monoords - Mathbox for Glauco Siliprandi

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Theorem monoords 45541

Description: Ordering relation for a strictly monotonic sequence, increasing case. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
monoords.fk	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)
monoords.flt	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝐹‘𝑘) < (𝐹‘(𝑘 + 1)))
monoords.i	⊢ (𝜑 → 𝐼 ∈ (𝑀...𝑁))
monoords.j	⊢ (𝜑 → 𝐽 ∈ (𝑀...𝑁))
monoords.iltj	⊢ (𝜑 → 𝐼 < 𝐽)

**Assertion**
Ref	Expression
monoords	⊢ (𝜑 → (𝐹‘𝐼) < (𝐹‘𝐽))

Distinct variable groups: 𝑘,𝐹 𝑘,𝐼 𝑘,𝐽 𝑘,𝑀 𝑘,𝑁 𝜑,𝑘

Proof of Theorem monoords Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		monoords.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ (𝑀...𝑁))
2	1	ancli 548	. . 3 ⊢ (𝜑 → (𝜑 ∧ 𝐼 ∈ (𝑀...𝑁)))
3		eleq1 2824	. . . . . 6 ⊢ (𝑘 = 𝐼 → (𝑘 ∈ (𝑀...𝑁) ↔ 𝐼 ∈ (𝑀...𝑁)))
4	3	anbi2d 630	. . . . 5 ⊢ (𝑘 = 𝐼 → ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ↔ (𝜑 ∧ 𝐼 ∈ (𝑀...𝑁))))
5		fveq2 6834	. . . . . 6 ⊢ (𝑘 = 𝐼 → (𝐹‘𝑘) = (𝐹‘𝐼))
6	5	eleq1d 2821	. . . . 5 ⊢ (𝑘 = 𝐼 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝐼) ∈ ℝ))
7	4, 6	imbi12d 344	. . . 4 ⊢ (𝑘 = 𝐼 → (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ) ↔ ((𝜑 ∧ 𝐼 ∈ (𝑀...𝑁)) → (𝐹‘𝐼) ∈ ℝ)))
8		monoords.fk	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ)
9	7, 8	vtoclg 3511	. . 3 ⊢ (𝐼 ∈ (𝑀...𝑁) → ((𝜑 ∧ 𝐼 ∈ (𝑀...𝑁)) → (𝐹‘𝐼) ∈ ℝ))
10	1, 2, 9	sylc 65	. 2 ⊢ (𝜑 → (𝐹‘𝐼) ∈ ℝ)
11		elfzel1 13439	. . . . . . 7 ⊢ (𝐼 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
12	1, 11	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
13	1	elfzelzd 13441	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ ℤ)
14		elfzle1 13443	. . . . . . 7 ⊢ (𝐼 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝐼)
15	1, 14	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ≤ 𝐼)
16		eluz2 12757	. . . . . 6 ⊢ (𝐼 ∈ (ℤ_≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼))
17	12, 13, 15, 16	syl3anbrc 1344	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (ℤ_≥‘𝑀))
18		elfzuz2 13445	. . . . . . 7 ⊢ (𝐼 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑀))
19	1, 18	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
20		eluzelz 12761	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
21	19, 20	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
22	13	zred 12596	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ ℝ)
23		monoords.j	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (𝑀...𝑁))
24	23	elfzelzd 13441	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℤ)
25	24	zred 12596	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ ℝ)
26	21	zred 12596	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
27		monoords.iltj	. . . . . 6 ⊢ (𝜑 → 𝐼 < 𝐽)
28		elfzle2 13444	. . . . . . 7 ⊢ (𝐽 ∈ (𝑀...𝑁) → 𝐽 ≤ 𝑁)
29	23, 28	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐽 ≤ 𝑁)
30	22, 25, 26, 27, 29	ltletrd 11293	. . . . 5 ⊢ (𝜑 → 𝐼 < 𝑁)
31		elfzo2 13578	. . . . 5 ⊢ (𝐼 ∈ (𝑀..^𝑁) ↔ (𝐼 ∈ (ℤ_≥‘𝑀) ∧ 𝑁 ∈ ℤ ∧ 𝐼 < 𝑁))
32	17, 21, 30, 31	syl3anbrc 1344	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (𝑀..^𝑁))
33		fzofzp1 13680	. . . 4 ⊢ (𝐼 ∈ (𝑀..^𝑁) → (𝐼 + 1) ∈ (𝑀...𝑁))
34	32, 33	syl 17	. . 3 ⊢ (𝜑 → (𝐼 + 1) ∈ (𝑀...𝑁))
35	34	ancli 548	. . 3 ⊢ (𝜑 → (𝜑 ∧ (𝐼 + 1) ∈ (𝑀...𝑁)))
36		eleq1 2824	. . . . . 6 ⊢ (𝑘 = (𝐼 + 1) → (𝑘 ∈ (𝑀...𝑁) ↔ (𝐼 + 1) ∈ (𝑀...𝑁)))
37	36	anbi2d 630	. . . . 5 ⊢ (𝑘 = (𝐼 + 1) → ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ↔ (𝜑 ∧ (𝐼 + 1) ∈ (𝑀...𝑁))))
38		fveq2 6834	. . . . . 6 ⊢ (𝑘 = (𝐼 + 1) → (𝐹‘𝑘) = (𝐹‘(𝐼 + 1)))
39	38	eleq1d 2821	. . . . 5 ⊢ (𝑘 = (𝐼 + 1) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(𝐼 + 1)) ∈ ℝ))
40	37, 39	imbi12d 344	. . . 4 ⊢ (𝑘 = (𝐼 + 1) → (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ) ↔ ((𝜑 ∧ (𝐼 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝐼 + 1)) ∈ ℝ)))
41	40, 8	vtoclg 3511	. . 3 ⊢ ((𝐼 + 1) ∈ (𝑀...𝑁) → ((𝜑 ∧ (𝐼 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝐼 + 1)) ∈ ℝ))
42	34, 35, 41	sylc 65	. 2 ⊢ (𝜑 → (𝐹‘(𝐼 + 1)) ∈ ℝ)
43	23	ancli 548	. . 3 ⊢ (𝜑 → (𝜑 ∧ 𝐽 ∈ (𝑀...𝑁)))
44		eleq1 2824	. . . . . 6 ⊢ (𝑘 = 𝐽 → (𝑘 ∈ (𝑀...𝑁) ↔ 𝐽 ∈ (𝑀...𝑁)))
45	44	anbi2d 630	. . . . 5 ⊢ (𝑘 = 𝐽 → ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ↔ (𝜑 ∧ 𝐽 ∈ (𝑀...𝑁))))
46		fveq2 6834	. . . . . 6 ⊢ (𝑘 = 𝐽 → (𝐹‘𝑘) = (𝐹‘𝐽))
47	46	eleq1d 2821	. . . . 5 ⊢ (𝑘 = 𝐽 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝐽) ∈ ℝ))
48	45, 47	imbi12d 344	. . . 4 ⊢ (𝑘 = 𝐽 → (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ) ↔ ((𝜑 ∧ 𝐽 ∈ (𝑀...𝑁)) → (𝐹‘𝐽) ∈ ℝ)))
49	48, 8	vtoclg 3511	. . 3 ⊢ (𝐽 ∈ (𝑀...𝑁) → ((𝜑 ∧ 𝐽 ∈ (𝑀...𝑁)) → (𝐹‘𝐽) ∈ ℝ))
50	23, 43, 49	sylc 65	. 2 ⊢ (𝜑 → (𝐹‘𝐽) ∈ ℝ)
51	32	ancli 548	. . 3 ⊢ (𝜑 → (𝜑 ∧ 𝐼 ∈ (𝑀..^𝑁)))
52		eleq1 2824	. . . . . 6 ⊢ (𝑘 = 𝐼 → (𝑘 ∈ (𝑀..^𝑁) ↔ 𝐼 ∈ (𝑀..^𝑁)))
53	52	anbi2d 630	. . . . 5 ⊢ (𝑘 = 𝐼 → ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ↔ (𝜑 ∧ 𝐼 ∈ (𝑀..^𝑁))))
54		fvoveq1 7381	. . . . . 6 ⊢ (𝑘 = 𝐼 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝐼 + 1)))
55	5, 54	breq12d 5111	. . . . 5 ⊢ (𝑘 = 𝐼 → ((𝐹‘𝑘) < (𝐹‘(𝑘 + 1)) ↔ (𝐹‘𝐼) < (𝐹‘(𝐼 + 1))))
56	53, 55	imbi12d 344	. . . 4 ⊢ (𝑘 = 𝐼 → (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝐹‘𝑘) < (𝐹‘(𝑘 + 1))) ↔ ((𝜑 ∧ 𝐼 ∈ (𝑀..^𝑁)) → (𝐹‘𝐼) < (𝐹‘(𝐼 + 1)))))
57		monoords.flt	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝐹‘𝑘) < (𝐹‘(𝑘 + 1)))
58	56, 57	vtoclg 3511	. . 3 ⊢ (𝐼 ∈ (𝑀..^𝑁) → ((𝜑 ∧ 𝐼 ∈ (𝑀..^𝑁)) → (𝐹‘𝐼) < (𝐹‘(𝐼 + 1))))
59	32, 51, 58	sylc 65	. 2 ⊢ (𝜑 → (𝐹‘𝐼) < (𝐹‘(𝐼 + 1)))
60	13	peano2zd 12599	. . . 4 ⊢ (𝜑 → (𝐼 + 1) ∈ ℤ)
61		zltp1le 12541	. . . . . 6 ⊢ ((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) → (𝐼 < 𝐽 ↔ (𝐼 + 1) ≤ 𝐽))
62	13, 24, 61	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐼 < 𝐽 ↔ (𝐼 + 1) ≤ 𝐽))
63	27, 62	mpbid 232	. . . 4 ⊢ (𝜑 → (𝐼 + 1) ≤ 𝐽)
64		eluz2 12757	. . . 4 ⊢ (𝐽 ∈ (ℤ_≥‘(𝐼 + 1)) ↔ ((𝐼 + 1) ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐼 + 1) ≤ 𝐽))
65	60, 24, 63, 64	syl3anbrc 1344	. . 3 ⊢ (𝜑 → 𝐽 ∈ (ℤ_≥‘(𝐼 + 1)))
66	12	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑀 ∈ ℤ)
67	21	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑁 ∈ ℤ)
68		elfzelz 13440	. . . . . 6 ⊢ (𝑘 ∈ ((𝐼 + 1)...𝐽) → 𝑘 ∈ ℤ)
69	68	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑘 ∈ ℤ)
70	66	zred 12596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑀 ∈ ℝ)
71	69	zred 12596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑘 ∈ ℝ)
72	60	zred 12596	. . . . . . . 8 ⊢ (𝜑 → (𝐼 + 1) ∈ ℝ)
73	72	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → (𝐼 + 1) ∈ ℝ)
74	22	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝐼 ∈ ℝ)
75	15	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑀 ≤ 𝐼)
76	74	ltp1d 12072	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝐼 < (𝐼 + 1))
77	70, 74, 73, 75, 76	lelttrd 11291	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑀 < (𝐼 + 1))
78		elfzle1 13443	. . . . . . . 8 ⊢ (𝑘 ∈ ((𝐼 + 1)...𝐽) → (𝐼 + 1) ≤ 𝑘)
79	78	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → (𝐼 + 1) ≤ 𝑘)
80	70, 73, 71, 77, 79	ltletrd 11293	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑀 < 𝑘)
81	70, 71, 80	ltled 11281	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑀 ≤ 𝑘)
82	25	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝐽 ∈ ℝ)
83	67	zred 12596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑁 ∈ ℝ)
84		elfzle2 13444	. . . . . . 7 ⊢ (𝑘 ∈ ((𝐼 + 1)...𝐽) → 𝑘 ≤ 𝐽)
85	84	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑘 ≤ 𝐽)
86	29	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝐽 ≤ 𝑁)
87	71, 82, 83, 85, 86	letrd 11290	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑘 ≤ 𝑁)
88	66, 67, 69, 81, 87	elfzd 13431	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑘 ∈ (𝑀...𝑁))
89	88, 8	syldan 591	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → (𝐹‘𝑘) ∈ ℝ)
90	12	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑀 ∈ ℤ)
91	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑁 ∈ ℤ)
92		elfzelz 13440	. . . . . . 7 ⊢ (𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1)) → 𝑘 ∈ ℤ)
93	92	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 ∈ ℤ)
94	90	zred 12596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑀 ∈ ℝ)
95	93	zred 12596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 ∈ ℝ)
96	72	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐼 + 1) ∈ ℝ)
97	12	zred 12596	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℝ)
98	22	ltp1d 12072	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 < (𝐼 + 1))
99	97, 22, 72, 15, 98	lelttrd 11291	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 < (𝐼 + 1))
100	99	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑀 < (𝐼 + 1))
101		elfzle1 13443	. . . . . . . . 9 ⊢ (𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1)) → (𝐼 + 1) ≤ 𝑘)
102	101	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐼 + 1) ≤ 𝑘)
103	94, 96, 95, 100, 102	ltletrd 11293	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑀 < 𝑘)
104	94, 95, 103	ltled 11281	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑀 ≤ 𝑘)
105	91	zred 12596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑁 ∈ ℝ)
106		peano2rem 11448	. . . . . . . . . 10 ⊢ (𝐽 ∈ ℝ → (𝐽 − 1) ∈ ℝ)
107	25, 106	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 − 1) ∈ ℝ)
108	107	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐽 − 1) ∈ ℝ)
109		elfzle2 13444	. . . . . . . . 9 ⊢ (𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1)) → 𝑘 ≤ (𝐽 − 1))
110	109	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 ≤ (𝐽 − 1))
111	25	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝐽 ∈ ℝ)
112	111	ltm1d 12074	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐽 − 1) < 𝐽)
113	29	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝐽 ≤ 𝑁)
114	108, 111, 105, 112, 113	ltletrd 11293	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐽 − 1) < 𝑁)
115	95, 108, 105, 110, 114	lelttrd 11291	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 < 𝑁)
116	95, 105, 115	ltled 11281	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 ≤ 𝑁)
117	90, 91, 93, 104, 116	elfzd 13431	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 ∈ (𝑀...𝑁))
118	117, 8	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐹‘𝑘) ∈ ℝ)
119		peano2zm 12534	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
120	91, 119	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝑁 − 1) ∈ ℤ)
121	120	zred 12596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝑁 − 1) ∈ ℝ)
122		1red 11133	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℝ)
123	25, 26, 122, 29	lesub1dd 11753	. . . . . . . 8 ⊢ (𝜑 → (𝐽 − 1) ≤ (𝑁 − 1))
124	123	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐽 − 1) ≤ (𝑁 − 1))
125	95, 108, 121, 110, 124	letrd 11290	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 ≤ (𝑁 − 1))
126	90, 120, 93, 104, 125	elfzd 13431	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 ∈ (𝑀...(𝑁 − 1)))
127		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → 𝑘 ∈ (𝑀...(𝑁 − 1)))
128		fzoval 13576	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1)))
129	21, 128	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀..^𝑁) = (𝑀...(𝑁 − 1)))
130	129	eqcomd 2742	. . . . . . . . 9 ⊢ (𝜑 → (𝑀...(𝑁 − 1)) = (𝑀..^𝑁))
131	130	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝑀...(𝑁 − 1)) = (𝑀..^𝑁))
132	127, 131	eleqtrd 2838	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → 𝑘 ∈ (𝑀..^𝑁))
133		fzofzp1 13680	. . . . . . 7 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
134	132, 133	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝑘 + 1) ∈ (𝑀...𝑁))
135		simpl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → 𝜑)
136	135, 134	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝜑 ∧ (𝑘 + 1) ∈ (𝑀...𝑁)))
137		eleq1 2824	. . . . . . . . 9 ⊢ (𝑗 = (𝑘 + 1) → (𝑗 ∈ (𝑀...𝑁) ↔ (𝑘 + 1) ∈ (𝑀...𝑁)))
138	137	anbi2d 630	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) ↔ (𝜑 ∧ (𝑘 + 1) ∈ (𝑀...𝑁))))
139		fveq2 6834	. . . . . . . . 9 ⊢ (𝑗 = (𝑘 + 1) → (𝐹‘𝑗) = (𝐹‘(𝑘 + 1)))
140	139	eleq1d 2821	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → ((𝐹‘𝑗) ∈ ℝ ↔ (𝐹‘(𝑘 + 1)) ∈ ℝ))
141	138, 140	imbi12d 344	. . . . . . 7 ⊢ (𝑗 = (𝑘 + 1) → (((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ∈ ℝ) ↔ ((𝜑 ∧ (𝑘 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝑘 + 1)) ∈ ℝ)))
142		eleq1 2824	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ (𝑀...𝑁) ↔ 𝑗 ∈ (𝑀...𝑁)))
143	142	anbi2d 630	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ↔ (𝜑 ∧ 𝑗 ∈ (𝑀...𝑁))))
144		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
145	144	eleq1d 2821	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑗) ∈ ℝ))
146	143, 145	imbi12d 344	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ) ↔ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ∈ ℝ)))
147	146, 8	chvarvv 1990	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ∈ ℝ)
148	141, 147	vtoclg 3511	. . . . . 6 ⊢ ((𝑘 + 1) ∈ (𝑀...𝑁) → ((𝜑 ∧ (𝑘 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝑘 + 1)) ∈ ℝ))
149	134, 136, 148	sylc 65	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ∈ ℝ)
150	126, 149	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐹‘(𝑘 + 1)) ∈ ℝ)
151	132, 57	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) < (𝐹‘(𝑘 + 1)))
152	126, 151	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐹‘𝑘) < (𝐹‘(𝑘 + 1)))
153	118, 150, 152	ltled 11281	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
154	65, 89, 153	monoord 13955	. 2 ⊢ (𝜑 → (𝐹‘(𝐼 + 1)) ≤ (𝐹‘𝐽))
155	10, 42, 50, 59, 154	ltletrd 11293	1 ⊢ (𝜑 → (𝐹‘𝐼) < (𝐹‘𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 class class class wbr 5098 ‘cfv 6492 (class class class)co 7358 ℝcr 11025 1c1 11027 + caddc 11029 < clt 11166 ≤ cle 11167 − cmin 11364 ℤcz 12488 ℤ_≥cuz 12751 ...cfz 13423 ..^cfzo 13570

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 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 3061 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-n0 12402 df-z 12489 df-uz 12752 df-fz 13424 df-fzo 13571

This theorem is referenced by: fourierdlem34 46381

W3C validator