4sqlem10 - Metamath Proof Explorer

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Theorem 4sqlem10 16925

Description: Lemma for 4sq 16942. (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 480	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℕ)
3	2	nnzd 12563	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℤ)
4		zsqcl 14101	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝑀↑2) ∈ ℤ)
5	3, 4	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∈ ℤ)
6		4sqlem5.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℤ)
7	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℤ)
8	2	nnred 12208	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℝ)
9	8	rehalfcld 12436	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℝ)
10	9	recnd 11209	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℂ)
11	10	negnegd 11531	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → --(𝑀 / 2) = (𝑀 / 2))
12		4sqlem5.4	. . . . . . . . . . . . . . 15 ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
13	6, 1, 12	4sqlem5 16920	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ))
14	13	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ))
15	14	simpld 494	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℤ)
16	15	zred 12645	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℝ)
17	6, 1, 12	4sqlem6 16921	. . . . . . . . . . . . 13 ⊢ (𝜑 → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2)))
18	17	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2)))
19	18	simprd 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 < (𝑀 / 2))
20	16, 19	ltned 11317	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ≠ (𝑀 / 2))
21	20	neneqd 2931	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝐵 = (𝑀 / 2))
22		2cnd 12271	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 2 ∈ ℂ)
23	22	sqvald 14115	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → (2↑2) = (2 · 2))
24	23	oveq2d 7406	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀↑2) / (2↑2)) = ((𝑀↑2) / (2 · 2)))
25	2	nncnd 12209	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℂ)
26		2ne0 12297	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
27	26	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 2 ≠ 0)
28	25, 22, 27	sqdivd 14131	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2)↑2) = ((𝑀↑2) / (2↑2)))
29	25	sqcld 14116	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∈ ℂ)
30	29, 22, 22, 27, 27	divdiv1d 11996	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) = ((𝑀↑2) / (2 · 2)))
31	24, 28, 30	3eqtr4d 2775	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2)↑2) = (((𝑀↑2) / 2) / 2))
32	29	halfcld 12434	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀↑2) / 2) ∈ ℂ)
33	32	halfcld 12434	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) ∈ ℂ)
34	15	zcnd 12646	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℂ)
35	34	sqcld 14116	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (𝐵↑2) ∈ ℂ)
36		4sqlem10.5	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ((((𝑀↑2) / 2) / 2) − (𝐵↑2)) = 0)
37	33, 35, 36	subeq0d 11548	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) = (𝐵↑2))
38	31, 37	eqtr2d 2766	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝐵↑2) = ((𝑀 / 2)↑2))
39		sqeqor 14188	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℂ ∧ (𝑀 / 2) ∈ ℂ) → ((𝐵↑2) = ((𝑀 / 2)↑2) ↔ (𝐵 = (𝑀 / 2) ∨ 𝐵 = -(𝑀 / 2))))
40	34, 10, 39	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → ((𝐵↑2) = ((𝑀 / 2)↑2) ↔ (𝐵 = (𝑀 / 2) ∨ 𝐵 = -(𝑀 / 2))))
41	38, 40	mpbid 232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (𝐵 = (𝑀 / 2) ∨ 𝐵 = -(𝑀 / 2)))
42	41	ord 864	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (¬ 𝐵 = (𝑀 / 2) → 𝐵 = -(𝑀 / 2)))
43	21, 42	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 = -(𝑀 / 2))
44	43, 15	eqeltrrd 2830	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → -(𝑀 / 2) ∈ ℤ)
45	44	znegcld 12647	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → --(𝑀 / 2) ∈ ℤ)
46	11, 45	eqeltrrd 2830	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℤ)
47	7, 46	zaddcld 12649	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℤ)
48		zsqcl 14101	. . . 4 ⊢ ((𝐴 + (𝑀 / 2)) ∈ ℤ → ((𝐴 + (𝑀 / 2))↑2) ∈ ℤ)
49	47, 48	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2))↑2) ∈ ℤ)
50	47, 3	zmulcld 12651	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · 𝑀) ∈ ℤ)
51	47	zred 12645	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℝ)
52	2	nnrpd 13000	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℝ⁺)
53	51, 52	modcld 13844	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℝ)
54	53	recnd 11209	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℂ)
55		0cnd 11174	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 0 ∈ ℂ)
56		df-neg 11415	. . . . . . 7 ⊢ -(𝑀 / 2) = (0 − (𝑀 / 2))
57	43, 12, 56	3eqtr3g 2788	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (0 − (𝑀 / 2)))
58	54, 55, 10, 57	subcan2d 11582	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) = 0)
59		dvdsval3 16233	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ (𝐴 + (𝑀 / 2)) ∈ ℤ) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ ((𝐴 + (𝑀 / 2)) mod 𝑀) = 0))
60	2, 47, 59	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ ((𝐴 + (𝑀 / 2)) mod 𝑀) = 0))
61	58, 60	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∥ (𝐴 + (𝑀 / 2)))
62		dvdssq 16544	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ (𝐴 + (𝑀 / 2)) ∈ ℤ) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2)))
63	3, 47, 62	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2)))
64	61, 63	mpbid 232	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2))
65	25	sqvald 14115	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) = (𝑀 · 𝑀))
66	2	nnne0d 12243	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ≠ 0)
67		dvdsmulcr 16262	. . . . . 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 1376	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 · 𝑀) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀) ↔ 𝑀 ∥ (𝐴 + (𝑀 / 2))))
69	61, 68	mpbird 257	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 · 𝑀) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀))
70	65, 69	eqbrtrd 5132	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀))
71	5, 49, 50, 64, 70	dvds2subd 16270	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ (((𝐴 + (𝑀 / 2))↑2) − ((𝐴 + (𝑀 / 2)) · 𝑀)))
72	47	zcnd 12646	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℂ)
73	72	sqvald 14115	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2))↑2) = ((𝐴 + (𝑀 / 2)) · (𝐴 + (𝑀 / 2))))
74	73	oveq1d 7405	. . 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 11641	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = (((𝐴 + (𝑀 / 2)) · (𝐴 + (𝑀 / 2))) − ((𝐴 + (𝑀 / 2)) · 𝑀)))
76	25	2halvesd 12435	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2) + (𝑀 / 2)) = 𝑀)
77	76	oveq2d 7406	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − ((𝑀 / 2) + (𝑀 / 2))) = ((𝐴 + (𝑀 / 2)) − 𝑀))
78	7	zcnd 12646	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℂ)
79	78, 10, 10	pnpcan2d 11578	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − ((𝑀 / 2) + (𝑀 / 2))) = (𝐴 − (𝑀 / 2)))
80	77, 79	eqtr3d 2767	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − 𝑀) = (𝐴 − (𝑀 / 2)))
81	80	oveq2d 7406	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = ((𝐴 + (𝑀 / 2)) · (𝐴 − (𝑀 / 2))))
82		subsq 14182	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 / 2) ∈ ℂ) → ((𝐴↑2) − ((𝑀 / 2)↑2)) = ((𝐴 + (𝑀 / 2)) · (𝐴 − (𝑀 / 2))))
83	78, 10, 82	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴↑2) − ((𝑀 / 2)↑2)) = ((𝐴 + (𝑀 / 2)) · (𝐴 − (𝑀 / 2))))
84	31	oveq2d 7406	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴↑2) − ((𝑀 / 2)↑2)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))
85	81, 83, 84	3eqtr2d 2771	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))
86	74, 75, 85	3eqtr2d 2771	. 2 ⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + (𝑀 / 2))↑2) − ((𝐴 + (𝑀 / 2)) · 𝑀)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))
87	71, 86	breqtrd 5136	1 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 class class class wbr 5110 (class class class)co 7390 ℂcc 11073 0cc0 11075 + caddc 11078 · cmul 11080 < clt 11215 ≤ cle 11216 − cmin 11412 -cneg 11413 / cdiv 11842 ℕcn 12193 2c2 12248 ℤcz 12536 mod cmo 13838 ↑cexp 14033 ∥ cdvds 16229

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-sup 9400 df-inf 9401 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-z 12537 df-uz 12801 df-rp 12959 df-fl 13761 df-mod 13839 df-seq 13974 df-exp 14034 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-dvds 16230 df-gcd 16472

This theorem is referenced by: 4sqlem16 16938

W3C validator