4sqlem10 - Metamath Proof Explorer

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Theorem 4sqlem10 16819

Description: Lemma for 4sq 16836. (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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℕ)
3	2	nnzd 12526	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℤ)
4		zsqcl 14034	. . . 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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℤ)
8	2	nnred 12168	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℝ)
9	8	rehalfcld 12400	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℝ)
10	9	recnd 11183	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℂ)
11	10	negnegd 11503	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → --(𝑀 / 2) = (𝑀 / 2))
12		4sqlem5.4	. . . . . . . . . . . . . . 15 ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
13	6, 1, 12	4sqlem5 16814	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ))
14	13	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ))
15	14	simpld 495	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℤ)
16	15	zred 12607	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℝ)
17	6, 1, 12	4sqlem6 16815	. . . . . . . . . . . . 13 ⊢ (𝜑 → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2)))
18	17	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2)))
19	18	simprd 496	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 < (𝑀 / 2))
20	16, 19	ltned 11291	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ≠ (𝑀 / 2))
21	20	neneqd 2948	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝐵 = (𝑀 / 2))
22		2cnd 12231	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 2 ∈ ℂ)
23	22	sqvald 14048	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → (2↑2) = (2 · 2))
24	23	oveq2d 7373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀↑2) / (2↑2)) = ((𝑀↑2) / (2 · 2)))
25	2	nncnd 12169	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℂ)
26		2ne0 12257	. . . . . . . . . . . . . . 15 ⊢ 2 ≠ 0
27	26	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 2 ≠ 0)
28	25, 22, 27	sqdivd 14064	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2)↑2) = ((𝑀↑2) / (2↑2)))
29	25	sqcld 14049	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∈ ℂ)
30	29, 22, 22, 27, 27	divdiv1d 11962	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) = ((𝑀↑2) / (2 · 2)))
31	24, 28, 30	3eqtr4d 2786	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2)↑2) = (((𝑀↑2) / 2) / 2))
32	29	halfcld 12398	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀↑2) / 2) ∈ ℂ)
33	32	halfcld 12398	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) ∈ ℂ)
34	15	zcnd 12608	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℂ)
35	34	sqcld 14049	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (𝐵↑2) ∈ ℂ)
36		4sqlem10.5	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → ((((𝑀↑2) / 2) / 2) − (𝐵↑2)) = 0)
37	33, 35, 36	subeq0d 11520	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) = (𝐵↑2))
38	31, 37	eqtr2d 2777	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → (𝐵↑2) = ((𝑀 / 2)↑2))
39		sqeqor 14120	. . . . . . . . . . . 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 231	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (𝐵 = (𝑀 / 2) ∨ 𝐵 = -(𝑀 / 2)))
42	41	ord 862	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (¬ 𝐵 = (𝑀 / 2) → 𝐵 = -(𝑀 / 2)))
43	21, 42	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 = -(𝑀 / 2))
44	43, 15	eqeltrrd 2839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → -(𝑀 / 2) ∈ ℤ)
45	44	znegcld 12609	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → --(𝑀 / 2) ∈ ℤ)
46	11, 45	eqeltrrd 2839	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℤ)
47	7, 46	zaddcld 12611	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℤ)
48		zsqcl 14034	. . . 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 12613	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · 𝑀) ∈ ℤ)
51	47	zred 12607	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℝ)
52	2	nnrpd 12955	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℝ⁺)
53	51, 52	modcld 13780	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℝ)
54	53	recnd 11183	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℂ)
55		0cnd 11148	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 0 ∈ ℂ)
56		df-neg 11388	. . . . . . 7 ⊢ -(𝑀 / 2) = (0 − (𝑀 / 2))
57	43, 12, 56	3eqtr3g 2799	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (0 − (𝑀 / 2)))
58	54, 55, 10, 57	subcan2d 11554	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) = 0)
59		dvdsval3 16140	. . . . . 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 256	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∥ (𝐴 + (𝑀 / 2)))
62		dvdssq 16443	. . . . 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 231	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2))
65	25	sqvald 14048	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) = (𝑀 · 𝑀))
66	2	nnne0d 12203	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ≠ 0)
67		dvdsmulcr 16168	. . . . . 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 1374	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 · 𝑀) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀) ↔ 𝑀 ∥ (𝐴 + (𝑀 / 2))))
69	61, 68	mpbird 256	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 · 𝑀) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀))
70	65, 69	eqbrtrd 5127	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀))
71	5, 49, 50, 64, 70	dvds2subd 16175	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ (((𝐴 + (𝑀 / 2))↑2) − ((𝐴 + (𝑀 / 2)) · 𝑀)))
72	47	zcnd 12608	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℂ)
73	72	sqvald 14048	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2))↑2) = ((𝐴 + (𝑀 / 2)) · (𝐴 + (𝑀 / 2))))
74	73	oveq1d 7372	. . 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 11611	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = (((𝐴 + (𝑀 / 2)) · (𝐴 + (𝑀 / 2))) − ((𝐴 + (𝑀 / 2)) · 𝑀)))
76	25	2halvesd 12399	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2) + (𝑀 / 2)) = 𝑀)
77	76	oveq2d 7373	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − ((𝑀 / 2) + (𝑀 / 2))) = ((𝐴 + (𝑀 / 2)) − 𝑀))
78	7	zcnd 12608	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℂ)
79	78, 10, 10	pnpcan2d 11550	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − ((𝑀 / 2) + (𝑀 / 2))) = (𝐴 − (𝑀 / 2)))
80	77, 79	eqtr3d 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − 𝑀) = (𝐴 − (𝑀 / 2)))
81	80	oveq2d 7373	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = ((𝐴 + (𝑀 / 2)) · (𝐴 − (𝑀 / 2))))
82		subsq 14114	. . . . 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 7373	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴↑2) − ((𝑀 / 2)↑2)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))
85	81, 83, 84	3eqtr2d 2782	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))
86	74, 75, 85	3eqtr2d 2782	. 2 ⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + (𝑀 / 2))↑2) − ((𝐴 + (𝑀 / 2)) · 𝑀)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))
87	71, 86	breqtrd 5131	1 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5105 (class class class)co 7357 ℂcc 11049 0cc0 11051 + caddc 11054 · cmul 11056 < clt 11189 ≤ cle 11190 − cmin 11385 -cneg 11386 / cdiv 11812 ℕcn 12153 2c2 12208 ℤcz 12499 mod cmo 13774 ↑cexp 13967 ∥ cdvds 16136

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-dvds 16137 df-gcd 16375

This theorem is referenced by: 4sqlem16 16832

W3C validator