fldiv4p1lem1div2 - Metamath Proof Explorer

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Theorem fldiv4p1lem1div2 13020

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 11069	. . . 4 ⊢ 1 ≤ 1
2	1	a1i 11	. . 3 ⊢ (𝑁 = 3 → 1 ≤ 1)
3		fvoveq1 6999	. . . . . 6 ⊢ (𝑁 = 3 → (⌊‘(𝑁 / 4)) = (⌊‘(3 / 4)))
4		3lt4 11621	. . . . . . 7 ⊢ 3 < 4
5		3nn0 11727	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
6		4nn 11524	. . . . . . . 8 ⊢ 4 ∈ ℕ
7		divfl0 13009	. . . . . . . 8 ⊢ ((3 ∈ ℕ₀ ∧ 4 ∈ ℕ) → (3 < 4 ↔ (⌊‘(3 / 4)) = 0))
8	5, 6, 7	mp2an 679	. . . . . . 7 ⊢ (3 < 4 ↔ (⌊‘(3 / 4)) = 0)
9	4, 8	mpbi 222	. . . . . 6 ⊢ (⌊‘(3 / 4)) = 0
10	3, 9	syl6eq 2831	. . . . 5 ⊢ (𝑁 = 3 → (⌊‘(𝑁 / 4)) = 0)
11	10	oveq1d 6991	. . . 4 ⊢ (𝑁 = 3 → ((⌊‘(𝑁 / 4)) + 1) = (0 + 1))
12		0p1e1 11569	. . . 4 ⊢ (0 + 1) = 1
13	11, 12	syl6eq 2831	. . 3 ⊢ (𝑁 = 3 → ((⌊‘(𝑁 / 4)) + 1) = 1)
14		oveq1 6983	. . . . . 6 ⊢ (𝑁 = 3 → (𝑁 − 1) = (3 − 1))
15		3m1e2 11575	. . . . . 6 ⊢ (3 − 1) = 2
16	14, 15	syl6eq 2831	. . . . 5 ⊢ (𝑁 = 3 → (𝑁 − 1) = 2)
17	16	oveq1d 6991	. . . 4 ⊢ (𝑁 = 3 → ((𝑁 − 1) / 2) = (2 / 2))
18		2div2e1 11588	. . . 4 ⊢ (2 / 2) = 1
19	17, 18	syl6eq 2831	. . 3 ⊢ (𝑁 = 3 → ((𝑁 − 1) / 2) = 1)
20	2, 13, 19	3brtr4d 4961	. 2 ⊢ (𝑁 = 3 → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
21		uzp1 12093	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘5) → (𝑁 = 5 ∨ 𝑁 ∈ (ℤ_≥‘(5 + 1))))
22		2re 11514	. . . . . . 7 ⊢ 2 ∈ ℝ
23	22	leidi 10975	. . . . . 6 ⊢ 2 ≤ 2
24	23	a1i 11	. . . . 5 ⊢ (𝑁 = 5 → 2 ≤ 2)
25		fvoveq1 6999	. . . . . . . 8 ⊢ (𝑁 = 5 → (⌊‘(𝑁 / 4)) = (⌊‘(5 / 4)))
26		df-5 11506	. . . . . . . . . . . 12 ⊢ 5 = (4 + 1)
27	26	oveq1i 6986	. . . . . . . . . . 11 ⊢ (5 / 4) = ((4 + 1) / 4)
28		4cn 11526	. . . . . . . . . . . 12 ⊢ 4 ∈ ℂ
29		ax-1cn 10393	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
30		4ne0 11555	. . . . . . . . . . . 12 ⊢ 4 ≠ 0
31	28, 29, 28, 30	divdiri 11198	. . . . . . . . . . 11 ⊢ ((4 + 1) / 4) = ((4 / 4) + (1 / 4))
32	28, 30	dividi 11174	. . . . . . . . . . . 12 ⊢ (4 / 4) = 1
33	32	oveq1i 6986	. . . . . . . . . . 11 ⊢ ((4 / 4) + (1 / 4)) = (1 + (1 / 4))
34	27, 31, 33	3eqtri 2807	. . . . . . . . . 10 ⊢ (5 / 4) = (1 + (1 / 4))
35	34	fveq2i 6502	. . . . . . . . 9 ⊢ (⌊‘(5 / 4)) = (⌊‘(1 + (1 / 4)))
36		1re 10439	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
37		0le1 10964	. . . . . . . . . . 11 ⊢ 0 ≤ 1
38		4re 11525	. . . . . . . . . . 11 ⊢ 4 ∈ ℝ
39		4pos 11554	. . . . . . . . . . 11 ⊢ 0 < 4
40		divge0 11310	. . . . . . . . . . 11 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ (4 ∈ ℝ ∧ 0 < 4)) → 0 ≤ (1 / 4))
41	36, 37, 38, 39, 40	mp4an 680	. . . . . . . . . 10 ⊢ 0 ≤ (1 / 4)
42		1lt4 11623	. . . . . . . . . . 11 ⊢ 1 < 4
43		recgt1 11337	. . . . . . . . . . . 12 ⊢ ((4 ∈ ℝ ∧ 0 < 4) → (1 < 4 ↔ (1 / 4) < 1))
44	38, 39, 43	mp2an 679	. . . . . . . . . . 11 ⊢ (1 < 4 ↔ (1 / 4) < 1)
45	42, 44	mpbi 222	. . . . . . . . . 10 ⊢ (1 / 4) < 1
46		1z 11825	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ
47	38, 30	rereccli 11206	. . . . . . . . . . 11 ⊢ (1 / 4) ∈ ℝ
48		flbi2 13002	. . . . . . . . . . 11 ⊢ ((1 ∈ ℤ ∧ (1 / 4) ∈ ℝ) → ((⌊‘(1 + (1 / 4))) = 1 ↔ (0 ≤ (1 / 4) ∧ (1 / 4) < 1)))
49	46, 47, 48	mp2an 679	. . . . . . . . . 10 ⊢ ((⌊‘(1 + (1 / 4))) = 1 ↔ (0 ≤ (1 / 4) ∧ (1 / 4) < 1))
50	41, 45, 49	mpbir2an 698	. . . . . . . . 9 ⊢ (⌊‘(1 + (1 / 4))) = 1
51	35, 50	eqtri 2803	. . . . . . . 8 ⊢ (⌊‘(5 / 4)) = 1
52	25, 51	syl6eq 2831	. . . . . . 7 ⊢ (𝑁 = 5 → (⌊‘(𝑁 / 4)) = 1)
53	52	oveq1d 6991	. . . . . 6 ⊢ (𝑁 = 5 → ((⌊‘(𝑁 / 4)) + 1) = (1 + 1))
54		1p1e2 11572	. . . . . 6 ⊢ (1 + 1) = 2
55	53, 54	syl6eq 2831	. . . . 5 ⊢ (𝑁 = 5 → ((⌊‘(𝑁 / 4)) + 1) = 2)
56		oveq1 6983	. . . . . . . 8 ⊢ (𝑁 = 5 → (𝑁 − 1) = (5 − 1))
57		5m1e4 11577	. . . . . . . 8 ⊢ (5 − 1) = 4
58	56, 57	syl6eq 2831	. . . . . . 7 ⊢ (𝑁 = 5 → (𝑁 − 1) = 4)
59	58	oveq1d 6991	. . . . . 6 ⊢ (𝑁 = 5 → ((𝑁 − 1) / 2) = (4 / 2))
60		4d2e2 11617	. . . . . 6 ⊢ (4 / 2) = 2
61	59, 60	syl6eq 2831	. . . . 5 ⊢ (𝑁 = 5 → ((𝑁 − 1) / 2) = 2)
62	24, 55, 61	3brtr4d 4961	. . . 4 ⊢ (𝑁 = 5 → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
63		eluz2 12064	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘6) ↔ (6 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 6 ≤ 𝑁))
64		zre 11797	. . . . . . . . . . 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 11269	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℝ → (𝑁 / 4) ∈ ℝ)
69		flle 12984	. . . . . . . . . . 11 ⊢ ((𝑁 / 4) ∈ ℝ → (⌊‘(𝑁 / 4)) ≤ (𝑁 / 4))
70	64, 68, 69	3syl 18	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (⌊‘(𝑁 / 4)) ≤ (𝑁 / 4))
71	70	adantr 473	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → (⌊‘(𝑁 / 4)) ≤ (𝑁 / 4))
72	68	flcld 12983	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℝ → (⌊‘(𝑁 / 4)) ∈ ℤ)
73	72	zred 11900	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℝ → (⌊‘(𝑁 / 4)) ∈ ℝ)
74	36	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℝ → 1 ∈ ℝ)
75	73, 68, 74	3jca 1108	. . . . . . . . . . . 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 473	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → ((⌊‘(𝑁 / 4)) ∈ ℝ ∧ (𝑁 / 4) ∈ ℝ ∧ 1 ∈ ℝ))
78		leadd1 10909	. . . . . . . . . 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 224	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 / 4) + 1))
81		div4p1lem1div2 11702	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℝ ∧ 6 ≤ 𝑁) → ((𝑁 / 4) + 1) ≤ ((𝑁 − 1) / 2))
82	64, 81	sylan 572	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → ((𝑁 / 4) + 1) ≤ ((𝑁 − 1) / 2))
83		peano2re 10613	. . . . . . . . . . . . 13 ⊢ ((⌊‘(𝑁 / 4)) ∈ ℝ → ((⌊‘(𝑁 / 4)) + 1) ∈ ℝ)
84	73, 83	syl 17	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → ((⌊‘(𝑁 / 4)) + 1) ∈ ℝ)
85		peano2re 10613	. . . . . . . . . . . . 13 ⊢ ((𝑁 / 4) ∈ ℝ → ((𝑁 / 4) + 1) ∈ ℝ)
86	68, 85	syl 17	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → ((𝑁 / 4) + 1) ∈ ℝ)
87		peano2rem 10754	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℝ → (𝑁 − 1) ∈ ℝ)
88	87	rehalfcld 11694	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → ((𝑁 − 1) / 2) ∈ ℝ)
89	84, 86, 88	3jca 1108	. . . . . . . . . . 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 473	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → (((⌊‘(𝑁 / 4)) + 1) ∈ ℝ ∧ ((𝑁 / 4) + 1) ∈ ℝ ∧ ((𝑁 − 1) / 2) ∈ ℝ))
92		letr 10534	. . . . . . . . 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 686	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
95	94	3adant1 1110	. . . . . 6 ⊢ ((6 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 6 ≤ 𝑁) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
96	63, 95	sylbi 209	. . . . 5 ⊢ (𝑁 ∈ (ℤ_≥‘6) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
97		5p1e6 11594	. . . . . 6 ⊢ (5 + 1) = 6
98	97	fveq2i 6502	. . . . 5 ⊢ (ℤ_≥‘(5 + 1)) = (ℤ_≥‘6)
99	96, 98	eleq2s 2885	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘(5 + 1)) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
100	62, 99	jaoi 843	. . 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 843	1 ⊢ ((𝑁 = 3 ∨ 𝑁 ∈ (ℤ_≥‘5)) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∨ wo 833 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ≠ wne 2968 class class class wbr 4929 ‘cfv 6188 (class class class)co 6976 ℝcr 10334 0cc0 10335 1c1 10336 + caddc 10338 < clt 10474 ≤ cle 10475 − cmin 10670 / cdiv 11098 ℕcn 11439 2c2 11495 3c3 11496 4c4 11497 5c5 11498 6c6 11499 ℕ₀cn0 11707 ℤcz 11793 ℤ_≥cuz 12058 ⌊cfl 12975

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 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413

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 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-sup 8701 df-inf 8702 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-n0 11708 df-z 11794 df-uz 12059 df-rp 12205 df-fl 12977

This theorem is referenced by: gausslemma2dlem0f 25639

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