fldiv4p1lem1div2 - Metamath Proof Explorer

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Theorem fldiv4p1lem1div2 13739

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 11748	. . . 4 ⊢ 1 ≤ 1
2	1	a1i 11	. . 3 ⊢ (𝑁 = 3 → 1 ≤ 1)
3		fvoveq1 7372	. . . . . 6 ⊢ (𝑁 = 3 → (⌊‘(𝑁 / 4)) = (⌊‘(3 / 4)))
4		3lt4 12297	. . . . . . 7 ⊢ 3 < 4
5		3nn0 12402	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
6		4nn 12211	. . . . . . . 8 ⊢ 4 ∈ ℕ
7		divfl0 13728	. . . . . . . 8 ⊢ ((3 ∈ ℕ₀ ∧ 4 ∈ ℕ) → (3 < 4 ↔ (⌊‘(3 / 4)) = 0))
8	5, 6, 7	mp2an 692	. . . . . . 7 ⊢ (3 < 4 ↔ (⌊‘(3 / 4)) = 0)
9	4, 8	mpbi 230	. . . . . 6 ⊢ (⌊‘(3 / 4)) = 0
10	3, 9	eqtrdi 2780	. . . . 5 ⊢ (𝑁 = 3 → (⌊‘(𝑁 / 4)) = 0)
11	10	oveq1d 7364	. . . 4 ⊢ (𝑁 = 3 → ((⌊‘(𝑁 / 4)) + 1) = (0 + 1))
12		0p1e1 12245	. . . 4 ⊢ (0 + 1) = 1
13	11, 12	eqtrdi 2780	. . 3 ⊢ (𝑁 = 3 → ((⌊‘(𝑁 / 4)) + 1) = 1)
14		oveq1 7356	. . . . . 6 ⊢ (𝑁 = 3 → (𝑁 − 1) = (3 − 1))
15		3m1e2 12251	. . . . . 6 ⊢ (3 − 1) = 2
16	14, 15	eqtrdi 2780	. . . . 5 ⊢ (𝑁 = 3 → (𝑁 − 1) = 2)
17	16	oveq1d 7364	. . . 4 ⊢ (𝑁 = 3 → ((𝑁 − 1) / 2) = (2 / 2))
18		2div2e1 12264	. . . 4 ⊢ (2 / 2) = 1
19	17, 18	eqtrdi 2780	. . 3 ⊢ (𝑁 = 3 → ((𝑁 − 1) / 2) = 1)
20	2, 13, 19	3brtr4d 5124	. 2 ⊢ (𝑁 = 3 → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
21		uzp1 12776	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘5) → (𝑁 = 5 ∨ 𝑁 ∈ (ℤ_≥‘(5 + 1))))
22		2re 12202	. . . . . . 7 ⊢ 2 ∈ ℝ
23	22	leidi 11654	. . . . . 6 ⊢ 2 ≤ 2
24	23	a1i 11	. . . . 5 ⊢ (𝑁 = 5 → 2 ≤ 2)
25		fvoveq1 7372	. . . . . . . 8 ⊢ (𝑁 = 5 → (⌊‘(𝑁 / 4)) = (⌊‘(5 / 4)))
26		df-5 12194	. . . . . . . . . . . 12 ⊢ 5 = (4 + 1)
27	26	oveq1i 7359	. . . . . . . . . . 11 ⊢ (5 / 4) = ((4 + 1) / 4)
28		4cn 12213	. . . . . . . . . . . 12 ⊢ 4 ∈ ℂ
29		ax-1cn 11067	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
30		4ne0 12236	. . . . . . . . . . . 12 ⊢ 4 ≠ 0
31	28, 29, 28, 30	divdiri 11881	. . . . . . . . . . 11 ⊢ ((4 + 1) / 4) = ((4 / 4) + (1 / 4))
32	28, 30	dividi 11857	. . . . . . . . . . . 12 ⊢ (4 / 4) = 1
33	32	oveq1i 7359	. . . . . . . . . . 11 ⊢ ((4 / 4) + (1 / 4)) = (1 + (1 / 4))
34	27, 31, 33	3eqtri 2756	. . . . . . . . . 10 ⊢ (5 / 4) = (1 + (1 / 4))
35	34	fveq2i 6825	. . . . . . . . 9 ⊢ (⌊‘(5 / 4)) = (⌊‘(1 + (1 / 4)))
36		1re 11115	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
37		0le1 11643	. . . . . . . . . . 11 ⊢ 0 ≤ 1
38		4re 12212	. . . . . . . . . . 11 ⊢ 4 ∈ ℝ
39		4pos 12235	. . . . . . . . . . 11 ⊢ 0 < 4
40		divge0 11994	. . . . . . . . . . 11 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ (4 ∈ ℝ ∧ 0 < 4)) → 0 ≤ (1 / 4))
41	36, 37, 38, 39, 40	mp4an 693	. . . . . . . . . 10 ⊢ 0 ≤ (1 / 4)
42		1lt4 12299	. . . . . . . . . . 11 ⊢ 1 < 4
43		recgt1 12021	. . . . . . . . . . . 12 ⊢ ((4 ∈ ℝ ∧ 0 < 4) → (1 < 4 ↔ (1 / 4) < 1))
44	38, 39, 43	mp2an 692	. . . . . . . . . . 11 ⊢ (1 < 4 ↔ (1 / 4) < 1)
45	42, 44	mpbi 230	. . . . . . . . . 10 ⊢ (1 / 4) < 1
46		1z 12505	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ
47	38, 30	rereccli 11889	. . . . . . . . . . 11 ⊢ (1 / 4) ∈ ℝ
48		flbi2 13721	. . . . . . . . . . 11 ⊢ ((1 ∈ ℤ ∧ (1 / 4) ∈ ℝ) → ((⌊‘(1 + (1 / 4))) = 1 ↔ (0 ≤ (1 / 4) ∧ (1 / 4) < 1)))
49	46, 47, 48	mp2an 692	. . . . . . . . . 10 ⊢ ((⌊‘(1 + (1 / 4))) = 1 ↔ (0 ≤ (1 / 4) ∧ (1 / 4) < 1))
50	41, 45, 49	mpbir2an 711	. . . . . . . . 9 ⊢ (⌊‘(1 + (1 / 4))) = 1
51	35, 50	eqtri 2752	. . . . . . . 8 ⊢ (⌊‘(5 / 4)) = 1
52	25, 51	eqtrdi 2780	. . . . . . 7 ⊢ (𝑁 = 5 → (⌊‘(𝑁 / 4)) = 1)
53	52	oveq1d 7364	. . . . . 6 ⊢ (𝑁 = 5 → ((⌊‘(𝑁 / 4)) + 1) = (1 + 1))
54		1p1e2 12248	. . . . . 6 ⊢ (1 + 1) = 2
55	53, 54	eqtrdi 2780	. . . . 5 ⊢ (𝑁 = 5 → ((⌊‘(𝑁 / 4)) + 1) = 2)
56		oveq1 7356	. . . . . . . 8 ⊢ (𝑁 = 5 → (𝑁 − 1) = (5 − 1))
57		5m1e4 12253	. . . . . . . 8 ⊢ (5 − 1) = 4
58	56, 57	eqtrdi 2780	. . . . . . 7 ⊢ (𝑁 = 5 → (𝑁 − 1) = 4)
59	58	oveq1d 7364	. . . . . 6 ⊢ (𝑁 = 5 → ((𝑁 − 1) / 2) = (4 / 2))
60		4d2e2 12293	. . . . . 6 ⊢ (4 / 2) = 2
61	59, 60	eqtrdi 2780	. . . . 5 ⊢ (𝑁 = 5 → ((𝑁 − 1) / 2) = 2)
62	24, 55, 61	3brtr4d 5124	. . . 4 ⊢ (𝑁 = 5 → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
63		eluz2 12741	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘6) ↔ (6 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 6 ≤ 𝑁))
64		zre 12475	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
65		id 22	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → 𝑁 ∈ ℝ)
66	38	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → 4 ∈ ℝ)
67	30	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → 4 ≠ 0)
68	65, 66, 67	redivcld 11952	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → (𝑁 / 4) ∈ ℝ)
69		flle 13703	. . . . . . . . . . 11 ⊢ ((𝑁 / 4) ∈ ℝ → (⌊‘(𝑁 / 4)) ≤ (𝑁 / 4))
70	64, 68, 69	3syl 18	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (⌊‘(𝑁 / 4)) ≤ (𝑁 / 4))
71	70	adantr 480	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → (⌊‘(𝑁 / 4)) ≤ (𝑁 / 4))
72	68	flcld 13702	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℝ → (⌊‘(𝑁 / 4)) ∈ ℤ)
73	72	zred 12580	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℝ → (⌊‘(𝑁 / 4)) ∈ ℝ)
74	36	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℝ → 1 ∈ ℝ)
75	73, 68, 74	3jca 1128	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → ((⌊‘(𝑁 / 4)) ∈ ℝ ∧ (𝑁 / 4) ∈ ℝ ∧ 1 ∈ ℝ))
76	64, 75	syl 17	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → ((⌊‘(𝑁 / 4)) ∈ ℝ ∧ (𝑁 / 4) ∈ ℝ ∧ 1 ∈ ℝ))
77	76	adantr 480	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → ((⌊‘(𝑁 / 4)) ∈ ℝ ∧ (𝑁 / 4) ∈ ℝ ∧ 1 ∈ ℝ))
78		leadd1 11588	. . . . . . . . . 10 ⊢ (((⌊‘(𝑁 / 4)) ∈ ℝ ∧ (𝑁 / 4) ∈ ℝ ∧ 1 ∈ ℝ) → ((⌊‘(𝑁 / 4)) ≤ (𝑁 / 4) ↔ ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 / 4) + 1)))
79	77, 78	syl 17	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → ((⌊‘(𝑁 / 4)) ≤ (𝑁 / 4) ↔ ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 / 4) + 1)))
80	71, 79	mpbid 232	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 / 4) + 1))
81		div4p1lem1div2 12379	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℝ ∧ 6 ≤ 𝑁) → ((𝑁 / 4) + 1) ≤ ((𝑁 − 1) / 2))
82	64, 81	sylan 580	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → ((𝑁 / 4) + 1) ≤ ((𝑁 − 1) / 2))
83		peano2re 11289	. . . . . . . . . . . . 13 ⊢ ((⌊‘(𝑁 / 4)) ∈ ℝ → ((⌊‘(𝑁 / 4)) + 1) ∈ ℝ)
84	73, 83	syl 17	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → ((⌊‘(𝑁 / 4)) + 1) ∈ ℝ)
85		peano2re 11289	. . . . . . . . . . . . 13 ⊢ ((𝑁 / 4) ∈ ℝ → ((𝑁 / 4) + 1) ∈ ℝ)
86	68, 85	syl 17	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → ((𝑁 / 4) + 1) ∈ ℝ)
87		peano2rem 11431	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℝ → (𝑁 − 1) ∈ ℝ)
88	87	rehalfcld 12371	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → ((𝑁 − 1) / 2) ∈ ℝ)
89	84, 86, 88	3jca 1128	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → (((⌊‘(𝑁 / 4)) + 1) ∈ ℝ ∧ ((𝑁 / 4) + 1) ∈ ℝ ∧ ((𝑁 − 1) / 2) ∈ ℝ))
90	64, 89	syl 17	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (((⌊‘(𝑁 / 4)) + 1) ∈ ℝ ∧ ((𝑁 / 4) + 1) ∈ ℝ ∧ ((𝑁 − 1) / 2) ∈ ℝ))
91	90	adantr 480	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → (((⌊‘(𝑁 / 4)) + 1) ∈ ℝ ∧ ((𝑁 / 4) + 1) ∈ ℝ ∧ ((𝑁 − 1) / 2) ∈ ℝ))
92		letr 11210	. . . . . . . . 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)))
93	91, 92	syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → ((((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 / 4) + 1) ∧ ((𝑁 / 4) + 1) ≤ ((𝑁 − 1) / 2)) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2)))
94	80, 82, 93	mp2and 699	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
95	94	3adant1 1130	. . . . . 6 ⊢ ((6 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
96	63, 95	sylbi 217	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘6) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
97		5p1e6 12270	. . . . . 6 ⊢ (5 + 1) = 6
98	97	fveq2i 6825	. . . . 5 ⊢ (ℤ_≥‘(5 + 1)) = (ℤ_≥‘6)
99	96, 98	eleq2s 2846	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘(5 + 1)) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
100	62, 99	jaoi 857	. . 3 ⊢ ((𝑁 = 5 ∨ 𝑁 ∈ (ℤ_≥‘(5 + 1))) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
101	21, 100	syl 17	. 2 ⊢ (𝑁 ∈ (ℤ_≥‘5) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
102	20, 101	jaoi 857	1 ⊢ ((𝑁 = 3 ∨ 𝑁 ∈ (ℤ_≥‘5)) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 class class class wbr 5092 ‘cfv 6482 (class class class)co 7349 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 < clt 11149 ≤ cle 11150 − cmin 11347 / cdiv 11777 ℕcn 12128 2c2 12183 3c3 12184 4c4 12185 5c5 12186 6c6 12187 ℕ₀cn0 12384 ℤcz 12471 ℤ_≥cuz 12735 ⌊cfl 13694

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 2701 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-sup 9332 df-inf 9333 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-n0 12385 df-z 12472 df-uz 12736 df-rp 12894 df-fl 13696

This theorem is referenced by: gausslemma2dlem0f 27270

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