4sqlem10 - Metamath Proof Explorer

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Theorem 4sqlem10 15651

Description: Lemma for 4sq 15668. (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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℤ)
3		4sqlem5.3	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℕ)
4	3	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℕ)
5	4	nnred 11035	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℝ)
6	5	rehalfcld 11279	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℝ)
7	6	recnd 10068	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℂ)
8	7	negnegd 10383	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → --(𝑀 / 2) = (𝑀 / 2))
9		4sqlem5.4	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
10	1, 3, 9	4sqlem5 15646	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ))
11	10	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ))
12	11	simpld 475	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℤ)
13	12	zred 11482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℝ)
14	1, 3, 9	4sqlem6 15647	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2)))
15	14	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2)))
16	15	simprd 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 < (𝑀 / 2))
17	13, 16	ltned 10173	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ≠ (𝑀 / 2))
18	17	neneqd 2799	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝐵 = (𝑀 / 2))
19		2cnd 11093	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝜓) → 2 ∈ ℂ)
20	19	sqvald 13005	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜓) → (2↑2) = (2 · 2))
21	20	oveq2d 6666	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀↑2) / (2↑2)) = ((𝑀↑2) / (2 · 2)))
22	4	nncnd 11036	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℂ)
23		2ne0 11113	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ≠ 0
24	23	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜓) → 2 ≠ 0)
25	22, 19, 24	sqdivd 13021	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2)↑2) = ((𝑀↑2) / (2↑2)))
26	22	sqcld 13006	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∈ ℂ)
27	26, 19, 19, 24, 24	divdiv1d 10832	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) = ((𝑀↑2) / (2 · 2)))
28	21, 25, 27	3eqtr4d 2666	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2)↑2) = (((𝑀↑2) / 2) / 2))
29	26	halfcld 11277	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀↑2) / 2) ∈ ℂ)
30	29	halfcld 11277	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) ∈ ℂ)
31	12	zcnd 11483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℂ)
32	31	sqcld 13006	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → (𝐵↑2) ∈ ℂ)
33		4sqlem10.5	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → ((((𝑀↑2) / 2) / 2) − (𝐵↑2)) = 0)
34	30, 32, 33	subeq0d 10400	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) = (𝐵↑2))
35	28, 34	eqtr2d 2657	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → (𝐵↑2) = ((𝑀 / 2)↑2))
36		sqeqor 12978	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ ℂ ∧ (𝑀 / 2) ∈ ℂ) → ((𝐵↑2) = ((𝑀 / 2)↑2) ↔ (𝐵 = (𝑀 / 2) ∨ 𝐵 = -(𝑀 / 2))))
37	31, 7, 36	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → ((𝐵↑2) = ((𝑀 / 2)↑2) ↔ (𝐵 = (𝑀 / 2) ∨ 𝐵 = -(𝑀 / 2))))
38	35, 37	mpbid 222	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → (𝐵 = (𝑀 / 2) ∨ 𝐵 = -(𝑀 / 2)))
39	38	ord 392	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → (¬ 𝐵 = (𝑀 / 2) → 𝐵 = -(𝑀 / 2)))
40	18, 39	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 = -(𝑀 / 2))
41	40, 12	eqeltrrd 2702	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → -(𝑀 / 2) ∈ ℤ)
42	41	znegcld 11484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → --(𝑀 / 2) ∈ ℤ)
43	8, 42	eqeltrrd 2702	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℤ)
44	2, 43	zaddcld 11486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℤ)
45	44	zred 11482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℝ)
46	4	nnrpd 11870	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℝ⁺)
47	45, 46	modcld 12674	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℝ)
48	47	recnd 10068	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℂ)
49		0cnd 10033	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 0 ∈ ℂ)
50		df-neg 10269	. . . . . . 7 ⊢ -(𝑀 / 2) = (0 − (𝑀 / 2))
51	40, 9, 50	3eqtr3g 2679	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (0 − (𝑀 / 2)))
52	48, 49, 7, 51	subcan2d 10434	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) = 0)
53		dvdsval3 14987	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ (𝐴 + (𝑀 / 2)) ∈ ℤ) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ ((𝐴 + (𝑀 / 2)) mod 𝑀) = 0))
54	4, 44, 53	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ ((𝐴 + (𝑀 / 2)) mod 𝑀) = 0))
55	52, 54	mpbird 247	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∥ (𝐴 + (𝑀 / 2)))
56	4	nnzd 11481	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℤ)
57		dvdssq 15280	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ (𝐴 + (𝑀 / 2)) ∈ ℤ) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2)))
58	56, 44, 57	syl2anc 693	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2)))
59	55, 58	mpbid 222	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2))
60	22	sqvald 13005	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) = (𝑀 · 𝑀))
61	4	nnne0d 11065	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ≠ 0)
62		dvdsmulcr 15011	. . . . . 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 1330	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 · 𝑀) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀) ↔ 𝑀 ∥ (𝐴 + (𝑀 / 2))))
64	55, 63	mpbird 247	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 · 𝑀) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀))
65	60, 64	eqbrtrd 4675	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀))
66		zsqcl 12934	. . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑀↑2) ∈ ℤ)
67	56, 66	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∈ ℤ)
68		zsqcl 12934	. . . . 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 11488	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · 𝑀) ∈ ℤ)
71		dvds2sub 15016	. . . 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 1326	. . 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 715	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ (((𝐴 + (𝑀 / 2))↑2) − ((𝐴 + (𝑀 / 2)) · 𝑀)))
74	44	zcnd 11483	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℂ)
75	74	sqvald 13005	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2))↑2) = ((𝐴 + (𝑀 / 2)) · (𝐴 + (𝑀 / 2))))
76	75	oveq1d 6665	. . 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 10486	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = (((𝐴 + (𝑀 / 2)) · (𝐴 + (𝑀 / 2))) − ((𝐴 + (𝑀 / 2)) · 𝑀)))
78	22	2halvesd 11278	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2) + (𝑀 / 2)) = 𝑀)
79	78	oveq2d 6666	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − ((𝑀 / 2) + (𝑀 / 2))) = ((𝐴 + (𝑀 / 2)) − 𝑀))
80	2	zcnd 11483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℂ)
81	80, 7, 7	pnpcan2d 10430	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − ((𝑀 / 2) + (𝑀 / 2))) = (𝐴 − (𝑀 / 2)))
82	79, 81	eqtr3d 2658	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − 𝑀) = (𝐴 − (𝑀 / 2)))
83	82	oveq2d 6666	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = ((𝐴 + (𝑀 / 2)) · (𝐴 − (𝑀 / 2))))
84		subsq 12972	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 / 2) ∈ ℂ) → ((𝐴↑2) − ((𝑀 / 2)↑2)) = ((𝐴 + (𝑀 / 2)) · (𝐴 − (𝑀 / 2))))
85	80, 7, 84	syl2anc 693	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴↑2) − ((𝑀 / 2)↑2)) = ((𝐴 + (𝑀 / 2)) · (𝐴 − (𝑀 / 2))))
86	28	oveq2d 6666	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴↑2) − ((𝑀 / 2)↑2)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))
87	83, 85, 86	3eqtr2d 2662	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))
88	76, 77, 87	3eqtr2d 2662	. 2 ⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + (𝑀 / 2))↑2) − ((𝐴 + (𝑀 / 2)) · 𝑀)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))
89	73, 88	breqtrd 4679	1 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 (class class class)co 6650 ℂcc 9934 0cc0 9936 + caddc 9939 · cmul 9941 < clt 10074 ≤ cle 10075 − cmin 10266 -cneg 10267 / cdiv 10684 ℕcn 11020 2c2 11070 ℤcz 11377 mod cmo 12668 ↑cexp 12860 ∥ cdvds 14983

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217

This theorem is referenced by: 4sqlem16 15664

W3C validator