fldiv4p1lem1div2 - Metamath Proof Explorer

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Theorem fldiv4p1lem1div2 13794

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 11778	. . . 4 ⊢ 1 ≤ 1
2	1	a1i 11	. . 3 ⊢ (𝑁 = 3 → 1 ≤ 1)
3		fvoveq1 7390	. . . . . 6 ⊢ (𝑁 = 3 → (⌊‘(𝑁 / 4)) = (⌊‘(3 / 4)))
4		3lt4 12350	. . . . . . 7 ⊢ 3 < 4
5		3nn0 12455	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
6		4nn 12264	. . . . . . . 8 ⊢ 4 ∈ ℕ
7		divfl0 13783	. . . . . . . 8 ⊢ ((3 ∈ ℕ₀ ∧ 4 ∈ ℕ) → (3 < 4 ↔ (⌊‘(3 / 4)) = 0))
8	5, 6, 7	mp2an 693	. . . . . . 7 ⊢ (3 < 4 ↔ (⌊‘(3 / 4)) = 0)
9	4, 8	mpbi 230	. . . . . 6 ⊢ (⌊‘(3 / 4)) = 0
10	3, 9	eqtrdi 2787	. . . . 5 ⊢ (𝑁 = 3 → (⌊‘(𝑁 / 4)) = 0)
11	10	oveq1d 7382	. . . 4 ⊢ (𝑁 = 3 → ((⌊‘(𝑁 / 4)) + 1) = (0 + 1))
12		0p1e1 12298	. . . 4 ⊢ (0 + 1) = 1
13	11, 12	eqtrdi 2787	. . 3 ⊢ (𝑁 = 3 → ((⌊‘(𝑁 / 4)) + 1) = 1)
14		oveq1 7374	. . . . . 6 ⊢ (𝑁 = 3 → (𝑁 − 1) = (3 − 1))
15		3m1e2 12304	. . . . . 6 ⊢ (3 − 1) = 2
16	14, 15	eqtrdi 2787	. . . . 5 ⊢ (𝑁 = 3 → (𝑁 − 1) = 2)
17	16	oveq1d 7382	. . . 4 ⊢ (𝑁 = 3 → ((𝑁 − 1) / 2) = (2 / 2))
18		2div2e1 12317	. . . 4 ⊢ (2 / 2) = 1
19	17, 18	eqtrdi 2787	. . 3 ⊢ (𝑁 = 3 → ((𝑁 − 1) / 2) = 1)
20	2, 13, 19	3brtr4d 5117	. 2 ⊢ (𝑁 = 3 → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
21		uzp1 12825	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘5) → (𝑁 = 5 ∨ 𝑁 ∈ (ℤ_≥‘(5 + 1))))
22		2re 12255	. . . . . . 7 ⊢ 2 ∈ ℝ
23	22	leidi 11684	. . . . . 6 ⊢ 2 ≤ 2
24	23	a1i 11	. . . . 5 ⊢ (𝑁 = 5 → 2 ≤ 2)
25		fvoveq1 7390	. . . . . . . 8 ⊢ (𝑁 = 5 → (⌊‘(𝑁 / 4)) = (⌊‘(5 / 4)))
26		df-5 12247	. . . . . . . . . . . 12 ⊢ 5 = (4 + 1)
27	26	oveq1i 7377	. . . . . . . . . . 11 ⊢ (5 / 4) = ((4 + 1) / 4)
28		4cn 12266	. . . . . . . . . . . 12 ⊢ 4 ∈ ℂ
29		ax-1cn 11096	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
30		4ne0 12289	. . . . . . . . . . . 12 ⊢ 4 ≠ 0
31	28, 29, 28, 30	divdiri 11912	. . . . . . . . . . 11 ⊢ ((4 + 1) / 4) = ((4 / 4) + (1 / 4))
32	28, 30	dividi 11888	. . . . . . . . . . . 12 ⊢ (4 / 4) = 1
33	32	oveq1i 7377	. . . . . . . . . . 11 ⊢ ((4 / 4) + (1 / 4)) = (1 + (1 / 4))
34	27, 31, 33	3eqtri 2763	. . . . . . . . . 10 ⊢ (5 / 4) = (1 + (1 / 4))
35	34	fveq2i 6843	. . . . . . . . 9 ⊢ (⌊‘(5 / 4)) = (⌊‘(1 + (1 / 4)))
36		1re 11144	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
37		0le1 11673	. . . . . . . . . . 11 ⊢ 0 ≤ 1
38		4re 12265	. . . . . . . . . . 11 ⊢ 4 ∈ ℝ
39		4pos 12288	. . . . . . . . . . 11 ⊢ 0 < 4
40		divge0 12025	. . . . . . . . . . 11 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ (4 ∈ ℝ ∧ 0 < 4)) → 0 ≤ (1 / 4))
41	36, 37, 38, 39, 40	mp4an 694	. . . . . . . . . 10 ⊢ 0 ≤ (1 / 4)
42		1lt4 12352	. . . . . . . . . . 11 ⊢ 1 < 4
43		recgt1 12052	. . . . . . . . . . . 12 ⊢ ((4 ∈ ℝ ∧ 0 < 4) → (1 < 4 ↔ (1 / 4) < 1))
44	38, 39, 43	mp2an 693	. . . . . . . . . . 11 ⊢ (1 < 4 ↔ (1 / 4) < 1)
45	42, 44	mpbi 230	. . . . . . . . . 10 ⊢ (1 / 4) < 1
46		1z 12557	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ
47	38, 30	rereccli 11920	. . . . . . . . . . 11 ⊢ (1 / 4) ∈ ℝ
48		flbi2 13776	. . . . . . . . . . 11 ⊢ ((1 ∈ ℤ ∧ (1 / 4) ∈ ℝ) → ((⌊‘(1 + (1 / 4))) = 1 ↔ (0 ≤ (1 / 4) ∧ (1 / 4) < 1)))
49	46, 47, 48	mp2an 693	. . . . . . . . . 10 ⊢ ((⌊‘(1 + (1 / 4))) = 1 ↔ (0 ≤ (1 / 4) ∧ (1 / 4) < 1))
50	41, 45, 49	mpbir2an 712	. . . . . . . . 9 ⊢ (⌊‘(1 + (1 / 4))) = 1
51	35, 50	eqtri 2759	. . . . . . . 8 ⊢ (⌊‘(5 / 4)) = 1
52	25, 51	eqtrdi 2787	. . . . . . 7 ⊢ (𝑁 = 5 → (⌊‘(𝑁 / 4)) = 1)
53	52	oveq1d 7382	. . . . . 6 ⊢ (𝑁 = 5 → ((⌊‘(𝑁 / 4)) + 1) = (1 + 1))
54		1p1e2 12301	. . . . . 6 ⊢ (1 + 1) = 2
55	53, 54	eqtrdi 2787	. . . . 5 ⊢ (𝑁 = 5 → ((⌊‘(𝑁 / 4)) + 1) = 2)
56		oveq1 7374	. . . . . . . 8 ⊢ (𝑁 = 5 → (𝑁 − 1) = (5 − 1))
57		5m1e4 12306	. . . . . . . 8 ⊢ (5 − 1) = 4
58	56, 57	eqtrdi 2787	. . . . . . 7 ⊢ (𝑁 = 5 → (𝑁 − 1) = 4)
59	58	oveq1d 7382	. . . . . 6 ⊢ (𝑁 = 5 → ((𝑁 − 1) / 2) = (4 / 2))
60		4div2e2 12346	. . . . . 6 ⊢ (4 / 2) = 2
61	59, 60	eqtrdi 2787	. . . . 5 ⊢ (𝑁 = 5 → ((𝑁 − 1) / 2) = 2)
62	24, 55, 61	3brtr4d 5117	. . . 4 ⊢ (𝑁 = 5 → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
63		eluz2 12794	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘6) ↔ (6 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 6 ≤ 𝑁))
64		zre 12528	. . . . . . . . . . 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 11983	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → (𝑁 / 4) ∈ ℝ)
69		flle 13758	. . . . . . . . . . 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 13757	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℝ → (⌊‘(𝑁 / 4)) ∈ ℤ)
73	72	zred 12633	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℝ → (⌊‘(𝑁 / 4)) ∈ ℝ)
74	36	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℝ → 1 ∈ ℝ)
75	73, 68, 74	3jca 1129	. . . . . . . . . . . 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 11618	. . . . . . . . . 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 12432	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℝ ∧ 6 ≤ 𝑁) → ((𝑁 / 4) + 1) ≤ ((𝑁 − 1) / 2))
82	64, 81	sylan 581	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → ((𝑁 / 4) + 1) ≤ ((𝑁 − 1) / 2))
83		peano2re 11319	. . . . . . . . . . . . 13 ⊢ ((⌊‘(𝑁 / 4)) ∈ ℝ → ((⌊‘(𝑁 / 4)) + 1) ∈ ℝ)
84	73, 83	syl 17	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → ((⌊‘(𝑁 / 4)) + 1) ∈ ℝ)
85		peano2re 11319	. . . . . . . . . . . . 13 ⊢ ((𝑁 / 4) ∈ ℝ → ((𝑁 / 4) + 1) ∈ ℝ)
86	68, 85	syl 17	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → ((𝑁 / 4) + 1) ∈ ℝ)
87		peano2rem 11461	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℝ → (𝑁 − 1) ∈ ℝ)
88	87	rehalfcld 12424	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → ((𝑁 − 1) / 2) ∈ ℝ)
89	84, 86, 88	3jca 1129	. . . . . . . . . . 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 11240	. . . . . . . . 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 700	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
95	94	3adant1 1131	. . . . . 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 12323	. . . . . 6 ⊢ (5 + 1) = 6
98	97	fveq2i 6843	. . . . 5 ⊢ (ℤ_≥‘(5 + 1)) = (ℤ_≥‘6)
99	96, 98	eleq2s 2854	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘(5 + 1)) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
100	62, 99	jaoi 858	. . 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 858	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 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 class class class wbr 5085 ‘cfv 6498 (class class class)co 7367 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 < clt 11179 ≤ cle 11180 − cmin 11377 / cdiv 11807 ℕcn 12174 2c2 12236 3c3 12237 4c4 12238 5c5 12239 6c6 12240 ℕ₀cn0 12437 ℤcz 12524 ℤ_≥cuz 12788 ⌊cfl 13749

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-sup 9355 df-inf 9356 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-n0 12438 df-z 12525 df-uz 12789 df-rp 12943 df-fl 13751

This theorem is referenced by: gausslemma2dlem0f 27324

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