monoords - Mathbox for Glauco Siliprandi

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Theorem monoords 44305

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 547	. . 3 ⊢ (𝜑 → (𝜑 ∧ 𝐼 ∈ (𝑀...𝑁)))
3		eleq1 2819	. . . . . 6 ⊢ (𝑘 = 𝐼 → (𝑘 ∈ (𝑀...𝑁) ↔ 𝐼 ∈ (𝑀...𝑁)))
4	3	anbi2d 627	. . . . 5 ⊢ (𝑘 = 𝐼 → ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ↔ (𝜑 ∧ 𝐼 ∈ (𝑀...𝑁))))
5		fveq2 6890	. . . . . 6 ⊢ (𝑘 = 𝐼 → (𝐹‘𝑘) = (𝐹‘𝐼))
6	5	eleq1d 2816	. . . . 5 ⊢ (𝑘 = 𝐼 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝐼) ∈ ℝ))
7	4, 6	imbi12d 343	. . . 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 3541	. . 3 ⊢ (𝐼 ∈ (𝑀...𝑁) → ((𝜑 ∧ 𝐼 ∈ (𝑀...𝑁)) → (𝐹‘𝐼) ∈ ℝ))
10	1, 2, 9	sylc 65	. 2 ⊢ (𝜑 → (𝐹‘𝐼) ∈ ℝ)
11		elfzel1 13504	. . . . . . 7 ⊢ (𝐼 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
12	1, 11	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
13	1	elfzelzd 13506	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ ℤ)
14		elfzle1 13508	. . . . . . 7 ⊢ (𝐼 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝐼)
15	1, 14	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ≤ 𝐼)
16		eluz2 12832	. . . . . 6 ⊢ (𝐼 ∈ (ℤ_≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼))
17	12, 13, 15, 16	syl3anbrc 1341	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (ℤ_≥‘𝑀))
18		elfzuz2 13510	. . . . . . 7 ⊢ (𝐼 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑀))
19	1, 18	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
20		eluzelz 12836	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
21	19, 20	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
22	13	zred 12670	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ ℝ)
23		monoords.j	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (𝑀...𝑁))
24	23	elfzelzd 13506	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℤ)
25	24	zred 12670	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ ℝ)
26	21	zred 12670	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
27		monoords.iltj	. . . . . 6 ⊢ (𝜑 → 𝐼 < 𝐽)
28		elfzle2 13509	. . . . . . 7 ⊢ (𝐽 ∈ (𝑀...𝑁) → 𝐽 ≤ 𝑁)
29	23, 28	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐽 ≤ 𝑁)
30	22, 25, 26, 27, 29	ltletrd 11378	. . . . 5 ⊢ (𝜑 → 𝐼 < 𝑁)
31		elfzo2 13639	. . . . 5 ⊢ (𝐼 ∈ (𝑀..^𝑁) ↔ (𝐼 ∈ (ℤ_≥‘𝑀) ∧ 𝑁 ∈ ℤ ∧ 𝐼 < 𝑁))
32	17, 21, 30, 31	syl3anbrc 1341	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (𝑀..^𝑁))
33		fzofzp1 13733	. . . 4 ⊢ (𝐼 ∈ (𝑀..^𝑁) → (𝐼 + 1) ∈ (𝑀...𝑁))
34	32, 33	syl 17	. . 3 ⊢ (𝜑 → (𝐼 + 1) ∈ (𝑀...𝑁))
35	34	ancli 547	. . 3 ⊢ (𝜑 → (𝜑 ∧ (𝐼 + 1) ∈ (𝑀...𝑁)))
36		eleq1 2819	. . . . . 6 ⊢ (𝑘 = (𝐼 + 1) → (𝑘 ∈ (𝑀...𝑁) ↔ (𝐼 + 1) ∈ (𝑀...𝑁)))
37	36	anbi2d 627	. . . . 5 ⊢ (𝑘 = (𝐼 + 1) → ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ↔ (𝜑 ∧ (𝐼 + 1) ∈ (𝑀...𝑁))))
38		fveq2 6890	. . . . . 6 ⊢ (𝑘 = (𝐼 + 1) → (𝐹‘𝑘) = (𝐹‘(𝐼 + 1)))
39	38	eleq1d 2816	. . . . 5 ⊢ (𝑘 = (𝐼 + 1) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(𝐼 + 1)) ∈ ℝ))
40	37, 39	imbi12d 343	. . . 4 ⊢ (𝑘 = (𝐼 + 1) → (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ) ↔ ((𝜑 ∧ (𝐼 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝐼 + 1)) ∈ ℝ)))
41	40, 8	vtoclg 3541	. . 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 547	. . 3 ⊢ (𝜑 → (𝜑 ∧ 𝐽 ∈ (𝑀...𝑁)))
44		eleq1 2819	. . . . . 6 ⊢ (𝑘 = 𝐽 → (𝑘 ∈ (𝑀...𝑁) ↔ 𝐽 ∈ (𝑀...𝑁)))
45	44	anbi2d 627	. . . . 5 ⊢ (𝑘 = 𝐽 → ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ↔ (𝜑 ∧ 𝐽 ∈ (𝑀...𝑁))))
46		fveq2 6890	. . . . . 6 ⊢ (𝑘 = 𝐽 → (𝐹‘𝑘) = (𝐹‘𝐽))
47	46	eleq1d 2816	. . . . 5 ⊢ (𝑘 = 𝐽 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝐽) ∈ ℝ))
48	45, 47	imbi12d 343	. . . 4 ⊢ (𝑘 = 𝐽 → (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ) ↔ ((𝜑 ∧ 𝐽 ∈ (𝑀...𝑁)) → (𝐹‘𝐽) ∈ ℝ)))
49	48, 8	vtoclg 3541	. . 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 547	. . 3 ⊢ (𝜑 → (𝜑 ∧ 𝐼 ∈ (𝑀..^𝑁)))
52		eleq1 2819	. . . . . 6 ⊢ (𝑘 = 𝐼 → (𝑘 ∈ (𝑀..^𝑁) ↔ 𝐼 ∈ (𝑀..^𝑁)))
53	52	anbi2d 627	. . . . 5 ⊢ (𝑘 = 𝐼 → ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ↔ (𝜑 ∧ 𝐼 ∈ (𝑀..^𝑁))))
54		fvoveq1 7434	. . . . . 6 ⊢ (𝑘 = 𝐼 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝐼 + 1)))
55	5, 54	breq12d 5160	. . . . 5 ⊢ (𝑘 = 𝐼 → ((𝐹‘𝑘) < (𝐹‘(𝑘 + 1)) ↔ (𝐹‘𝐼) < (𝐹‘(𝐼 + 1))))
56	53, 55	imbi12d 343	. . . 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 3541	. . 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 12673	. . . 4 ⊢ (𝜑 → (𝐼 + 1) ∈ ℤ)
61		zltp1le 12616	. . . . . 6 ⊢ ((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) → (𝐼 < 𝐽 ↔ (𝐼 + 1) ≤ 𝐽))
62	13, 24, 61	syl2anc 582	. . . . 5 ⊢ (𝜑 → (𝐼 < 𝐽 ↔ (𝐼 + 1) ≤ 𝐽))
63	27, 62	mpbid 231	. . . 4 ⊢ (𝜑 → (𝐼 + 1) ≤ 𝐽)
64		eluz2 12832	. . . 4 ⊢ (𝐽 ∈ (ℤ_≥‘(𝐼 + 1)) ↔ ((𝐼 + 1) ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐼 + 1) ≤ 𝐽))
65	60, 24, 63, 64	syl3anbrc 1341	. . 3 ⊢ (𝜑 → 𝐽 ∈ (ℤ_≥‘(𝐼 + 1)))
66	12	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑀 ∈ ℤ)
67	21	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑁 ∈ ℤ)
68		elfzelz 13505	. . . . . 6 ⊢ (𝑘 ∈ ((𝐼 + 1)...𝐽) → 𝑘 ∈ ℤ)
69	68	adantl 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑘 ∈ ℤ)
70	66	zred 12670	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑀 ∈ ℝ)
71	69	zred 12670	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑘 ∈ ℝ)
72	60	zred 12670	. . . . . . . 8 ⊢ (𝜑 → (𝐼 + 1) ∈ ℝ)
73	72	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → (𝐼 + 1) ∈ ℝ)
74	22	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝐼 ∈ ℝ)
75	15	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑀 ≤ 𝐼)
76	74	ltp1d 12148	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝐼 < (𝐼 + 1))
77	70, 74, 73, 75, 76	lelttrd 11376	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑀 < (𝐼 + 1))
78		elfzle1 13508	. . . . . . . 8 ⊢ (𝑘 ∈ ((𝐼 + 1)...𝐽) → (𝐼 + 1) ≤ 𝑘)
79	78	adantl 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → (𝐼 + 1) ≤ 𝑘)
80	70, 73, 71, 77, 79	ltletrd 11378	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑀 < 𝑘)
81	70, 71, 80	ltled 11366	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑀 ≤ 𝑘)
82	25	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝐽 ∈ ℝ)
83	67	zred 12670	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑁 ∈ ℝ)
84		elfzle2 13509	. . . . . . 7 ⊢ (𝑘 ∈ ((𝐼 + 1)...𝐽) → 𝑘 ≤ 𝐽)
85	84	adantl 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑘 ≤ 𝐽)
86	29	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝐽 ≤ 𝑁)
87	71, 82, 83, 85, 86	letrd 11375	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑘 ≤ 𝑁)
88	66, 67, 69, 81, 87	elfzd 13496	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑘 ∈ (𝑀...𝑁))
89	88, 8	syldan 589	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → (𝐹‘𝑘) ∈ ℝ)
90	12	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑀 ∈ ℤ)
91	21	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑁 ∈ ℤ)
92		elfzelz 13505	. . . . . . 7 ⊢ (𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1)) → 𝑘 ∈ ℤ)
93	92	adantl 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 ∈ ℤ)
94	90	zred 12670	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑀 ∈ ℝ)
95	93	zred 12670	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 ∈ ℝ)
96	72	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐼 + 1) ∈ ℝ)
97	12	zred 12670	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℝ)
98	22	ltp1d 12148	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 < (𝐼 + 1))
99	97, 22, 72, 15, 98	lelttrd 11376	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 < (𝐼 + 1))
100	99	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑀 < (𝐼 + 1))
101		elfzle1 13508	. . . . . . . . 9 ⊢ (𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1)) → (𝐼 + 1) ≤ 𝑘)
102	101	adantl 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐼 + 1) ≤ 𝑘)
103	94, 96, 95, 100, 102	ltletrd 11378	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑀 < 𝑘)
104	94, 95, 103	ltled 11366	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑀 ≤ 𝑘)
105	91	zred 12670	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑁 ∈ ℝ)
106		peano2rem 11531	. . . . . . . . . 10 ⊢ (𝐽 ∈ ℝ → (𝐽 − 1) ∈ ℝ)
107	25, 106	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 − 1) ∈ ℝ)
108	107	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐽 − 1) ∈ ℝ)
109		elfzle2 13509	. . . . . . . . 9 ⊢ (𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1)) → 𝑘 ≤ (𝐽 − 1))
110	109	adantl 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 ≤ (𝐽 − 1))
111	25	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝐽 ∈ ℝ)
112	111	ltm1d 12150	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐽 − 1) < 𝐽)
113	29	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝐽 ≤ 𝑁)
114	108, 111, 105, 112, 113	ltletrd 11378	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐽 − 1) < 𝑁)
115	95, 108, 105, 110, 114	lelttrd 11376	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 < 𝑁)
116	95, 105, 115	ltled 11366	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 ≤ 𝑁)
117	90, 91, 93, 104, 116	elfzd 13496	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 ∈ (𝑀...𝑁))
118	117, 8	syldan 589	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐹‘𝑘) ∈ ℝ)
119		peano2zm 12609	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
120	91, 119	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝑁 − 1) ∈ ℤ)
121	120	zred 12670	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝑁 − 1) ∈ ℝ)
122		1red 11219	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℝ)
123	25, 26, 122, 29	lesub1dd 11834	. . . . . . . 8 ⊢ (𝜑 → (𝐽 − 1) ≤ (𝑁 − 1))
124	123	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐽 − 1) ≤ (𝑁 − 1))
125	95, 108, 121, 110, 124	letrd 11375	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 ≤ (𝑁 − 1))
126	90, 120, 93, 104, 125	elfzd 13496	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 ∈ (𝑀...(𝑁 − 1)))
127		simpr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → 𝑘 ∈ (𝑀...(𝑁 − 1)))
128		fzoval 13637	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1)))
129	21, 128	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀..^𝑁) = (𝑀...(𝑁 − 1)))
130	129	eqcomd 2736	. . . . . . . . 9 ⊢ (𝜑 → (𝑀...(𝑁 − 1)) = (𝑀..^𝑁))
131	130	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝑀...(𝑁 − 1)) = (𝑀..^𝑁))
132	127, 131	eleqtrd 2833	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → 𝑘 ∈ (𝑀..^𝑁))
133		fzofzp1 13733	. . . . . . 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 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → 𝜑)
136	135, 134	jca 510	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝜑 ∧ (𝑘 + 1) ∈ (𝑀...𝑁)))
137		eleq1 2819	. . . . . . . . 9 ⊢ (𝑗 = (𝑘 + 1) → (𝑗 ∈ (𝑀...𝑁) ↔ (𝑘 + 1) ∈ (𝑀...𝑁)))
138	137	anbi2d 627	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) ↔ (𝜑 ∧ (𝑘 + 1) ∈ (𝑀...𝑁))))
139		fveq2 6890	. . . . . . . . 9 ⊢ (𝑗 = (𝑘 + 1) → (𝐹‘𝑗) = (𝐹‘(𝑘 + 1)))
140	139	eleq1d 2816	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → ((𝐹‘𝑗) ∈ ℝ ↔ (𝐹‘(𝑘 + 1)) ∈ ℝ))
141	138, 140	imbi12d 343	. . . . . . 7 ⊢ (𝑗 = (𝑘 + 1) → (((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ∈ ℝ) ↔ ((𝜑 ∧ (𝑘 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝑘 + 1)) ∈ ℝ)))
142		eleq1 2819	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ (𝑀...𝑁) ↔ 𝑗 ∈ (𝑀...𝑁)))
143	142	anbi2d 627	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ↔ (𝜑 ∧ 𝑗 ∈ (𝑀...𝑁))))
144		fveq2 6890	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
145	144	eleq1d 2816	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑗) ∈ ℝ))
146	143, 145	imbi12d 343	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ) ↔ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ∈ ℝ)))
147	146, 8	chvarvv 2000	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ∈ ℝ)
148	141, 147	vtoclg 3541	. . . . . 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 589	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐹‘(𝑘 + 1)) ∈ ℝ)
151	132, 57	syldan 589	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) < (𝐹‘(𝑘 + 1)))
152	126, 151	syldan 589	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐹‘𝑘) < (𝐹‘(𝑘 + 1)))
153	118, 150, 152	ltled 11366	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
154	65, 89, 153	monoord 14002	. 2 ⊢ (𝜑 → (𝐹‘(𝐼 + 1)) ≤ (𝐹‘𝐽))
155	10, 42, 50, 59, 154	ltletrd 11378	1 ⊢ (𝜑 → (𝐹‘𝐼) < (𝐹‘𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 class class class wbr 5147 ‘cfv 6542 (class class class)co 7411 ℝcr 11111 1c1 11113 + caddc 11115 < clt 11252 ≤ cle 11253 − cmin 11448 ℤcz 12562 ℤ_≥cuz 12826 ...cfz 13488 ..^cfzo 13631

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632

This theorem is referenced by: fourierdlem34 45155

W3C validator