4sqlem10 - Metamath Proof Explorer

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Theorem 4sqlem10 16133

Description: Lemma for 4sq 16150. (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 473	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℕ)
3	2	nnzd 11896	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℤ)
4		zsqcl 13306	. . . 4 ⊢ (𝑀 ∈ ℤ → (𝑀↑2) ∈ ℤ)
5	3, 4	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∈ ℤ)
6		4sqlem5.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℤ)
7	6	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℤ)
8	2	nnred 11452	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℝ)
9	8	rehalfcld 11691	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℝ)
10	9	recnd 10464	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℂ)
11	10	negnegd 10785	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → --(𝑀 / 2) = (𝑀 / 2))
12		4sqlem5.4	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐵 = (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2))
13	6, 1, 12	4sqlem5 16128	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ))
14	13	adantr 473	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → (𝐵 ∈ ℤ ∧ ((𝐴 − 𝐵) / 𝑀) ∈ ℤ))
15	14	simpld 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℤ)
16	15	zred 11897	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℝ)
17	6, 1, 12	4sqlem6 16129	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2)))
18	17	adantr 473	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → (-(𝑀 / 2) ≤ 𝐵 ∧ 𝐵 < (𝑀 / 2)))
19	18	simprd 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 < (𝑀 / 2))
20	16, 19	ltned 10572	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ≠ (𝑀 / 2))
21	20	neneqd 2969	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝐵 = (𝑀 / 2))
22		2cnd 11515	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝜓) → 2 ∈ ℂ)
23	22	sqvald 13319	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜓) → (2↑2) = (2 · 2))
24	23	oveq2d 6990	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀↑2) / (2↑2)) = ((𝑀↑2) / (2 · 2)))
25	2	nncnd 11453	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℂ)
26		2ne0 11548	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ≠ 0
27	26	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜓) → 2 ≠ 0)
28	25, 22, 27	sqdivd 13335	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2)↑2) = ((𝑀↑2) / (2↑2)))
29	25	sqcld 13320	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∈ ℂ)
30	29, 22, 22, 27, 27	divdiv1d 11244	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) = ((𝑀↑2) / (2 · 2)))
31	24, 28, 30	3eqtr4d 2821	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2)↑2) = (((𝑀↑2) / 2) / 2))
32	29	halfcld 11689	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀↑2) / 2) ∈ ℂ)
33	32	halfcld 11689	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) ∈ ℂ)
34	15	zcnd 11898	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 ∈ ℂ)
35	34	sqcld 13320	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → (𝐵↑2) ∈ ℂ)
36		4sqlem10.5	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝜓) → ((((𝑀↑2) / 2) / 2) − (𝐵↑2)) = 0)
37	33, 35, 36	subeq0d 10802	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝜓) → (((𝑀↑2) / 2) / 2) = (𝐵↑2))
38	31, 37	eqtr2d 2812	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → (𝐵↑2) = ((𝑀 / 2)↑2))
39		sqeqor 13390	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐵 ∈ ℂ ∧ (𝑀 / 2) ∈ ℂ) → ((𝐵↑2) = ((𝑀 / 2)↑2) ↔ (𝐵 = (𝑀 / 2) ∨ 𝐵 = -(𝑀 / 2))))
40	34, 10, 39	syl2anc 576	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝜓) → ((𝐵↑2) = ((𝑀 / 2)↑2) ↔ (𝐵 = (𝑀 / 2) ∨ 𝐵 = -(𝑀 / 2))))
41	38, 40	mpbid 224	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝜓) → (𝐵 = (𝑀 / 2) ∨ 𝐵 = -(𝑀 / 2)))
42	41	ord 850	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝜓) → (¬ 𝐵 = (𝑀 / 2) → 𝐵 = -(𝑀 / 2)))
43	21, 42	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 = -(𝑀 / 2))
44	43, 15	eqeltrrd 2864	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝜓) → -(𝑀 / 2) ∈ ℤ)
45	44	znegcld 11899	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → --(𝑀 / 2) ∈ ℤ)
46	11, 45	eqeltrrd 2864	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 / 2) ∈ ℤ)
47	7, 46	zaddcld 11901	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℤ)
48	47	zred 11897	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℝ)
49	2	nnrpd 12243	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∈ ℝ⁺)
50	48, 49	modcld 13055	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℝ)
51	50	recnd 10464	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) ∈ ℂ)
52		0cnd 10428	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 0 ∈ ℂ)
53		df-neg 10669	. . . . . . 7 ⊢ -(𝑀 / 2) = (0 − (𝑀 / 2))
54	43, 12, 53	3eqtr3g 2834	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + (𝑀 / 2)) mod 𝑀) − (𝑀 / 2)) = (0 − (𝑀 / 2)))
55	51, 52, 10, 54	subcan2d 10836	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) mod 𝑀) = 0)
56		dvdsval3 15465	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ (𝐴 + (𝑀 / 2)) ∈ ℤ) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ ((𝐴 + (𝑀 / 2)) mod 𝑀) = 0))
57	2, 47, 56	syl2anc 576	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ ((𝐴 + (𝑀 / 2)) mod 𝑀) = 0))
58	55, 57	mpbird 249	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ∥ (𝐴 + (𝑀 / 2)))
59		dvdssq 15761	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ (𝐴 + (𝑀 / 2)) ∈ ℤ) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2)))
60	3, 47, 59	syl2anc 576	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 ∥ (𝐴 + (𝑀 / 2)) ↔ (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2)))
61	58, 60	mpbid 224	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2))↑2))
62	25	sqvald 13319	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) = (𝑀 · 𝑀))
63	2	nnne0d 11487	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑀 ≠ 0)
64		dvdsmulcr 15493	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ (𝐴 + (𝑀 / 2)) ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑀 ≠ 0)) → ((𝑀 · 𝑀) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀) ↔ 𝑀 ∥ (𝐴 + (𝑀 / 2))))
65	3, 47, 3, 63, 64	syl112anc 1354	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 · 𝑀) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀) ↔ 𝑀 ∥ (𝐴 + (𝑀 / 2))))
66	58, 65	mpbird 249	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑀 · 𝑀) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀))
67	62, 66	eqbrtrd 4949	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴 + (𝑀 / 2)) · 𝑀))
68		zsqcl 13306	. . . 4 ⊢ ((𝐴 + (𝑀 / 2)) ∈ ℤ → ((𝐴 + (𝑀 / 2))↑2) ∈ ℤ)
69	47, 68	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2))↑2) ∈ ℤ)
70	47, 3	zmulcld 11903	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · 𝑀) ∈ ℤ)
71	5, 61, 67, 69, 70	dvds2subd 15499	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ (((𝐴 + (𝑀 / 2))↑2) − ((𝐴 + (𝑀 / 2)) · 𝑀)))
72	47	zcnd 11898	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 + (𝑀 / 2)) ∈ ℂ)
73	72	sqvald 13319	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2))↑2) = ((𝐴 + (𝑀 / 2)) · (𝐴 + (𝑀 / 2))))
74	73	oveq1d 6989	. . 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 10893	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = (((𝐴 + (𝑀 / 2)) · (𝐴 + (𝑀 / 2))) − ((𝐴 + (𝑀 / 2)) · 𝑀)))
76	25	2halvesd 11690	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ((𝑀 / 2) + (𝑀 / 2)) = 𝑀)
77	76	oveq2d 6990	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − ((𝑀 / 2) + (𝑀 / 2))) = ((𝐴 + (𝑀 / 2)) − 𝑀))
78	7	zcnd 11898	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ∈ ℂ)
79	78, 10, 10	pnpcan2d 10832	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − ((𝑀 / 2) + (𝑀 / 2))) = (𝐴 − (𝑀 / 2)))
80	77, 79	eqtr3d 2813	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) − 𝑀) = (𝐴 − (𝑀 / 2)))
81	80	oveq2d 6990	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = ((𝐴 + (𝑀 / 2)) · (𝐴 − (𝑀 / 2))))
82		subsq 13384	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑀 / 2) ∈ ℂ) → ((𝐴↑2) − ((𝑀 / 2)↑2)) = ((𝐴 + (𝑀 / 2)) · (𝐴 − (𝑀 / 2))))
83	78, 10, 82	syl2anc 576	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴↑2) − ((𝑀 / 2)↑2)) = ((𝐴 + (𝑀 / 2)) · (𝐴 − (𝑀 / 2))))
84	31	oveq2d 6990	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴↑2) − ((𝑀 / 2)↑2)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))
85	81, 83, 84	3eqtr2d 2817	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + (𝑀 / 2)) · ((𝐴 + (𝑀 / 2)) − 𝑀)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))
86	74, 75, 85	3eqtr2d 2817	. 2 ⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + (𝑀 / 2))↑2) − ((𝐴 + (𝑀 / 2)) · 𝑀)) = ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))
87	71, 86	breqtrd 4953	1 ⊢ ((𝜑 ∧ 𝜓) → (𝑀↑2) ∥ ((𝐴↑2) − (((𝑀↑2) / 2) / 2)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∨ wo 833 = wceq 1507 ∈ wcel 2048 ≠ wne 2964 class class class wbr 4927 (class class class)co 6974 ℂcc 10329 0cc0 10331 + caddc 10334 · cmul 10336 < clt 10470 ≤ cle 10471 − cmin 10666 -cneg 10667 / cdiv 11094 ℕcn 11435 2c2 11492 ℤcz 11790 mod cmo 13049 ↑cexp 13241 ∥ cdvds 15461

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2747 ax-sep 5058 ax-nul 5065 ax-pow 5117 ax-pr 5184 ax-un 7277 ax-cnex 10387 ax-resscn 10388 ax-1cn 10389 ax-icn 10390 ax-addcl 10391 ax-addrcl 10392 ax-mulcl 10393 ax-mulrcl 10394 ax-mulcom 10395 ax-addass 10396 ax-mulass 10397 ax-distr 10398 ax-i2m1 10399 ax-1ne0 10400 ax-1rid 10401 ax-rnegex 10402 ax-rrecex 10403 ax-cnre 10404 ax-pre-lttri 10405 ax-pre-lttrn 10406 ax-pre-ltadd 10407 ax-pre-mulgt0 10408 ax-pre-sup 10409

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2756 df-cleq 2768 df-clel 2843 df-nfc 2915 df-ne 2965 df-nel 3071 df-ral 3090 df-rex 3091 df-reu 3092 df-rmo 3093 df-rab 3094 df-v 3414 df-sbc 3681 df-csb 3786 df-dif 3831 df-un 3833 df-in 3835 df-ss 3842 df-pss 3844 df-nul 4178 df-if 4349 df-pw 4422 df-sn 4440 df-pr 4442 df-tp 4444 df-op 4446 df-uni 4711 df-iun 4792 df-br 4928 df-opab 4990 df-mpt 5007 df-tr 5029 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-2nd 7499 df-wrecs 7747 df-recs 7809 df-rdg 7847 df-er 8085 df-en 8303 df-dom 8304 df-sdom 8305 df-sup 8697 df-inf 8698 df-pnf 10472 df-mnf 10473 df-xr 10474 df-ltxr 10475 df-le 10476 df-sub 10668 df-neg 10669 df-div 11095 df-nn 11436 df-2 11500 df-3 11501 df-n0 11705 df-z 11791 df-uz 12056 df-rp 12202 df-fl 12974 df-mod 13050 df-seq 13182 df-exp 13242 df-cj 14313 df-re 14314 df-im 14315 df-sqrt 14449 df-abs 14450 df-dvds 15462 df-gcd 15698

This theorem is referenced by: 4sqlem16 16146

W3C validator