monoords - Mathbox for Glauco Siliprandi

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Theorem monoords 41556

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 551	. . 3 ⊢ (𝜑 → (𝜑 ∧ 𝐼 ∈ (𝑀...𝑁)))
3		eleq1 2900	. . . . . 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 6665	. . . . . 6 ⊢ (𝑘 = 𝐼 → (𝐹‘𝑘) = (𝐹‘𝐼))
6	5	eleq1d 2897	. . . . 5 ⊢ (𝑘 = 𝐼 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝐼) ∈ ℝ))
7	4, 6	imbi12d 347	. . . 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 3568	. . 3 ⊢ (𝐼 ∈ (𝑀...𝑁) → ((𝜑 ∧ 𝐼 ∈ (𝑀...𝑁)) → (𝐹‘𝐼) ∈ ℝ))
10	1, 2, 9	sylc 65	. 2 ⊢ (𝜑 → (𝐹‘𝐼) ∈ ℝ)
11		elfzel1 12901	. . . . . . 7 ⊢ (𝐼 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
12	1, 11	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
13		elfzelz 12902	. . . . . . 7 ⊢ (𝐼 ∈ (𝑀...𝑁) → 𝐼 ∈ ℤ)
14	1, 13	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ ℤ)
15		elfzle1 12904	. . . . . . 7 ⊢ (𝐼 ∈ (𝑀...𝑁) → 𝑀 ≤ 𝐼)
16	1, 15	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ≤ 𝐼)
17		eluz2 12243	. . . . . 6 ⊢ (𝐼 ∈ (ℤ_≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀 ≤ 𝐼))
18	12, 14, 16, 17	syl3anbrc 1339	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (ℤ_≥‘𝑀))
19		elfzuz2 12906	. . . . . . 7 ⊢ (𝐼 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑀))
20	1, 19	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
21		eluzelz 12247	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
22	20, 21	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
23	14	zred 12081	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ ℝ)
24		monoords.j	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (𝑀...𝑁))
25		elfzelz 12902	. . . . . . . 8 ⊢ (𝐽 ∈ (𝑀...𝑁) → 𝐽 ∈ ℤ)
26	24, 25	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℤ)
27	26	zred 12081	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ ℝ)
28	22	zred 12081	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℝ)
29		monoords.iltj	. . . . . 6 ⊢ (𝜑 → 𝐼 < 𝐽)
30		elfzle2 12905	. . . . . . 7 ⊢ (𝐽 ∈ (𝑀...𝑁) → 𝐽 ≤ 𝑁)
31	24, 30	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐽 ≤ 𝑁)
32	23, 27, 28, 29, 31	ltletrd 10794	. . . . 5 ⊢ (𝜑 → 𝐼 < 𝑁)
33		elfzo2 13035	. . . . 5 ⊢ (𝐼 ∈ (𝑀..^𝑁) ↔ (𝐼 ∈ (ℤ_≥‘𝑀) ∧ 𝑁 ∈ ℤ ∧ 𝐼 < 𝑁))
34	18, 22, 32, 33	syl3anbrc 1339	. . . 4 ⊢ (𝜑 → 𝐼 ∈ (𝑀..^𝑁))
35		fzofzp1 13128	. . . 4 ⊢ (𝐼 ∈ (𝑀..^𝑁) → (𝐼 + 1) ∈ (𝑀...𝑁))
36	34, 35	syl 17	. . 3 ⊢ (𝜑 → (𝐼 + 1) ∈ (𝑀...𝑁))
37	36	ancli 551	. . 3 ⊢ (𝜑 → (𝜑 ∧ (𝐼 + 1) ∈ (𝑀...𝑁)))
38		eleq1 2900	. . . . . 6 ⊢ (𝑘 = (𝐼 + 1) → (𝑘 ∈ (𝑀...𝑁) ↔ (𝐼 + 1) ∈ (𝑀...𝑁)))
39	38	anbi2d 630	. . . . 5 ⊢ (𝑘 = (𝐼 + 1) → ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ↔ (𝜑 ∧ (𝐼 + 1) ∈ (𝑀...𝑁))))
40		fveq2 6665	. . . . . 6 ⊢ (𝑘 = (𝐼 + 1) → (𝐹‘𝑘) = (𝐹‘(𝐼 + 1)))
41	40	eleq1d 2897	. . . . 5 ⊢ (𝑘 = (𝐼 + 1) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(𝐼 + 1)) ∈ ℝ))
42	39, 41	imbi12d 347	. . . 4 ⊢ (𝑘 = (𝐼 + 1) → (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ) ↔ ((𝜑 ∧ (𝐼 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝐼 + 1)) ∈ ℝ)))
43	42, 8	vtoclg 3568	. . 3 ⊢ ((𝐼 + 1) ∈ (𝑀...𝑁) → ((𝜑 ∧ (𝐼 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝐼 + 1)) ∈ ℝ))
44	36, 37, 43	sylc 65	. 2 ⊢ (𝜑 → (𝐹‘(𝐼 + 1)) ∈ ℝ)
45	24	ancli 551	. . 3 ⊢ (𝜑 → (𝜑 ∧ 𝐽 ∈ (𝑀...𝑁)))
46		eleq1 2900	. . . . . 6 ⊢ (𝑘 = 𝐽 → (𝑘 ∈ (𝑀...𝑁) ↔ 𝐽 ∈ (𝑀...𝑁)))
47	46	anbi2d 630	. . . . 5 ⊢ (𝑘 = 𝐽 → ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ↔ (𝜑 ∧ 𝐽 ∈ (𝑀...𝑁))))
48		fveq2 6665	. . . . . 6 ⊢ (𝑘 = 𝐽 → (𝐹‘𝑘) = (𝐹‘𝐽))
49	48	eleq1d 2897	. . . . 5 ⊢ (𝑘 = 𝐽 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝐽) ∈ ℝ))
50	47, 49	imbi12d 347	. . . 4 ⊢ (𝑘 = 𝐽 → (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ) ↔ ((𝜑 ∧ 𝐽 ∈ (𝑀...𝑁)) → (𝐹‘𝐽) ∈ ℝ)))
51	50, 8	vtoclg 3568	. . 3 ⊢ (𝐽 ∈ (𝑀...𝑁) → ((𝜑 ∧ 𝐽 ∈ (𝑀...𝑁)) → (𝐹‘𝐽) ∈ ℝ))
52	24, 45, 51	sylc 65	. 2 ⊢ (𝜑 → (𝐹‘𝐽) ∈ ℝ)
53	34	ancli 551	. . 3 ⊢ (𝜑 → (𝜑 ∧ 𝐼 ∈ (𝑀..^𝑁)))
54		eleq1 2900	. . . . . 6 ⊢ (𝑘 = 𝐼 → (𝑘 ∈ (𝑀..^𝑁) ↔ 𝐼 ∈ (𝑀..^𝑁)))
55	54	anbi2d 630	. . . . 5 ⊢ (𝑘 = 𝐼 → ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ↔ (𝜑 ∧ 𝐼 ∈ (𝑀..^𝑁))))
56		fvoveq1 7173	. . . . . 6 ⊢ (𝑘 = 𝐼 → (𝐹‘(𝑘 + 1)) = (𝐹‘(𝐼 + 1)))
57	5, 56	breq12d 5072	. . . . 5 ⊢ (𝑘 = 𝐼 → ((𝐹‘𝑘) < (𝐹‘(𝑘 + 1)) ↔ (𝐹‘𝐼) < (𝐹‘(𝐼 + 1))))
58	55, 57	imbi12d 347	. . . 4 ⊢ (𝑘 = 𝐼 → (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝐹‘𝑘) < (𝐹‘(𝑘 + 1))) ↔ ((𝜑 ∧ 𝐼 ∈ (𝑀..^𝑁)) → (𝐹‘𝐼) < (𝐹‘(𝐼 + 1)))))
59		monoords.flt	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝐹‘𝑘) < (𝐹‘(𝑘 + 1)))
60	58, 59	vtoclg 3568	. . 3 ⊢ (𝐼 ∈ (𝑀..^𝑁) → ((𝜑 ∧ 𝐼 ∈ (𝑀..^𝑁)) → (𝐹‘𝐼) < (𝐹‘(𝐼 + 1))))
61	34, 53, 60	sylc 65	. 2 ⊢ (𝜑 → (𝐹‘𝐼) < (𝐹‘(𝐼 + 1)))
62	14	peano2zd 12084	. . . 4 ⊢ (𝜑 → (𝐼 + 1) ∈ ℤ)
63		zltp1le 12026	. . . . . 6 ⊢ ((𝐼 ∈ ℤ ∧ 𝐽 ∈ ℤ) → (𝐼 < 𝐽 ↔ (𝐼 + 1) ≤ 𝐽))
64	14, 26, 63	syl2anc 586	. . . . 5 ⊢ (𝜑 → (𝐼 < 𝐽 ↔ (𝐼 + 1) ≤ 𝐽))
65	29, 64	mpbid 234	. . . 4 ⊢ (𝜑 → (𝐼 + 1) ≤ 𝐽)
66		eluz2 12243	. . . 4 ⊢ (𝐽 ∈ (ℤ_≥‘(𝐼 + 1)) ↔ ((𝐼 + 1) ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐼 + 1) ≤ 𝐽))
67	62, 26, 65, 66	syl3anbrc 1339	. . 3 ⊢ (𝜑 → 𝐽 ∈ (ℤ_≥‘(𝐼 + 1)))
68	12	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑀 ∈ ℤ)
69	22	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑁 ∈ ℤ)
70		elfzelz 12902	. . . . . . . 8 ⊢ (𝑘 ∈ ((𝐼 + 1)...𝐽) → 𝑘 ∈ ℤ)
71	70	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑘 ∈ ℤ)
72	68, 69, 71	3jca 1124	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ))
73	68	zred 12081	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑀 ∈ ℝ)
74	71	zred 12081	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑘 ∈ ℝ)
75	62	zred 12081	. . . . . . . . 9 ⊢ (𝜑 → (𝐼 + 1) ∈ ℝ)
76	75	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → (𝐼 + 1) ∈ ℝ)
77	23	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝐼 ∈ ℝ)
78	16	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑀 ≤ 𝐼)
79	77	ltp1d 11564	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝐼 < (𝐼 + 1))
80	73, 77, 76, 78, 79	lelttrd 10792	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑀 < (𝐼 + 1))
81		elfzle1 12904	. . . . . . . . 9 ⊢ (𝑘 ∈ ((𝐼 + 1)...𝐽) → (𝐼 + 1) ≤ 𝑘)
82	81	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → (𝐼 + 1) ≤ 𝑘)
83	73, 76, 74, 80, 82	ltletrd 10794	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑀 < 𝑘)
84	73, 74, 83	ltled 10782	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑀 ≤ 𝑘)
85	27	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝐽 ∈ ℝ)
86	69	zred 12081	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑁 ∈ ℝ)
87		elfzle2 12905	. . . . . . . 8 ⊢ (𝑘 ∈ ((𝐼 + 1)...𝐽) → 𝑘 ≤ 𝐽)
88	87	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑘 ≤ 𝐽)
89	31	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝐽 ≤ 𝑁)
90	74, 85, 86, 88, 89	letrd 10791	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑘 ≤ 𝑁)
91	72, 84, 90	jca32 518	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)))
92		elfz2 12893	. . . . 5 ⊢ (𝑘 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)))
93	91, 92	sylibr 236	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → 𝑘 ∈ (𝑀...𝑁))
94	93, 8	syldan 593	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...𝐽)) → (𝐹‘𝑘) ∈ ℝ)
95	12	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑀 ∈ ℤ)
96	22	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑁 ∈ ℤ)
97		elfzelz 12902	. . . . . . . . 9 ⊢ (𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1)) → 𝑘 ∈ ℤ)
98	97	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 ∈ ℤ)
99	95, 96, 98	3jca 1124	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ))
100	95	zred 12081	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑀 ∈ ℝ)
101	98	zred 12081	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 ∈ ℝ)
102	75	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐼 + 1) ∈ ℝ)
103	12	zred 12081	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
104	23	ltp1d 11564	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 < (𝐼 + 1))
105	103, 23, 75, 16, 104	lelttrd 10792	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 < (𝐼 + 1))
106	105	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑀 < (𝐼 + 1))
107		elfzle1 12904	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1)) → (𝐼 + 1) ≤ 𝑘)
108	107	adantl 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐼 + 1) ≤ 𝑘)
109	100, 102, 101, 106, 108	ltletrd 10794	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑀 < 𝑘)
110	100, 101, 109	ltled 10782	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑀 ≤ 𝑘)
111	96	zred 12081	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑁 ∈ ℝ)
112		peano2rem 10947	. . . . . . . . . . 11 ⊢ (𝐽 ∈ ℝ → (𝐽 − 1) ∈ ℝ)
113	27, 112	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽 − 1) ∈ ℝ)
114	113	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐽 − 1) ∈ ℝ)
115		elfzle2 12905	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1)) → 𝑘 ≤ (𝐽 − 1))
116	115	adantl 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 ≤ (𝐽 − 1))
117	27	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝐽 ∈ ℝ)
118	117	ltm1d 11566	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐽 − 1) < 𝐽)
119	31	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝐽 ≤ 𝑁)
120	114, 117, 111, 118, 119	ltletrd 10794	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐽 − 1) < 𝑁)
121	101, 114, 111, 116, 120	lelttrd 10792	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 < 𝑁)
122	101, 111, 121	ltled 10782	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 ≤ 𝑁)
123	99, 110, 122	jca32 518	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁)))
124	123, 92	sylibr 236	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 ∈ (𝑀...𝑁))
125	124, 8	syldan 593	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐹‘𝑘) ∈ ℝ)
126		peano2zm 12019	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
127	96, 126	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝑁 − 1) ∈ ℤ)
128	95, 127, 98	3jca 1124	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ ∧ 𝑘 ∈ ℤ))
129	127	zred 12081	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝑁 − 1) ∈ ℝ)
130		1red 10636	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℝ)
131	27, 28, 130, 31	lesub1dd 11250	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 − 1) ≤ (𝑁 − 1))
132	131	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐽 − 1) ≤ (𝑁 − 1))
133	101, 114, 129, 116, 132	letrd 10791	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 ≤ (𝑁 − 1))
134	128, 110, 133	jca32 518	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ (𝑁 − 1))))
135		elfz2 12893	. . . . . 6 ⊢ (𝑘 ∈ (𝑀...(𝑁 − 1)) ↔ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (𝑀 ≤ 𝑘 ∧ 𝑘 ≤ (𝑁 − 1))))
136	134, 135	sylibr 236	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → 𝑘 ∈ (𝑀...(𝑁 − 1)))
137		simpr 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → 𝑘 ∈ (𝑀...(𝑁 − 1)))
138		fzoval 13033	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → (𝑀..^𝑁) = (𝑀...(𝑁 − 1)))
139	22, 138	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀..^𝑁) = (𝑀...(𝑁 − 1)))
140	139	eqcomd 2827	. . . . . . . . 9 ⊢ (𝜑 → (𝑀...(𝑁 − 1)) = (𝑀..^𝑁))
141	140	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝑀...(𝑁 − 1)) = (𝑀..^𝑁))
142	137, 141	eleqtrd 2915	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → 𝑘 ∈ (𝑀..^𝑁))
143		fzofzp1 13128	. . . . . . 7 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
144	142, 143	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝑘 + 1) ∈ (𝑀...𝑁))
145		simpl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → 𝜑)
146	145, 144	jca 514	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝜑 ∧ (𝑘 + 1) ∈ (𝑀...𝑁)))
147		eleq1 2900	. . . . . . . . 9 ⊢ (𝑗 = (𝑘 + 1) → (𝑗 ∈ (𝑀...𝑁) ↔ (𝑘 + 1) ∈ (𝑀...𝑁)))
148	147	anbi2d 630	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) ↔ (𝜑 ∧ (𝑘 + 1) ∈ (𝑀...𝑁))))
149		fveq2 6665	. . . . . . . . 9 ⊢ (𝑗 = (𝑘 + 1) → (𝐹‘𝑗) = (𝐹‘(𝑘 + 1)))
150	149	eleq1d 2897	. . . . . . . 8 ⊢ (𝑗 = (𝑘 + 1) → ((𝐹‘𝑗) ∈ ℝ ↔ (𝐹‘(𝑘 + 1)) ∈ ℝ))
151	148, 150	imbi12d 347	. . . . . . 7 ⊢ (𝑗 = (𝑘 + 1) → (((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ∈ ℝ) ↔ ((𝜑 ∧ (𝑘 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝑘 + 1)) ∈ ℝ)))
152		eleq1 2900	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ (𝑀...𝑁) ↔ 𝑗 ∈ (𝑀...𝑁)))
153	152	anbi2d 630	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ↔ (𝜑 ∧ 𝑗 ∈ (𝑀...𝑁))))
154		fveq2 6665	. . . . . . . . . 10 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
155	154	eleq1d 2897	. . . . . . . . 9 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘𝑗) ∈ ℝ))
156	153, 155	imbi12d 347	. . . . . . . 8 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℝ) ↔ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ∈ ℝ)))
157	156, 8	chvarvv 2001	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝑁)) → (𝐹‘𝑗) ∈ ℝ)
158	151, 157	vtoclg 3568	. . . . . 6 ⊢ ((𝑘 + 1) ∈ (𝑀...𝑁) → ((𝜑 ∧ (𝑘 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝑘 + 1)) ∈ ℝ))
159	144, 146, 158	sylc 65	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘(𝑘 + 1)) ∈ ℝ)
160	136, 159	syldan 593	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐹‘(𝑘 + 1)) ∈ ℝ)
161	142, 59	syldan 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...(𝑁 − 1))) → (𝐹‘𝑘) < (𝐹‘(𝑘 + 1)))
162	136, 161	syldan 593	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐹‘𝑘) < (𝐹‘(𝑘 + 1)))
163	125, 160, 162	ltled 10782	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐼 + 1)...(𝐽 − 1))) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1)))
164	67, 94, 163	monoord 13394	. 2 ⊢ (𝜑 → (𝐹‘(𝐼 + 1)) ≤ (𝐹‘𝐽))
165	10, 44, 52, 61, 164	ltletrd 10794	1 ⊢ (𝜑 → (𝐹‘𝐼) < (𝐹‘𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 ℝcr 10530 1c1 10532 + caddc 10534 < clt 10669 ≤ cle 10670 − cmin 10864 ℤcz 11975 ℤ_≥cuz 12237 ...cfz 12886 ..^cfzo 13027

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028

This theorem is referenced by: fourierdlem34 42419

W3C validator