 2) ∈ ℤ)
43	8, 42	eqeltrrd 2689	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℤ)
44	2, 43	zaddcld 11362	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℤ)
45	44	zred 11358	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℝ)
46	4	nnrpd 11746	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℝ⁺)
47	45, 46	modcld 12536	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℝ)
48	47	recnd 9947	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℂ)
49		0cnd 9912	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 0 ∈ ℂ)
50		df-neg 10148	. . . . . . 7 ⊢ -(𝑀 / 2) = (0 − (𝑀 / 2))
51	40, 9, 50	3eqtr3g 2667	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (0 − (𝑀 / 2)))
52	48, 49, 7, 51	subcan2d 10313	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) = 0)
53		dvdsval3 14825	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ (𝐴 + (𝑀 / 2)) ∈ ℤ) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ ((𝐴 + (𝑀 / 2)) mod 𝑀) = 0))
54	4, 44, 53	syl2anc 691	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ ((𝐴 + (𝑀 / 2)) mod 𝑀) = 0))
55	52, 54	mpbird 246	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∥ (𝐴 + (𝑀 / 2)))
56	4	nnzd 11357	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℤ)
57		dvdssq 15118	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ (𝐴 + (𝑀 / 2)) ∈ ℤ) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2)))
58	56, 44, 57	syl2anc 691	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2)))
59	55, 58	mpbid 221	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2))
60	22	sqvald 12867	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) = (𝑀 · 𝑀))
61	4	nnne0d 10942	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ≠ 0)
62		dvdsmulcr 14849	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ (𝐴 + (𝑀 / 2)) ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → ((𝑀 · 𝑀) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀) ↔ 𝑀 ∥ (𝐴 + (𝑀 / 2))))
63	56, 44, 56, 61, 62	syl112anc 1322	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 · 𝑀) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀) ↔ 𝑀 ∥ (𝐴 + (𝑀 / 2))))
64	55, 63	mpbird 246	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 · 𝑀) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀))
65	60, 64	eqbrtrd 4605	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀))
66		zsqcl 12796	. . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑀↑2) ∈ ℤ)
67	56, 66	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∈ ℤ)
68		zsqcl 12796	. . . . 5 ⊢ ((𝐴 + (𝑀 / 2)) ∈ ℤ → ((𝐴 + (𝑀 / 2))↑2) ∈ ℤ)
69	44, 68	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2))↑2) ∈ ℤ)
70	44, 56	zmulcld 11364	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · 𝑀) ∈ ℤ)
71		dvds2sub 14854	. . . 4 ⊢ (((𝑀↑2) ∈ ℤ ∧ ((𝐴 + (𝑀 / 2))↑2) ∈ ℤ ∧ ((𝐴 + (𝑀 / 2)) · 𝑀) ∈ ℤ) → (((𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2) ∧ (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀)) → (𝑀↑2) ∥ (((𝐴 + (𝑀 / 2))↑2) − ((𝐴 + (𝑀 / 2)) · 𝑀))))
72	67, 69, 70, 71	syl3anc 1318	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2) ∧ (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀)) → (𝑀↑2) ∥ (((𝐴 + (𝑀 / 2))↑2) − ((𝐴 + (𝑀 / 2)) · 𝑀))))
73	59, 65, 72	mp2and 711	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ (((𝐴 + (𝑀 / 2))↑2) − ((𝐴 + (𝑀 / 2)) · 𝑀)))
74	44	zcnd 11359	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℂ)
75	74	sqvald 12867	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2))↑2) = ((𝐴 + (𝑀 / 2)) · (𝐴 + (𝑀 / 2))))
76	75	oveq1d 6564	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + (𝑀 / 2))↑2) − ((𝐴 + (𝑀 / 2)) · 𝑀)) = (((𝐴 + (𝑀 / 2)) · (𝐴 + (𝑀 / 2))) − ((𝐴 + (𝑀 / 2)) · 𝑀)))
77	74, 74, 22	subdid 10365	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = (((𝐴 + (𝑀 / 2)) · (𝐴 + (𝑀 / 2))) − ((𝐴 + (𝑀 / 2)) · 𝑀)))
78	22	2halvesd 11155	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2) + (𝑀 / 2)) = 𝑀)
79	78	oveq2d 6565	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − ((𝑀 / 2) + (𝑀 / 2))) = ((𝐴 + (𝑀 / 2)) − 𝑀))
80	2	zcnd 11359	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℂ)
81	80, 7, 7	pnpcan2d 10309	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − ((𝑀 / 2) + (𝑀 / 2))) = (𝐴 − (𝑀 / 2)))
82	79, 81	eqtr3d 2646	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − 𝑀) = (𝐴 − (𝑀 / 2)))
83	82	oveq2d 6565	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = ((𝐴 + (𝑀 / 2)) · (𝐴 − (𝑀 / 2))))
84		subsq 12834	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 / 2) ∈ ℂ) → ((𝐴↑2) − ((𝑀 / 2)↑2)) = ((𝐴 + (𝑀 / 2)) · (𝐴 − (𝑀 / 2))))
85	80, 7, 84	syl2anc 691	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴↑2) − ((𝑀 / 2)↑2)) = ((𝐴 + (𝑀 / 2)) · (𝐴 − (𝑀 / 2))))
86	28	oveq2d 6565	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴↑2) − ((𝑀 / 2)↑2)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))
87	83, 85, 86	3eqtr2d 2650	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))
88	76, 77, 87	3eqtr2d 2650	. 2 ⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + (𝑀 / 2))↑2) − ((𝐴 + (𝑀 / 2)) · 𝑀)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))
89	73, 88	breqtrd 4609	1 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 (class class class)co 6549 ℂcc 9813 0cc0 9815 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 -cneg 10146 / cdiv 10563 ℕcn 10897 2c2 10947 ℤcz 11254 mod cmo 12530 ↑cexp 12722 ∥ cdvds 14821

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-gcd 15055

This theorem is referenced by: 4sqlem16 15502

W3C validator