Theorem fourierdlem14 46141

Description: Given the partition 𝑉, 𝑄 is the partition shifted to the left by 𝑋. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
fourierdlem14.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
fourierdlem14.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
fourierdlem14.x	⊢ (𝜑 → 𝑋 ∈ ℝ)
fourierdlem14.p	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑋) ∧ (𝑝‘𝑚) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem14.o	⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem14.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
fourierdlem14.v	⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀))
fourierdlem14.q	⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋))

Assertion

Ref	Expression
fourierdlem14	⊢ (𝜑 → 𝑄 ∈ (𝑂‘𝑀))

Distinct variable groups:   𝐴,𝑚,𝑝   𝐵,𝑚,𝑝   𝑖,𝑀,𝑚,𝑝   𝑄,𝑖,𝑝   𝑖,𝑉,𝑝   𝑖,𝑋,𝑚,𝑝   𝜑,𝑖

Allowed substitution hints:   𝜑(𝑚,𝑝)   𝐴(𝑖)   𝐵(𝑖)   𝑃(𝑖,𝑚,𝑝)   𝑄(𝑚)   𝑂(𝑖,𝑚,𝑝)   𝑉(𝑚)

Proof of Theorem fourierdlem14

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem14.v	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ (𝑃‘𝑀))
2		fourierdlem14.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3		fourierdlem14.p	. . . . . . . . . . . 12 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑋) ∧ (𝑝‘𝑚) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
4	3	fourierdlem2 46129	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → (𝑉 ∈ (𝑃‘𝑀) ↔ (𝑉 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉‘𝑀) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1))))))
5	2, 4	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑉 ∈ (𝑃‘𝑀) ↔ (𝑉 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉‘𝑀) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1))))))
6	1, 5	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → (𝑉 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉‘𝑀) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1)))))
7	6	simpld 494	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ (ℝ ↑m (0...𝑀)))
8		elmapi 8890	. . . . . . . 8 ⊢ (𝑉 ∈ (ℝ ↑m (0...𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑉:(0...𝑀)⟶ℝ)
10	9	ffvelcdmda 7103	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑉‘𝑖) ∈ ℝ)
11		fourierdlem14.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℝ)
12	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑋 ∈ ℝ)
13	10, 12	resubcld 11692	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑉‘𝑖) − 𝑋) ∈ ℝ)
14		fourierdlem14.q	. . . . 5 ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋))
15	13, 14	fmptd 7133	. . . 4 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
16		reex 11247	. . . . . 6 ⊢ ℝ ∈ V
17	16	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ V)
18		ovex 7465	. . . . . 6 ⊢ (0...𝑀) ∈ V
19	18	a1i 11	. . . . 5 ⊢ (𝜑 → (0...𝑀) ∈ V)
20	17, 19	elmapd 8881	. . . 4 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ))
21	15, 20	mpbird 257	. . 3 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑m (0...𝑀)))
22	14	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)))
23		fveq2 6905	. . . . . . . 8 ⊢ (𝑖 = 0 → (𝑉‘𝑖) = (𝑉‘0))
24	23	oveq1d 7447	. . . . . . 7 ⊢ (𝑖 = 0 → ((𝑉‘𝑖) − 𝑋) = ((𝑉‘0) − 𝑋))
25	24	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 = 0) → ((𝑉‘𝑖) − 𝑋) = ((𝑉‘0) − 𝑋))
26		0zd 12627	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℤ)
27	2	nnzd 12642	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
28		0le0 12368	. . . . . . . 8 ⊢ 0 ≤ 0
29	28	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 0)
30		0red 11265	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
31	2	nnred 12282	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ)
32	2	nngt0d 12316	. . . . . . . 8 ⊢ (𝜑 → 0 < 𝑀)
33	30, 31, 32	ltled 11410	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑀)
34	26, 27, 26, 29, 33	elfzd 13556	. . . . . 6 ⊢ (𝜑 → 0 ∈ (0...𝑀))
35	9, 34	ffvelcdmd 7104	. . . . . . 7 ⊢ (𝜑 → (𝑉‘0) ∈ ℝ)
36	35, 11	resubcld 11692	. . . . . 6 ⊢ (𝜑 → ((𝑉‘0) − 𝑋) ∈ ℝ)
37	22, 25, 34, 36	fvmptd 7022	. . . . 5 ⊢ (𝜑 → (𝑄‘0) = ((𝑉‘0) − 𝑋))
38	6	simprd 495	. . . . . . . 8 ⊢ (𝜑 → (((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉‘𝑀) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1))))
39	38	simpld 494	. . . . . . 7 ⊢ (𝜑 → ((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉‘𝑀) = (𝐵 + 𝑋)))
40	39	simpld 494	. . . . . 6 ⊢ (𝜑 → (𝑉‘0) = (𝐴 + 𝑋))
41	40	oveq1d 7447	. . . . 5 ⊢ (𝜑 → ((𝑉‘0) − 𝑋) = ((𝐴 + 𝑋) − 𝑋))
42		fourierdlem14.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
43	42	recnd 11290	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ)
44	11	recnd 11290	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℂ)
45	43, 44	pncand 11622	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝑋) − 𝑋) = 𝐴)
46	37, 41, 45	3eqtrd 2780	. . . 4 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
47		fveq2 6905	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → (𝑉‘𝑖) = (𝑉‘𝑀))
48	47	oveq1d 7447	. . . . . . 7 ⊢ (𝑖 = 𝑀 → ((𝑉‘𝑖) − 𝑋) = ((𝑉‘𝑀) − 𝑋))
49	48	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 = 𝑀) → ((𝑉‘𝑖) − 𝑋) = ((𝑉‘𝑀) − 𝑋))
50	31	leidd 11830	. . . . . . 7 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
51	26, 27, 27, 33, 50	elfzd 13556	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
52	9, 51	ffvelcdmd 7104	. . . . . . 7 ⊢ (𝜑 → (𝑉‘𝑀) ∈ ℝ)
53	52, 11	resubcld 11692	. . . . . 6 ⊢ (𝜑 → ((𝑉‘𝑀) − 𝑋) ∈ ℝ)
54	22, 49, 51, 53	fvmptd 7022	. . . . 5 ⊢ (𝜑 → (𝑄‘𝑀) = ((𝑉‘𝑀) − 𝑋))
55	39	simprd 495	. . . . . 6 ⊢ (𝜑 → (𝑉‘𝑀) = (𝐵 + 𝑋))
56	55	oveq1d 7447	. . . . 5 ⊢ (𝜑 → ((𝑉‘𝑀) − 𝑋) = ((𝐵 + 𝑋) − 𝑋))
57		fourierdlem14.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
58	57	recnd 11290	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℂ)
59	58, 44	pncand 11622	. . . . 5 ⊢ (𝜑 → ((𝐵 + 𝑋) − 𝑋) = 𝐵)
60	54, 56, 59	3eqtrd 2780	. . . 4 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
61	46, 60	jca 511	. . 3 ⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵))
62		elfzofz 13716	. . . . . . 7 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
63	62, 10	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℝ)
64	9	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
65		fzofzp1 13804	. . . . . . . 8 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
66	65	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
67	64, 66	ffvelcdmd 7104	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
68	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
69	38	simprd 495	. . . . . . 7 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1)))
70	69	r19.21bi 3250	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) < (𝑉‘(𝑖 + 1)))
71	63, 67, 68, 70	ltsub1dd 11876	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘𝑖) − 𝑋) < ((𝑉‘(𝑖 + 1)) − 𝑋))
72	62	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
73	62, 13	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘𝑖) − 𝑋) ∈ ℝ)
74	14	fvmpt2 7026	. . . . . 6 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ((𝑉‘𝑖) − 𝑋) ∈ ℝ) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋))
75	72, 73, 74	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋))
76		fveq2 6905	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑉‘𝑖) = (𝑉‘𝑗))
77	76	oveq1d 7447	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝑉‘𝑖) − 𝑋) = ((𝑉‘𝑗) − 𝑋))
78	77	cbvmptv 5254	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋))
79	14, 78	eqtri 2764	. . . . . . 7 ⊢ 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋))
80	79	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋)))
81		fveq2 6905	. . . . . . . 8 ⊢ (𝑗 = (𝑖 + 1) → (𝑉‘𝑗) = (𝑉‘(𝑖 + 1)))
82	81	oveq1d 7447	. . . . . . 7 ⊢ (𝑗 = (𝑖 + 1) → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
83	82	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
84	67, 68	resubcld 11692	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
85	80, 83, 66, 84	fvmptd 7022	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
86	71, 75, 85	3brtr4d 5174	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
87	86	ralrimiva 3145	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
88	21, 61, 87	jca32 515	. 2 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
89		fourierdlem14.o	. . . 4 ⊢ 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝‘𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
90	89	fourierdlem2 46129	. . 3 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑂‘𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
91	2, 90	syl 17	. 2 ⊢ (𝜑 → (𝑄 ∈ (𝑂‘𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))
92	88, 91	mpbird 257	1 ⊢ (𝜑 → 𝑄 ∈ (𝑂‘𝑀))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3060  {crab 3435  Vcvv 3479   class class class wbr 5142   ↦ cmpt 5224  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432   ↑_m cmap 8867  ℝcr 11155  0cc0 11156  1c1 11157   + caddc 11159   < clt 11296   ≤ cle 11297   − cmin 11493  ℕcn 12267  ...cfz 13548  ..^cfzo 13695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-n0 12529  df-z 12616  df-uz 12880  df-fz 13549  df-fzo 13696

This theorem is referenced by:  fourierdlem74  46200  fourierdlem75  46201  fourierdlem84  46210  fourierdlem85  46211  fourierdlem88  46214  fourierdlem103  46229  fourierdlem104  46230