fldiv4p1lem1div2 - Intuitionistic Logic Explorer

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Theorem fldiv4p1lem1div2 10448

Description: The floor of an integer equal to 3 or greater than 4, increased by 1, is less than or equal to the half of the integer minus 1. (Contributed by AV, 8-Jul-2021.)

**Assertion**
Ref	Expression
fldiv4p1lem1div2	⊢ ((𝑁 = 3 ∨ 𝑁 ∈ (ℤ_≥‘5)) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))

**Proof of Theorem fldiv4p1lem1div2**
Step	Hyp	Ref	Expression
1		1le1 8645	. . . 4 ⊢ 1 ≤ 1
2	1	a1i 9	. . 3 ⊢ (𝑁 = 3 → 1 ≤ 1)
3		oveq1 5951	. . . . . . 7 ⊢ (𝑁 = 3 → (𝑁 / 4) = (3 / 4))
4	3	fveq2d 5580	. . . . . 6 ⊢ (𝑁 = 3 → (⌊‘(𝑁 / 4)) = (⌊‘(3 / 4)))
5		3lt4 9209	. . . . . . 7 ⊢ 3 < 4
6		3nn0 9313	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
7		4nn 9200	. . . . . . . 8 ⊢ 4 ∈ ℕ
8		divfl0 10439	. . . . . . . 8 ⊢ ((3 ∈ ℕ₀ ∧ 4 ∈ ℕ) → (3 < 4 ↔ (⌊‘(3 / 4)) = 0))
9	6, 7, 8	mp2an 426	. . . . . . 7 ⊢ (3 < 4 ↔ (⌊‘(3 / 4)) = 0)
10	5, 9	mpbi 145	. . . . . 6 ⊢ (⌊‘(3 / 4)) = 0
11	4, 10	eqtrdi 2254	. . . . 5 ⊢ (𝑁 = 3 → (⌊‘(𝑁 / 4)) = 0)
12	11	oveq1d 5959	. . . 4 ⊢ (𝑁 = 3 → ((⌊‘(𝑁 / 4)) + 1) = (0 + 1))
13		0p1e1 9150	. . . 4 ⊢ (0 + 1) = 1
14	12, 13	eqtrdi 2254	. . 3 ⊢ (𝑁 = 3 → ((⌊‘(𝑁 / 4)) + 1) = 1)
15		oveq1 5951	. . . . . 6 ⊢ (𝑁 = 3 → (𝑁 − 1) = (3 − 1))
16		3m1e2 9156	. . . . . 6 ⊢ (3 − 1) = 2
17	15, 16	eqtrdi 2254	. . . . 5 ⊢ (𝑁 = 3 → (𝑁 − 1) = 2)
18	17	oveq1d 5959	. . . 4 ⊢ (𝑁 = 3 → ((𝑁 − 1) / 2) = (2 / 2))
19		2div2e1 9169	. . . 4 ⊢ (2 / 2) = 1
20	18, 19	eqtrdi 2254	. . 3 ⊢ (𝑁 = 3 → ((𝑁 − 1) / 2) = 1)
21	2, 14, 20	3brtr4d 4076	. 2 ⊢ (𝑁 = 3 → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
22		uzp1 9682	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘5) → (𝑁 = 5 ∨ 𝑁 ∈ (ℤ_≥‘(5 + 1))))
23		2re 9106	. . . . . . 7 ⊢ 2 ∈ ℝ
24	23	leidi 8558	. . . . . 6 ⊢ 2 ≤ 2
25	24	a1i 9	. . . . 5 ⊢ (𝑁 = 5 → 2 ≤ 2)
26		oveq1 5951	. . . . . . . . 9 ⊢ (𝑁 = 5 → (𝑁 / 4) = (5 / 4))
27	26	fveq2d 5580	. . . . . . . 8 ⊢ (𝑁 = 5 → (⌊‘(𝑁 / 4)) = (⌊‘(5 / 4)))
28		df-5 9098	. . . . . . . . . . . 12 ⊢ 5 = (4 + 1)
29	28	oveq1i 5954	. . . . . . . . . . 11 ⊢ (5 / 4) = ((4 + 1) / 4)
30		4cn 9114	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℂ
31		ax-1cn 8018	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
32		4ap0 9135	. . . . . . . . . . . . 13 ⊢ 4 # 0
33	30, 31, 30, 32	divdirapi 8842	. . . . . . . . . . . 12 ⊢ ((4 + 1) / 4) = ((4 / 4) + (1 / 4))
34	30, 32	dividapi 8818	. . . . . . . . . . . . 13 ⊢ (4 / 4) = 1
35	34	oveq1i 5954	. . . . . . . . . . . 12 ⊢ ((4 / 4) + (1 / 4)) = (1 + (1 / 4))
36	33, 35	eqtri 2226	. . . . . . . . . . 11 ⊢ ((4 + 1) / 4) = (1 + (1 / 4))
37	29, 36	eqtri 2226	. . . . . . . . . 10 ⊢ (5 / 4) = (1 + (1 / 4))
38	37	fveq2i 5579	. . . . . . . . 9 ⊢ (⌊‘(5 / 4)) = (⌊‘(1 + (1 / 4)))
39		1re 8071	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
40		0le1 8554	. . . . . . . . . . 11 ⊢ 0 ≤ 1
41		4re 9113	. . . . . . . . . . 11 ⊢ 4 ∈ ℝ
42		4pos 9133	. . . . . . . . . . 11 ⊢ 0 < 4
43		divge0 8946	. . . . . . . . . . 11 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ (4 ∈ ℝ ∧ 0 < 4)) → 0 ≤ (1 / 4))
44	39, 40, 41, 42, 43	mp4an 427	. . . . . . . . . 10 ⊢ 0 ≤ (1 / 4)
45		1lt4 9211	. . . . . . . . . . 11 ⊢ 1 < 4
46		recgt1 8970	. . . . . . . . . . . 12 ⊢ ((4 ∈ ℝ ∧ 0 < 4) → (1 < 4 ↔ (1 / 4) < 1))
47	41, 42, 46	mp2an 426	. . . . . . . . . . 11 ⊢ (1 < 4 ↔ (1 / 4) < 1)
48	45, 47	mpbi 145	. . . . . . . . . 10 ⊢ (1 / 4) < 1
49		1z 9398	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ
50		znq 9745	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ 4 ∈ ℕ) → (1 / 4) ∈ ℚ)
51	49, 7, 50	mp2an 426	. . . . . . . . . . 11 ⊢ (1 / 4) ∈ ℚ
52		flqbi2 10434	. . . . . . . . . . 11 ⊢ ((1 ∈ ℤ ∧ (1 / 4) ∈ ℚ) → ((⌊‘(1 + (1 / 4))) = 1 ↔ (0 ≤ (1 / 4) ∧ (1 / 4) < 1)))
53	49, 51, 52	mp2an 426	. . . . . . . . . 10 ⊢ ((⌊‘(1 + (1 / 4))) = 1 ↔ (0 ≤ (1 / 4) ∧ (1 / 4) < 1))
54	44, 48, 53	mpbir2an 945	. . . . . . . . 9 ⊢ (⌊‘(1 + (1 / 4))) = 1
55	38, 54	eqtri 2226	. . . . . . . 8 ⊢ (⌊‘(5 / 4)) = 1
56	27, 55	eqtrdi 2254	. . . . . . 7 ⊢ (𝑁 = 5 → (⌊‘(𝑁 / 4)) = 1)
57	56	oveq1d 5959	. . . . . 6 ⊢ (𝑁 = 5 → ((⌊‘(𝑁 / 4)) + 1) = (1 + 1))
58		1p1e2 9153	. . . . . 6 ⊢ (1 + 1) = 2
59	57, 58	eqtrdi 2254	. . . . 5 ⊢ (𝑁 = 5 → ((⌊‘(𝑁 / 4)) + 1) = 2)
60		oveq1 5951	. . . . . . . 8 ⊢ (𝑁 = 5 → (𝑁 − 1) = (5 − 1))
61	30, 31, 28	mvrraddi 8289	. . . . . . . 8 ⊢ (5 − 1) = 4
62	60, 61	eqtrdi 2254	. . . . . . 7 ⊢ (𝑁 = 5 → (𝑁 − 1) = 4)
63	62	oveq1d 5959	. . . . . 6 ⊢ (𝑁 = 5 → ((𝑁 − 1) / 2) = (4 / 2))
64		4d2e2 9197	. . . . . 6 ⊢ (4 / 2) = 2
65	63, 64	eqtrdi 2254	. . . . 5 ⊢ (𝑁 = 5 → ((𝑁 − 1) / 2) = 2)
66	25, 59, 65	3brtr4d 4076	. . . 4 ⊢ (𝑁 = 5 → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
67		eluz2 9654	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘6) ↔ (6 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 6 ≤ 𝑁))
68		znq 9745	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 4 ∈ ℕ) → (𝑁 / 4) ∈ ℚ)
69	7, 68	mpan2 425	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (𝑁 / 4) ∈ ℚ)
70		flqle 10421	. . . . . . . . . . 11 ⊢ ((𝑁 / 4) ∈ ℚ → (⌊‘(𝑁 / 4)) ≤ (𝑁 / 4))
71	69, 70	syl 14	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (⌊‘(𝑁 / 4)) ≤ (𝑁 / 4))
72	71	adantr 276	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → (⌊‘(𝑁 / 4)) ≤ (𝑁 / 4))
73	69	flqcld 10420	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → (⌊‘(𝑁 / 4)) ∈ ℤ)
74	73	zred 9495	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℤ → (⌊‘(𝑁 / 4)) ∈ ℝ)
75		zre 9376	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
76		id 19	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℝ)
77	41	a1i 9	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℝ → 4 ∈ ℝ)
78	32	a1i 9	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℝ → 4 # 0)
79	76, 77, 78	redivclapd 8908	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℝ → (𝑁 / 4) ∈ ℝ)
80	75, 79	syl 14	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℤ → (𝑁 / 4) ∈ ℝ)
81	39	a1i 9	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℝ)
82	74, 80, 81	3jca 1180	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → ((⌊‘(𝑁 / 4)) ∈ ℝ ∧ (𝑁 / 4) ∈ ℝ ∧ 1 ∈ ℝ))
83	82	adantr 276	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → ((⌊‘(𝑁 / 4)) ∈ ℝ ∧ (𝑁 / 4) ∈ ℝ ∧ 1 ∈ ℝ))
84		leadd1 8503	. . . . . . . . . 10 ⊢ (((⌊‘(𝑁 / 4)) ∈ ℝ ∧ (𝑁 / 4) ∈ ℝ ∧ 1 ∈ ℝ) → ((⌊‘(𝑁 / 4)) ≤ (𝑁 / 4) ↔ ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 / 4) + 1)))
85	83, 84	syl 14	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → ((⌊‘(𝑁 / 4)) ≤ (𝑁 / 4) ↔ ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 / 4) + 1)))
86	72, 85	mpbid 147	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 / 4) + 1))
87		div4p1lem1div2 9291	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℝ ∧ 6 ≤ 𝑁) → ((𝑁 / 4) + 1) ≤ ((𝑁 − 1) / 2))
88	75, 87	sylan 283	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → ((𝑁 / 4) + 1) ≤ ((𝑁 − 1) / 2))
89		peano2re 8208	. . . . . . . . . . . 12 ⊢ ((⌊‘(𝑁 / 4)) ∈ ℝ → ((⌊‘(𝑁 / 4)) + 1) ∈ ℝ)
90	74, 89	syl 14	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → ((⌊‘(𝑁 / 4)) + 1) ∈ ℝ)
91		peano2re 8208	. . . . . . . . . . . 12 ⊢ ((𝑁 / 4) ∈ ℝ → ((𝑁 / 4) + 1) ∈ ℝ)
92	80, 91	syl 14	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 4) + 1) ∈ ℝ)
93		peano2rem 8339	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℝ → (𝑁 − 1) ∈ ℝ)
94	93	rehalfcld 9284	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → ((𝑁 − 1) / 2) ∈ ℝ)
95	75, 94	syl 14	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → ((𝑁 − 1) / 2) ∈ ℝ)
96	90, 92, 95	3jca 1180	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (((⌊‘(𝑁 / 4)) + 1) ∈ ℝ ∧ ((𝑁 / 4) + 1) ∈ ℝ ∧ ((𝑁 − 1) / 2) ∈ ℝ))
97	96	adantr 276	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → (((⌊‘(𝑁 / 4)) + 1) ∈ ℝ ∧ ((𝑁 / 4) + 1) ∈ ℝ ∧ ((𝑁 − 1) / 2) ∈ ℝ))
98		letr 8155	. . . . . . . . 9 ⊢ ((((⌊‘(𝑁 / 4)) + 1) ∈ ℝ ∧ ((𝑁 / 4) + 1) ∈ ℝ ∧ ((𝑁 − 1) / 2) ∈ ℝ) → ((((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 / 4) + 1) ∧ ((𝑁 / 4) + 1) ≤ ((𝑁 − 1) / 2)) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2)))
99	97, 98	syl 14	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → ((((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 / 4) + 1) ∧ ((𝑁 / 4) + 1) ≤ ((𝑁 − 1) / 2)) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2)))
100	86, 88, 99	mp2and 433	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
101	100	3adant1 1018	. . . . . 6 ⊢ ((6 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
102	67, 101	sylbi 121	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘6) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
103		5p1e6 9174	. . . . . 6 ⊢ (5 + 1) = 6
104	103	fveq2i 5579	. . . . 5 ⊢ (ℤ_≥‘(5 + 1)) = (ℤ_≥‘6)
105	102, 104	eleq2s 2300	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘(5 + 1)) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
106	66, 105	jaoi 718	. . 3 ⊢ ((𝑁 = 5 ∨ 𝑁 ∈ (ℤ_≥‘(5 + 1))) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
107	22, 106	syl 14	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘5) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
108	21, 107	jaoi 718	1 ⊢ ((𝑁 = 3 ∨ 𝑁 ∈ (ℤ_≥‘5)) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 710 ∧ w3a 981 = wceq 1373 ∈ wcel 2176 class class class wbr 4044 ‘cfv 5271 (class class class)co 5944 ℝcr 7924 0cc0 7925 1c1 7926 + caddc 7928 < clt 8107 ≤ cle 8108 − cmin 8243 # cap 8654 / cdiv 8745 ℕcn 9036 2c2 9087 3c3 9088 4c4 9089 5c5 9090 6c6 9091 ℕ₀cn0 9295 ℤcz 9372 ℤ_≥cuz 9648 ℚcq 9740 ⌊cfl 10411

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 ax-pre-mulgt0 8042 ax-pre-mulext 8043 ax-arch 8044

This theorem depends on definitions: df-bi 117 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-id 4340 df-po 4343 df-iso 4344 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-reap 8648 df-ap 8655 df-div 8746 df-inn 9037 df-2 9095 df-3 9096 df-4 9097 df-5 9098 df-6 9099 df-n0 9296 df-z 9373 df-uz 9649 df-q 9741 df-rp 9776 df-fl 10413

This theorem is referenced by: gausslemma2dlem0f 15531

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