fldiv4p1lem1div2 - Metamath Proof Explorer

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Theorem fldiv4p1lem1div2 13886

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 11918	. . . 4 ⊢ 1 ≤ 1
2	1	a1i 11	. . 3 ⊢ (𝑁 = 3 → 1 ≤ 1)
3		fvoveq1 7471	. . . . . 6 ⊢ (𝑁 = 3 → (⌊‘(𝑁 / 4)) = (⌊‘(3 / 4)))
4		3lt4 12467	. . . . . . 7 ⊢ 3 < 4
5		3nn0 12571	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
6		4nn 12376	. . . . . . . 8 ⊢ 4 ∈ ℕ
7		divfl0 13875	. . . . . . . 8 ⊢ ((3 ∈ ℕ₀ ∧ 4 ∈ ℕ) → (3 < 4 ↔ (⌊‘(3 / 4)) = 0))
8	5, 6, 7	mp2an 691	. . . . . . 7 ⊢ (3 < 4 ↔ (⌊‘(3 / 4)) = 0)
9	4, 8	mpbi 230	. . . . . 6 ⊢ (⌊‘(3 / 4)) = 0
10	3, 9	eqtrdi 2796	. . . . 5 ⊢ (𝑁 = 3 → (⌊‘(𝑁 / 4)) = 0)
11	10	oveq1d 7463	. . . 4 ⊢ (𝑁 = 3 → ((⌊‘(𝑁 / 4)) + 1) = (0 + 1))
12		0p1e1 12415	. . . 4 ⊢ (0 + 1) = 1
13	11, 12	eqtrdi 2796	. . 3 ⊢ (𝑁 = 3 → ((⌊‘(𝑁 / 4)) + 1) = 1)
14		oveq1 7455	. . . . . 6 ⊢ (𝑁 = 3 → (𝑁 − 1) = (3 − 1))
15		3m1e2 12421	. . . . . 6 ⊢ (3 − 1) = 2
16	14, 15	eqtrdi 2796	. . . . 5 ⊢ (𝑁 = 3 → (𝑁 − 1) = 2)
17	16	oveq1d 7463	. . . 4 ⊢ (𝑁 = 3 → ((𝑁 − 1) / 2) = (2 / 2))
18		2div2e1 12434	. . . 4 ⊢ (2 / 2) = 1
19	17, 18	eqtrdi 2796	. . 3 ⊢ (𝑁 = 3 → ((𝑁 − 1) / 2) = 1)
20	2, 13, 19	3brtr4d 5198	. 2 ⊢ (𝑁 = 3 → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
21		uzp1 12944	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘5) → (𝑁 = 5 ∨ 𝑁 ∈ (ℤ_≥‘(5 + 1))))
22		2re 12367	. . . . . . 7 ⊢ 2 ∈ ℝ
23	22	leidi 11824	. . . . . 6 ⊢ 2 ≤ 2
24	23	a1i 11	. . . . 5 ⊢ (𝑁 = 5 → 2 ≤ 2)
25		fvoveq1 7471	. . . . . . . 8 ⊢ (𝑁 = 5 → (⌊‘(𝑁 / 4)) = (⌊‘(5 / 4)))
26		df-5 12359	. . . . . . . . . . . 12 ⊢ 5 = (4 + 1)
27	26	oveq1i 7458	. . . . . . . . . . 11 ⊢ (5 / 4) = ((4 + 1) / 4)
28		4cn 12378	. . . . . . . . . . . 12 ⊢ 4 ∈ ℂ
29		ax-1cn 11242	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
30		4ne0 12401	. . . . . . . . . . . 12 ⊢ 4 ≠ 0
31	28, 29, 28, 30	divdiri 12051	. . . . . . . . . . 11 ⊢ ((4 + 1) / 4) = ((4 / 4) + (1 / 4))
32	28, 30	dividi 12027	. . . . . . . . . . . 12 ⊢ (4 / 4) = 1
33	32	oveq1i 7458	. . . . . . . . . . 11 ⊢ ((4 / 4) + (1 / 4)) = (1 + (1 / 4))
34	27, 31, 33	3eqtri 2772	. . . . . . . . . 10 ⊢ (5 / 4) = (1 + (1 / 4))
35	34	fveq2i 6923	. . . . . . . . 9 ⊢ (⌊‘(5 / 4)) = (⌊‘(1 + (1 / 4)))
36		1re 11290	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
37		0le1 11813	. . . . . . . . . . 11 ⊢ 0 ≤ 1
38		4re 12377	. . . . . . . . . . 11 ⊢ 4 ∈ ℝ
39		4pos 12400	. . . . . . . . . . 11 ⊢ 0 < 4
40		divge0 12164	. . . . . . . . . . 11 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ (4 ∈ ℝ ∧ 0 < 4)) → 0 ≤ (1 / 4))
41	36, 37, 38, 39, 40	mp4an 692	. . . . . . . . . 10 ⊢ 0 ≤ (1 / 4)
42		1lt4 12469	. . . . . . . . . . 11 ⊢ 1 < 4
43		recgt1 12191	. . . . . . . . . . . 12 ⊢ ((4 ∈ ℝ ∧ 0 < 4) → (1 < 4 ↔ (1 / 4) < 1))
44	38, 39, 43	mp2an 691	. . . . . . . . . . 11 ⊢ (1 < 4 ↔ (1 / 4) < 1)
45	42, 44	mpbi 230	. . . . . . . . . 10 ⊢ (1 / 4) < 1
46		1z 12673	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ
47	38, 30	rereccli 12059	. . . . . . . . . . 11 ⊢ (1 / 4) ∈ ℝ
48		flbi2 13868	. . . . . . . . . . 11 ⊢ ((1 ∈ ℤ ∧ (1 / 4) ∈ ℝ) → ((⌊‘(1 + (1 / 4))) = 1 ↔ (0 ≤ (1 / 4) ∧ (1 / 4) < 1)))
49	46, 47, 48	mp2an 691	. . . . . . . . . 10 ⊢ ((⌊‘(1 + (1 / 4))) = 1 ↔ (0 ≤ (1 / 4) ∧ (1 / 4) < 1))
50	41, 45, 49	mpbir2an 710	. . . . . . . . 9 ⊢ (⌊‘(1 + (1 / 4))) = 1
51	35, 50	eqtri 2768	. . . . . . . 8 ⊢ (⌊‘(5 / 4)) = 1
52	25, 51	eqtrdi 2796	. . . . . . 7 ⊢ (𝑁 = 5 → (⌊‘(𝑁 / 4)) = 1)
53	52	oveq1d 7463	. . . . . 6 ⊢ (𝑁 = 5 → ((⌊‘(𝑁 / 4)) + 1) = (1 + 1))
54		1p1e2 12418	. . . . . 6 ⊢ (1 + 1) = 2
55	53, 54	eqtrdi 2796	. . . . 5 ⊢ (𝑁 = 5 → ((⌊‘(𝑁 / 4)) + 1) = 2)
56		oveq1 7455	. . . . . . . 8 ⊢ (𝑁 = 5 → (𝑁 − 1) = (5 − 1))
57		5m1e4 12423	. . . . . . . 8 ⊢ (5 − 1) = 4
58	56, 57	eqtrdi 2796	. . . . . . 7 ⊢ (𝑁 = 5 → (𝑁 − 1) = 4)
59	58	oveq1d 7463	. . . . . 6 ⊢ (𝑁 = 5 → ((𝑁 − 1) / 2) = (4 / 2))
60		4d2e2 12463	. . . . . 6 ⊢ (4 / 2) = 2
61	59, 60	eqtrdi 2796	. . . . 5 ⊢ (𝑁 = 5 → ((𝑁 − 1) / 2) = 2)
62	24, 55, 61	3brtr4d 5198	. . . 4 ⊢ (𝑁 = 5 → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
63		eluz2 12909	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘6) ↔ (6 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 6 ≤ 𝑁))
64		zre 12643	. . . . . . . . . . 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 12122	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → (𝑁 / 4) ∈ ℝ)
69		flle 13850	. . . . . . . . . . 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 13849	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℝ → (⌊‘(𝑁 / 4)) ∈ ℤ)
73	72	zred 12747	. . . . . . . . . . . . 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 11758	. . . . . . . . . 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 12548	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℝ ∧ 6 ≤ 𝑁) → ((𝑁 / 4) + 1) ≤ ((𝑁 − 1) / 2))
82	64, 81	sylan 579	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → ((𝑁 / 4) + 1) ≤ ((𝑁 − 1) / 2))
83		peano2re 11463	. . . . . . . . . . . . 13 ⊢ ((⌊‘(𝑁 / 4)) ∈ ℝ → ((⌊‘(𝑁 / 4)) + 1) ∈ ℝ)
84	73, 83	syl 17	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → ((⌊‘(𝑁 / 4)) + 1) ∈ ℝ)
85		peano2re 11463	. . . . . . . . . . . . 13 ⊢ ((𝑁 / 4) ∈ ℝ → ((𝑁 / 4) + 1) ∈ ℝ)
86	68, 85	syl 17	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → ((𝑁 / 4) + 1) ∈ ℝ)
87		peano2rem 11603	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℝ → (𝑁 − 1) ∈ ℝ)
88	87	rehalfcld 12540	. . . . . . . . . . . 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 11384	. . . . . . . . 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 698	. . . . . . 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 12440	. . . . . 6 ⊢ (5 + 1) = 6
98	97	fveq2i 6923	. . . . 5 ⊢ (ℤ_≥‘(5 + 1)) = (ℤ_≥‘6)
99	96, 98	eleq2s 2862	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘(5 + 1)) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
100	62, 99	jaoi 856	. . 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 856	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 846 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 ℕcn 12293 2c2 12348 3c3 12349 4c4 12350 5c5 12351 6c6 12352 ℕ₀cn0 12553 ℤcz 12639 ℤ_≥cuz 12903 ⌊cfl 13841

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-inf 9512 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-fl 13843

This theorem is referenced by: gausslemma2dlem0f 27423

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