fldiv4p1lem1div2 - Intuitionistic Logic Explorer

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Theorem fldiv4p1lem1div2 10414

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 8618	. . . 4 ⊢ 1 ≤ 1
2	1	a1i 9	. . 3 ⊢ (𝑁 = 3 → 1 ≤ 1)
3		oveq1 5932	. . . . . . 7 ⊢ (𝑁 = 3 → (𝑁 / 4) = (3 / 4))
4	3	fveq2d 5565	. . . . . 6 ⊢ (𝑁 = 3 → (⌊‘(𝑁 / 4)) = (⌊‘(3 / 4)))
5		3lt4 9182	. . . . . . 7 ⊢ 3 < 4
6		3nn0 9286	. . . . . . . 8 ⊢ 3 ∈ ℕ₀
7		4nn 9173	. . . . . . . 8 ⊢ 4 ∈ ℕ
8		divfl0 10405	. . . . . . . 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 2245	. . . . 5 ⊢ (𝑁 = 3 → (⌊‘(𝑁 / 4)) = 0)
12	11	oveq1d 5940	. . . 4 ⊢ (𝑁 = 3 → ((⌊‘(𝑁 / 4)) + 1) = (0 + 1))
13		0p1e1 9123	. . . 4 ⊢ (0 + 1) = 1
14	12, 13	eqtrdi 2245	. . 3 ⊢ (𝑁 = 3 → ((⌊‘(𝑁 / 4)) + 1) = 1)
15		oveq1 5932	. . . . . 6 ⊢ (𝑁 = 3 → (𝑁 − 1) = (3 − 1))
16		3m1e2 9129	. . . . . 6 ⊢ (3 − 1) = 2
17	15, 16	eqtrdi 2245	. . . . 5 ⊢ (𝑁 = 3 → (𝑁 − 1) = 2)
18	17	oveq1d 5940	. . . 4 ⊢ (𝑁 = 3 → ((𝑁 − 1) / 2) = (2 / 2))
19		2div2e1 9142	. . . 4 ⊢ (2 / 2) = 1
20	18, 19	eqtrdi 2245	. . 3 ⊢ (𝑁 = 3 → ((𝑁 − 1) / 2) = 1)
21	2, 14, 20	3brtr4d 4066	. 2 ⊢ (𝑁 = 3 → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
22		uzp1 9654	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘5) → (𝑁 = 5 ∨ 𝑁 ∈ (ℤ_≥‘(5 + 1))))
23		2re 9079	. . . . . . 7 ⊢ 2 ∈ ℝ
24	23	leidi 8531	. . . . . 6 ⊢ 2 ≤ 2
25	24	a1i 9	. . . . 5 ⊢ (𝑁 = 5 → 2 ≤ 2)
26		oveq1 5932	. . . . . . . . 9 ⊢ (𝑁 = 5 → (𝑁 / 4) = (5 / 4))
27	26	fveq2d 5565	. . . . . . . 8 ⊢ (𝑁 = 5 → (⌊‘(𝑁 / 4)) = (⌊‘(5 / 4)))
28		df-5 9071	. . . . . . . . . . . 12 ⊢ 5 = (4 + 1)
29	28	oveq1i 5935	. . . . . . . . . . 11 ⊢ (5 / 4) = ((4 + 1) / 4)
30		4cn 9087	. . . . . . . . . . . . 13 ⊢ 4 ∈ ℂ
31		ax-1cn 7991	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
32		4ap0 9108	. . . . . . . . . . . . 13 ⊢ 4 # 0
33	30, 31, 30, 32	divdirapi 8815	. . . . . . . . . . . 12 ⊢ ((4 + 1) / 4) = ((4 / 4) + (1 / 4))
34	30, 32	dividapi 8791	. . . . . . . . . . . . 13 ⊢ (4 / 4) = 1
35	34	oveq1i 5935	. . . . . . . . . . . 12 ⊢ ((4 / 4) + (1 / 4)) = (1 + (1 / 4))
36	33, 35	eqtri 2217	. . . . . . . . . . 11 ⊢ ((4 + 1) / 4) = (1 + (1 / 4))
37	29, 36	eqtri 2217	. . . . . . . . . 10 ⊢ (5 / 4) = (1 + (1 / 4))
38	37	fveq2i 5564	. . . . . . . . 9 ⊢ (⌊‘(5 / 4)) = (⌊‘(1 + (1 / 4)))
39		1re 8044	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
40		0le1 8527	. . . . . . . . . . 11 ⊢ 0 ≤ 1
41		4re 9086	. . . . . . . . . . 11 ⊢ 4 ∈ ℝ
42		4pos 9106	. . . . . . . . . . 11 ⊢ 0 < 4
43		divge0 8919	. . . . . . . . . . 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 9184	. . . . . . . . . . 11 ⊢ 1 < 4
46		recgt1 8943	. . . . . . . . . . . 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 9371	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ
50		znq 9717	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ 4 ∈ ℕ) → (1 / 4) ∈ ℚ)
51	49, 7, 50	mp2an 426	. . . . . . . . . . 11 ⊢ (1 / 4) ∈ ℚ
52		flqbi2 10400	. . . . . . . . . . 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 944	. . . . . . . . 9 ⊢ (⌊‘(1 + (1 / 4))) = 1
55	38, 54	eqtri 2217	. . . . . . . 8 ⊢ (⌊‘(5 / 4)) = 1
56	27, 55	eqtrdi 2245	. . . . . . 7 ⊢ (𝑁 = 5 → (⌊‘(𝑁 / 4)) = 1)
57	56	oveq1d 5940	. . . . . 6 ⊢ (𝑁 = 5 → ((⌊‘(𝑁 / 4)) + 1) = (1 + 1))
58		1p1e2 9126	. . . . . 6 ⊢ (1 + 1) = 2
59	57, 58	eqtrdi 2245	. . . . 5 ⊢ (𝑁 = 5 → ((⌊‘(𝑁 / 4)) + 1) = 2)
60		oveq1 5932	. . . . . . . 8 ⊢ (𝑁 = 5 → (𝑁 − 1) = (5 − 1))
61	30, 31, 28	mvrraddi 8262	. . . . . . . 8 ⊢ (5 − 1) = 4
62	60, 61	eqtrdi 2245	. . . . . . 7 ⊢ (𝑁 = 5 → (𝑁 − 1) = 4)
63	62	oveq1d 5940	. . . . . 6 ⊢ (𝑁 = 5 → ((𝑁 − 1) / 2) = (4 / 2))
64		4d2e2 9170	. . . . . 6 ⊢ (4 / 2) = 2
65	63, 64	eqtrdi 2245	. . . . 5 ⊢ (𝑁 = 5 → ((𝑁 − 1) / 2) = 2)
66	25, 59, 65	3brtr4d 4066	. . . 4 ⊢ (𝑁 = 5 → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
67		eluz2 9626	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘6) ↔ (6 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 6 ≤ 𝑁))
68		znq 9717	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 4 ∈ ℕ) → (𝑁 / 4) ∈ ℚ)
69	7, 68	mpan2 425	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (𝑁 / 4) ∈ ℚ)
70		flqle 10387	. . . . . . . . . . 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 10386	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℤ → (⌊‘(𝑁 / 4)) ∈ ℤ)
74	73	zred 9467	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℤ → (⌊‘(𝑁 / 4)) ∈ ℝ)
75		zre 9349	. . . . . . . . . . . . 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 8881	. . . . . . . . . . . . 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 1179	. . . . . . . . . . 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 8476	. . . . . . . . . 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 9264	. . . . . . . . 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 8181	. . . . . . . . . . . 12 ⊢ ((⌊‘(𝑁 / 4)) ∈ ℝ → ((⌊‘(𝑁 / 4)) + 1) ∈ ℝ)
90	74, 89	syl 14	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → ((⌊‘(𝑁 / 4)) + 1) ∈ ℝ)
91		peano2re 8181	. . . . . . . . . . . 12 ⊢ ((𝑁 / 4) ∈ ℝ → ((𝑁 / 4) + 1) ∈ ℝ)
92	80, 91	syl 14	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → ((𝑁 / 4) + 1) ∈ ℝ)
93		peano2rem 8312	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℝ → (𝑁 − 1) ∈ ℝ)
94	93	rehalfcld 9257	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℝ → ((𝑁 − 1) / 2) ∈ ℝ)
95	75, 94	syl 14	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → ((𝑁 − 1) / 2) ∈ ℝ)
96	90, 92, 95	3jca 1179	. . . . . . . . . 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 8128	. . . . . . . . 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 1017	. . . . . 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 9147	. . . . . 6 ⊢ (5 + 1) = 6
104	103	fveq2i 5564	. . . . 5 ⊢ (ℤ_≥‘(5 + 1)) = (ℤ_≥‘6)
105	102, 104	eleq2s 2291	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘(5 + 1)) → ((⌊‘(𝑁 / 4)) + 1) ≤ ((𝑁 − 1) / 2))
106	66, 105	jaoi 717	. . 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 717	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 709 ∧ w3a 980 = wceq 1364 ∈ wcel 2167 class class class wbr 4034 ‘cfv 5259 (class class class)co 5925 ℝcr 7897 0cc0 7898 1c1 7899 + caddc 7901 < clt 8080 ≤ cle 8081 − cmin 8216 # cap 8627 / cdiv 8718 ℕcn 9009 2c2 9060 3c3 9061 4c4 9062 5c5 9063 6c6 9064 ℕ₀cn0 9268 ℤcz 9345 ℤ_≥cuz 9620 ℚcq 9712 ⌊cfl 10377

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 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-cnex 7989 ax-resscn 7990 ax-1cn 7991 ax-1re 7992 ax-icn 7993 ax-addcl 7994 ax-addrcl 7995 ax-mulcl 7996 ax-mulrcl 7997 ax-addcom 7998 ax-mulcom 7999 ax-addass 8000 ax-mulass 8001 ax-distr 8002 ax-i2m1 8003 ax-0lt1 8004 ax-1rid 8005 ax-0id 8006 ax-rnegex 8007 ax-precex 8008 ax-cnre 8009 ax-pre-ltirr 8010 ax-pre-ltwlin 8011 ax-pre-lttrn 8012 ax-pre-apti 8013 ax-pre-ltadd 8014 ax-pre-mulgt0 8015 ax-pre-mulext 8016 ax-arch 8017

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-id 4329 df-po 4332 df-iso 4333 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-pnf 8082 df-mnf 8083 df-xr 8084 df-ltxr 8085 df-le 8086 df-sub 8218 df-neg 8219 df-reap 8621 df-ap 8628 df-div 8719 df-inn 9010 df-2 9068 df-3 9069 df-4 9070 df-5 9071 df-6 9072 df-n0 9269 df-z 9346 df-uz 9621 df-q 9713 df-rp 9748 df-fl 10379

This theorem is referenced by: gausslemma2dlem0f 15403

W3C validator