fourierdlem15 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem15 46093

Description: The range of the partition is between its starting point and its ending point. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem15.1	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem15.2	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem15.3	⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))

**Assertion**
Ref	Expression
fourierdlem15	⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))

Distinct variable groups: 𝐴,𝑖,𝑚,𝑝 𝐵,𝑖,𝑚,𝑝 𝑖,𝑀,𝑚,𝑝 𝑄,𝑖,𝑝 𝜑,𝑖 Allowed substitution hints: 𝜑(𝑚,𝑝) 𝑃(𝑖,𝑚,𝑝) 𝑄(𝑚)

Proof of Theorem fourierdlem15 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem15.3	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ (𝑃‘𝑀))
2		fourierdlem15.2	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3		fourierdlem15.1	. . . . . . . 8 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑_m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
4	3	fourierdlem2 46080	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
5	2, 4	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
6	1, 5	mpbid 232	. . . . 5 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
7	6	simpld 494	. . . 4 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑_m (0...𝑀)))
8		reex 11135	. . . . . 6 ⊢ ℝ ∈ V
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ V)
10		ovex 7402	. . . . . 6 ⊢ (0...𝑀) ∈ V
11	10	a1i 11	. . . . 5 ⊢ (𝜑 → (0...𝑀) ∈ V)
12	9, 11	elmapd 8790	. . . 4 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑_m (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ))
13	7, 12	mpbid 232	. . 3 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
14		ffn 6670	. . 3 ⊢ (𝑄:(0...𝑀)⟶ℝ → 𝑄 Fn (0...𝑀))
15	13, 14	syl 17	. 2 ⊢ (𝜑 → 𝑄 Fn (0...𝑀))
16	6	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
17	16	simpld 494	. . . . . . . 8 ⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵))
18	17	simpld 494	. . . . . . 7 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
19		nnnn0 12425	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
20		nn0uz 12811	. . . . . . . . . . 11 ⊢ ℕ₀ = (ℤ_≥‘0)
21	19, 20	eleqtrdi 2838	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ_≥‘0))
22	2, 21	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
23		eluzfz1 13468	. . . . . . . . 9 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑀))
24	22, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (0...𝑀))
25	13, 24	ffvelcdmd 7039	. . . . . . 7 ⊢ (𝜑 → (𝑄‘0) ∈ ℝ)
26	18, 25	eqeltrrd 2829	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
27	26	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝐴 ∈ ℝ)
28	17	simprd 495	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
29		eluzfz2 13469	. . . . . . . . 9 ⊢ (𝑀 ∈ (ℤ_≥‘0) → 𝑀 ∈ (0...𝑀))
30	22, 29	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
31	13, 30	ffvelcdmd 7039	. . . . . . 7 ⊢ (𝜑 → (𝑄‘𝑀) ∈ ℝ)
32	28, 31	eqeltrrd 2829	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ)
33	32	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝐵 ∈ ℝ)
34	13	ffvelcdmda 7038	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ ℝ)
35	18	eqcomd 2735	. . . . . . 7 ⊢ (𝜑 → 𝐴 = (𝑄‘0))
36	35	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝐴 = (𝑄‘0))
37		elfzuz 13457	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ (ℤ_≥‘0))
38	37	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (ℤ_≥‘0))
39	13	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → 𝑄:(0...𝑀)⟶ℝ)
40		0zd 12517	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 0 ∈ ℤ)
41		elfzel2 13459	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℤ)
42	41	adantr 480	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑀 ∈ ℤ)
43		elfzelz 13461	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℤ)
44	43	adantl 481	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ ℤ)
45		elfzle1 13464	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...𝑖) → 0 ≤ 𝑗)
46	45	adantl 481	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 0 ≤ 𝑗)
47	43	zred 12614	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ∈ ℝ)
48	47	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ ℝ)
49		elfzelz 13461	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℤ)
50	49	zred 12614	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℝ)
51	50	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑖 ∈ ℝ)
52	41	zred 12614	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 𝑀 ∈ ℝ)
53	52	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑀 ∈ ℝ)
54		elfzle2 13465	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...𝑖) → 𝑗 ≤ 𝑖)
55	54	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ≤ 𝑖)
56		elfzle2 13465	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (0...𝑀) → 𝑖 ≤ 𝑀)
57	56	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑖 ≤ 𝑀)
58	48, 51, 53, 55, 57	letrd 11307	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ≤ 𝑀)
59	40, 42, 44, 46, 58	elfzd 13452	. . . . . . . . 9 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ (0...𝑀))
60	59	adantll 714	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → 𝑗 ∈ (0...𝑀))
61	39, 60	ffvelcdmd 7039	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) → (𝑄‘𝑗) ∈ ℝ)
62		simpll 766	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝜑)
63		elfzle1 13464	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...(𝑖 − 1)) → 0 ≤ 𝑗)
64	63	adantl 481	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 0 ≤ 𝑗)
65		elfzelz 13461	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ∈ ℤ)
66	65	zred 12614	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ∈ ℝ)
67	66	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ ℝ)
68	50	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑖 ∈ ℝ)
69	52	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑀 ∈ ℝ)
70		peano2rem 11465	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ ℝ → (𝑖 − 1) ∈ ℝ)
71	68, 70	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑖 − 1) ∈ ℝ)
72		elfzle2 13465	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (0...(𝑖 − 1)) → 𝑗 ≤ (𝑖 − 1))
73	72	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ≤ (𝑖 − 1))
74	68	ltm1d 12091	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑖 − 1) < 𝑖)
75	67, 71, 68, 73, 74	lelttrd 11308	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 < 𝑖)
76	56	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑖 ≤ 𝑀)
77	67, 68, 69, 75, 76	ltletrd 11310	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 < 𝑀)
78	65	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ ℤ)
79		0zd 12517	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 0 ∈ ℤ)
80	41	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑀 ∈ ℤ)
81		elfzo 13598	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗 ∧ 𝑗 < 𝑀)))
82	78, 79, 80, 81	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗 ∧ 𝑗 < 𝑀)))
83	64, 77, 82	mpbir2and 713	. . . . . . . . 9 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ (0..^𝑀))
84	83	adantll 714	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → 𝑗 ∈ (0..^𝑀))
85	13	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
86		elfzofz 13612	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
87	86	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀))
88	85, 87	ffvelcdmd 7039	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘𝑗) ∈ ℝ)
89		fzofzp1 13701	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
90	89	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀))
91	85, 90	ffvelcdmd 7039	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘(𝑗 + 1)) ∈ ℝ)
92		eleq1w 2811	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ (0..^𝑀) ↔ 𝑗 ∈ (0..^𝑀)))
93	92	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ↔ (𝜑 ∧ 𝑗 ∈ (0..^𝑀))))
94		fveq2 6840	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝑄‘𝑖) = (𝑄‘𝑗))
95		oveq1 7376	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
96	95	fveq2d 6844	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑗 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑗 + 1)))
97	94, 96	breq12d 5115	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑗 → ((𝑄‘𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄‘𝑗) < (𝑄‘(𝑗 + 1))))
98	93, 97	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘𝑗) < (𝑄‘(𝑗 + 1)))))
99	16	simprd 495	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
100	99	r19.21bi 3227	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
101	98, 100	chvarvv 1989	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘𝑗) < (𝑄‘(𝑗 + 1)))
102	88, 91, 101	ltled 11298	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑄‘𝑗) ≤ (𝑄‘(𝑗 + 1)))
103	62, 84, 102	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 − 1))) → (𝑄‘𝑗) ≤ (𝑄‘(𝑗 + 1)))
104	38, 61, 103	monoord 13973	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘0) ≤ (𝑄‘𝑖))
105	36, 104	eqbrtrd 5124	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝐴 ≤ (𝑄‘𝑖))
106		elfzuz3 13458	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝑀) → 𝑀 ∈ (ℤ_≥‘𝑖))
107	106	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑀 ∈ (ℤ_≥‘𝑖))
108	13	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
109		fz0fzelfz0 13571	. . . . . . . . 9 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑗 ∈ (0...𝑀))
110	109	adantll 714	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → 𝑗 ∈ (0...𝑀))
111	108, 110	ffvelcdmd 7039	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) → (𝑄‘𝑗) ∈ ℝ)
112	13	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑄:(0...𝑀)⟶ℝ)
113		0zd 12517	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ∈ ℤ)
114	41	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℤ)
115		elfzelz 13461	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ∈ ℤ)
116	115	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℤ)
117		0red 11153	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ∈ ℝ)
118	50	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑖 ∈ ℝ)
119	115	zred 12614	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ∈ ℝ)
120	119	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
121		elfzle1 13464	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → 0 ≤ 𝑖)
122	121	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑖)
123		elfzle1 13464	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑖 ≤ 𝑗)
124	123	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑖 ≤ 𝑗)
125	117, 118, 120, 122, 124	letrd 11307	. . . . . . . . . . 11 ⊢ ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑗)
126	125	adantll 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 𝑗)
127	119	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
128	2	nnred 12177	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ ℝ)
129	128	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℝ)
130		1red 11151	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℝ)
131	129, 130	resubcld 11582	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑀 − 1) ∈ ℝ)
132		elfzle2 13465	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ (𝑖...(𝑀 − 1)) → 𝑗 ≤ (𝑀 − 1))
133	132	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ≤ (𝑀 − 1))
134	129	ltm1d 12091	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑀 − 1) < 𝑀)
135	127, 131, 129, 133, 134	lelttrd 11308	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 < 𝑀)
136	127, 129, 135	ltled 11298	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ≤ 𝑀)
137	136	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ≤ 𝑀)
138	113, 114, 116, 126, 137	elfzd 13452	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ (0...𝑀))
139	112, 138	ffvelcdmd 7039	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄‘𝑗) ∈ ℝ)
140	116	peano2zd 12617	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ∈ ℤ)
141	119	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ ℝ)
142		1red 11151	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℝ)
143		0le1 11677	. . . . . . . . . . . 12 ⊢ 0 ≤ 1
144	143	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ 1)
145	141, 142, 126, 144	addge0d 11730	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 0 ≤ (𝑗 + 1))
146	127, 131, 130, 133	leadd1dd 11768	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ ((𝑀 − 1) + 1))
147	2	nncnd 12178	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℂ)
148	147	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℂ)
149		1cnd 11145	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 1 ∈ ℂ)
150	148, 149	npcand 11513	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → ((𝑀 − 1) + 1) = 𝑀)
151	146, 150	breqtrd 5128	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ 𝑀)
152	151	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ≤ 𝑀)
153	113, 114, 140, 145, 152	elfzd 13452	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 + 1) ∈ (0...𝑀))
154	112, 153	ffvelcdmd 7039	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄‘(𝑗 + 1)) ∈ ℝ)
155		simpll 766	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝜑)
156	135	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 < 𝑀)
157	116, 113, 114, 81	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑗 ∈ (0..^𝑀) ↔ (0 ≤ 𝑗 ∧ 𝑗 < 𝑀)))
158	126, 156, 157	mpbir2and 713	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → 𝑗 ∈ (0..^𝑀))
159	155, 158, 101	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄‘𝑗) < (𝑄‘(𝑗 + 1)))
160	139, 154, 159	ltled 11298	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 − 1))) → (𝑄‘𝑗) ≤ (𝑄‘(𝑗 + 1)))
161	107, 111, 160	monoord 13973	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ≤ (𝑄‘𝑀))
162	28	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑀) = 𝐵)
163	161, 162	breqtrd 5128	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ≤ 𝐵)
164	27, 33, 34, 105, 163	eliccd 45475	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑄‘𝑖) ∈ (𝐴[,]𝐵))
165	164	ralrimiva 3125	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)(𝑄‘𝑖) ∈ (𝐴[,]𝐵))
166		fnfvrnss 7075	. . 3 ⊢ ((𝑄 Fn (0...𝑀) ∧ ∀𝑖 ∈ (0...𝑀)(𝑄‘𝑖) ∈ (𝐴[,]𝐵)) → ran 𝑄 ⊆ (𝐴[,]𝐵))
167	15, 165, 166	syl2anc 584	. 2 ⊢ (𝜑 → ran 𝑄 ⊆ (𝐴[,]𝐵))
168		df-f 6503	. 2 ⊢ (𝑄:(0...𝑀)⟶(𝐴[,]𝐵) ↔ (𝑄 Fn (0...𝑀) ∧ ran 𝑄 ⊆ (𝐴[,]𝐵)))
169	15, 167, 168	sylanbrc 583	1 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶(𝐴[,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∀wral 3044 {crab 3402 Vcvv 3444 ⊆ wss 3911 class class class wbr 5102 ↦ cmpt 5183 ran crn 5632 Fn wfn 6494 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ↑_m cmap 8776 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 + caddc 11047 < clt 11184 ≤ cle 11185 − cmin 11381 ℕcn 12162 ℕ₀cn0 12418 ℤcz 12505 ℤ_≥cuz 12769 [,]cicc 13285 ...cfz 13444 ..^cfzo 13591

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 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-n0 12419 df-z 12506 df-uz 12770 df-icc 13289 df-fz 13445 df-fzo 13592

This theorem is referenced by: fourierdlem38 46116 fourierdlem50 46127 fourierdlem54 46131 fourierdlem63 46140 fourierdlem65 46142 fourierdlem69 46146 fourierdlem70 46147 fourierdlem74 46151 fourierdlem75 46152 fourierdlem76 46153 fourierdlem79 46156 fourierdlem81 46158 fourierdlem84 46161 fourierdlem85 46162 fourierdlem88 46165 fourierdlem89 46166 fourierdlem90 46167 fourierdlem91 46168 fourierdlem92 46169 fourierdlem93 46170 fourierdlem100 46177 fourierdlem101 46178 fourierdlem103 46180 fourierdlem104 46181 fourierdlem107 46184 fourierdlem111 46188 fourierdlem112 46189 fourierdlem113 46190

W3C validator