4sqlem10 - Metamath Proof Explorer

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Theorem 4sqlem10 17007

Description: Lemma for 4sq 17024. (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		4sqlem5.3	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
2	1	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℕ)
3	2	nnzd 12617	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℤ)
4		zsqcl 14165	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝑀↑2) ∈ ℤ)
5	3, 4	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∈ ℤ)
6		4sqlem5.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℤ)
7	6	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℤ)
8	2	nnred 12248	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℝ)
9	8	rehalfcld 12491	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℝ)
10	9	recnd 11237	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℂ)
11	10	negnegd 11560	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → --(𝑀 / 2) = (𝑀 / 2))
12		4sqlem5.4	. . . . . . . . . . . . . . 15 ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
13	6, 1, 12	4sqlem5 17002	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ))
14	13	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ))
15	14	simpld 499	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℤ)
16	15	zred 12700	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℝ)
17	6, 1, 12	4sqlem6 17003	. . . . . . . . . . . . 13 ⊢ (𝜑 → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2)))
18	17	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2)))
19	18	simprd 500	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 < (𝑀 / 2))
20	16, 19	ltned 11346	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ≠ (𝑀 / 2))
21	20	neneqd 2969	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝐵 = (𝑀 / 2))
22		2cnd 12319	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 2 ∈ ℂ)
23	22	sqvald 14179	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → (2↑2) = (2 · 2))
24	23	oveq2d 7427	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀↑2) / (2↑2)) = ((𝑀↑2) / (2 · 2)))
25	2	nncnd 12249	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℂ)
26		2ne0 12347	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
27	26	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 2 ≠ 0)
28	25, 22, 27	sqdivd 14195	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2)↑2) = ((𝑀↑2) / (2↑2)))
29	25	sqcld 14180	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∈ ℂ)
30	29, 22, 22, 27, 27	divdiv1d 12022	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) = ((𝑀↑2) / (2 · 2)))
31	24, 28, 30	3eqtr4d 2814	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2)↑2) = (((𝑀↑2) / 2) / 2))
32	29	halfcld 12489	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀↑2) / 2) ∈ ℂ)
33	32	halfcld 12489	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) ∈ ℂ)
34	15	zcnd 12701	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℂ)
35	34	sqcld 14180	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (𝐵↑2) ∈ ℂ)
36		4sqlem10.5	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ((((𝑀↑2) / 2) / 2) − (𝐵↑2)) = 0)
37	33, 35, 36	subeq0d 11577	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) = (𝐵↑2))
38	31, 37	eqtr2d 2805	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝐵↑2) = ((𝑀 / 2)↑2))
39		sqeqor 14252	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℂ ∧ (𝑀 / 2) ∈ ℂ) → ((𝐵↑2) = ((𝑀 / 2)↑2) ↔ (𝐵 = (𝑀 / 2) ∨ 𝐵 = -(𝑀 / 2))))
40	34, 10, 39	syl2anc 595	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → ((𝐵↑2) = ((𝑀 / 2)↑2) ↔ (𝐵 = (𝑀 / 2) ∨ 𝐵 = -(𝑀 / 2))))
41	38, 40	mpbid 235	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (𝐵 = (𝑀 / 2) ∨ 𝐵 = -(𝑀 / 2)))
42	41	ord 877	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (¬ 𝐵 = (𝑀 / 2) → 𝐵 = -(𝑀 / 2)))
43	21, 42	mpd 16	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 = -(𝑀 / 2))
44	43, 15	eqeltrrd 2870	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → -(𝑀 / 2) ∈ ℤ)
45	44	znegcld 12702	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → --(𝑀 / 2) ∈ ℤ)
46	11, 45	eqeltrrd 2870	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℤ)
47	7, 46	zaddcld 12704	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℤ)
48		zsqcl 14165	. . . 4 ⊢ ((𝐴 + (𝑀 / 2)) ∈ ℤ → ((𝐴 + (𝑀 / 2))↑2) ∈ ℤ)
49	47, 48	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2))↑2) ∈ ℤ)
50	47, 3	zmulcld 12706	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · 𝑀) ∈ ℤ)
51	47	zred 12700	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℝ)
52	2	nnrpd 13058	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℝ⁺)
53	51, 52	modcld 13908	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℝ)
54	53	recnd 11237	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℂ)
55		0cnd 11199	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 0 ∈ ℂ)
56		df-neg 11444	. . . . . . 7 ⊢ -(𝑀 / 2) = (0 − (𝑀 / 2))
57	43, 12, 56	3eqtr3g 2827	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (0 − (𝑀 / 2)))
58	54, 55, 10, 57	subcan2d 11611	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) = 0)
59		dvdsval3 16314	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ (𝐴 + (𝑀 / 2)) ∈ ℤ) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ ((𝐴 + (𝑀 / 2)) mod 𝑀) = 0))
60	2, 47, 59	syl2anc 595	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ ((𝐴 + (𝑀 / 2)) mod 𝑀) = 0))
61	58, 60	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∥ (𝐴 + (𝑀 / 2)))
62		dvdssq 16625	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ (𝐴 + (𝑀 / 2)) ∈ ℤ) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2)))
63	3, 47, 62	syl2anc 595	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2)))
64	61, 63	mpbid 235	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2))
65	25	sqvald 14179	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) = (𝑀 · 𝑀))
66	2	nnne0d 12286	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ≠ 0)
67		dvdsmulcr 16343	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ (𝐴 + (𝑀 / 2)) ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → ((𝑀 · 𝑀) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀) ↔ 𝑀 ∥ (𝐴 + (𝑀 / 2))))
68	3, 47, 3, 66, 67	syl112anc 1399	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 · 𝑀) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀) ↔ 𝑀 ∥ (𝐴 + (𝑀 / 2))))
69	61, 68	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 · 𝑀) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀))
70	65, 69	eqbrtrd 5137	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀))
71	5, 49, 50, 64, 70	dvds2subd 16351	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ (((𝐴 + (𝑀 / 2))↑2) − ((𝐴 + (𝑀 / 2)) · 𝑀)))
72	47	zcnd 12701	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℂ)
73	72	sqvald 14179	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2))↑2) = ((𝐴 + (𝑀 / 2)) · (𝐴 + (𝑀 / 2))))
74	73	oveq1d 7426	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + (𝑀 / 2))↑2) − ((𝐴 + (𝑀 / 2)) · 𝑀)) = (((𝐴 + (𝑀 / 2)) · (𝐴 + (𝑀 / 2))) − ((𝐴 + (𝑀 / 2)) · 𝑀)))
75	72, 72, 25	subdid 11670	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = (((𝐴 + (𝑀 / 2)) · (𝐴 + (𝑀 / 2))) − ((𝐴 + (𝑀 / 2)) · 𝑀)))
76	25	2halvesd 12490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2) + (𝑀 / 2)) = 𝑀)
77	76	oveq2d 7427	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − ((𝑀 / 2) + (𝑀 / 2))) = ((𝐴 + (𝑀 / 2)) − 𝑀))
78	7	zcnd 12701	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℂ)
79	78, 10, 10	pnpcan2d 11607	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − ((𝑀 / 2) + (𝑀 / 2))) = (𝐴 − (𝑀 / 2)))
80	77, 79	eqtr3d 2806	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − 𝑀) = (𝐴 − (𝑀 / 2)))
81	80	oveq2d 7427	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = ((𝐴 + (𝑀 / 2)) · (𝐴 − (𝑀 / 2))))
82		subsq 14246	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 / 2) ∈ ℂ) → ((𝐴↑2) − ((𝑀 / 2)↑2)) = ((𝐴 + (𝑀 / 2)) · (𝐴 − (𝑀 / 2))))
83	78, 10, 82	syl2anc 595	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴↑2) − ((𝑀 / 2)↑2)) = ((𝐴 + (𝑀 / 2)) · (𝐴 − (𝑀 / 2))))
84	31	oveq2d 7427	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴↑2) − ((𝑀 / 2)↑2)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))
85	81, 83, 84	3eqtr2d 2810	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))
86	74, 75, 85	3eqtr2d 2810	. 2 ⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + (𝑀 / 2))↑2) − ((𝐴 + (𝑀 / 2)) · 𝑀)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))
87	71, 86	breqtrd 5141	1 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 class class class wbr 5113 (class class class)co 7411 ℂcc 11098 0cc0 11100 + caddc 11103 · cmul 11105 < clt 11243 ≤ cle 11244 − cmin 11441 -cneg 11442 / cdiv 11871 ℕcn 12233 2c2 12295 ℤcz 12591 mod cmo 13902 ↑cexp 14097 ∥ cdvds 16310

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-er 8694 df-en 8944 df-dom 8945 df-sdom 8946 df-sup 9402 df-inf 9403 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-n0 12505 df-z 12592 df-uz 12863 df-rp 13017 df-fl 13825 df-mod 13903 df-seq 14038 df-exp 14098 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-dvds 16311 df-gcd 16553

This theorem is referenced by: 4sqlem16 17020

W3C validator