Theorem fourierdlem14 46875

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 46863	. . . . . . . . . . 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 18	. . . . . . . . . 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 235	. . . . . . . . 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 500	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ (ℝ ↑m (0...𝑀)))
8		elmapi 8855	. . . . . . . 8 ⊢ (𝑉 ∈ (ℝ ↑m (0...𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
9	7, 8	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑉:(0...𝑀)⟶ℝ)
10	9	ffvelcdmda 7086	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑉‘𝑖) ∈ ℝ)
11		fourierdlem14.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℝ)
12	11	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑋 ∈ ℝ)
13	10, 12	resubcld 11660	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑉‘𝑖) − 𝑋) ∈ ℝ)
14		fourierdlem14.q	. . . . 5 ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋))
15	13, 14	fmptd 7116	. . . 4 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
16		reex 11209	. . . . . 6 ⊢ ℝ ∈ V
17	16	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ V)
18		ovex 7456	. . . . . 6 ⊢ (0...𝑀) ∈ V
19	18	a1i 11	. . . . 5 ⊢ (𝜑 → (0...𝑀) ∈ V)
20	17, 19	elmapd 8846	. . . 4 ⊢ (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ))
21	15, 20	mpbird 260	. . 3 ⊢ (𝜑 → 𝑄 ∈ (ℝ ↑m (0...𝑀)))
22	14	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)))
23		fveq2 6888	. . . . . . . 8 ⊢ (𝑖 = 0 → (𝑉‘𝑖) = (𝑉‘0))
24	23	oveq1d 7438	. . . . . . 7 ⊢ (𝑖 = 0 → ((𝑉‘𝑖) − 𝑋) = ((𝑉‘0) − 𝑋))
25	24	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 = 0) → ((𝑉‘𝑖) − 𝑋) = ((𝑉‘0) − 𝑋))
26		0zd 12621	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℤ)
27	2	nnzd 12635	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
28		0le0 12360	. . . . . . . 8 ⊢ 0 ≤ 0
29	28	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 0)
30		0red 11229	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
31	2	nnred 12266	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ)
32	2	nngt0d 12303	. . . . . . . 8 ⊢ (𝜑 → 0 < 𝑀)
33	30, 31, 32	ltled 11376	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑀)
34	26, 27, 26, 29, 33	elfzd 13561	. . . . . 6 ⊢ (𝜑 → 0 ∈ (0...𝑀))
35	9, 34	ffvelcdmd 7087	. . . . . . 7 ⊢ (𝜑 → (𝑉‘0) ∈ ℝ)
36	35, 11	resubcld 11660	. . . . . 6 ⊢ (𝜑 → ((𝑉‘0) − 𝑋) ∈ ℝ)
37	22, 25, 34, 36	fvmptd 7004	. . . . 5 ⊢ (𝜑 → (𝑄‘0) = ((𝑉‘0) − 𝑋))
38	6	simprd 501	. . . . . . . 8 ⊢ (𝜑 → (((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉‘𝑀) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1))))
39	38	simpld 500	. . . . . . 7 ⊢ (𝜑 → ((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉‘𝑀) = (𝐵 + 𝑋)))
40	39	simpld 500	. . . . . 6 ⊢ (𝜑 → (𝑉‘0) = (𝐴 + 𝑋))
41	40	oveq1d 7438	. . . . 5 ⊢ (𝜑 → ((𝑉‘0) − 𝑋) = ((𝐴 + 𝑋) − 𝑋))
42		fourierdlem14.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
43	42	recnd 11255	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ)
44	11	recnd 11255	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℂ)
45	43, 44	pncand 11588	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝑋) − 𝑋) = 𝐴)
46	37, 41, 45	3eqtrd 2805	. . . 4 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
47		fveq2 6888	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → (𝑉‘𝑖) = (𝑉‘𝑀))
48	47	oveq1d 7438	. . . . . . 7 ⊢ (𝑖 = 𝑀 → ((𝑉‘𝑖) − 𝑋) = ((𝑉‘𝑀) − 𝑋))
49	48	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 = 𝑀) → ((𝑉‘𝑖) − 𝑋) = ((𝑉‘𝑀) − 𝑋))
50	31	leidd 11798	. . . . . . 7 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
51	26, 27, 27, 33, 50	elfzd 13561	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
52	9, 51	ffvelcdmd 7087	. . . . . . 7 ⊢ (𝜑 → (𝑉‘𝑀) ∈ ℝ)
53	52, 11	resubcld 11660	. . . . . 6 ⊢ (𝜑 → ((𝑉‘𝑀) − 𝑋) ∈ ℝ)
54	22, 49, 51, 53	fvmptd 7004	. . . . 5 ⊢ (𝜑 → (𝑄‘𝑀) = ((𝑉‘𝑀) − 𝑋))
55	39	simprd 501	. . . . . 6 ⊢ (𝜑 → (𝑉‘𝑀) = (𝐵 + 𝑋))
56	55	oveq1d 7438	. . . . 5 ⊢ (𝜑 → ((𝑉‘𝑀) − 𝑋) = ((𝐵 + 𝑋) − 𝑋))
57		fourierdlem14.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
58	57	recnd 11255	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℂ)
59	58, 44	pncand 11588	. . . . 5 ⊢ (𝜑 → ((𝐵 + 𝑋) − 𝑋) = 𝐵)
60	54, 56, 59	3eqtrd 2805	. . . 4 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
61	46, 60	jca 521	. . 3 ⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵))
62		elfzofz 13723	. . . . . . 7 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
63	62, 10	sylan2 605	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℝ)
64	9	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
65		fzofzp1 13812	. . . . . . . 8 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
66	65	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
67	64, 66	ffvelcdmd 7087	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
68	11	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
69	38	simprd 501	. . . . . . 7 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑉‘𝑖) < (𝑉‘(𝑖 + 1)))
70	69	r19.21bi 3260	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) < (𝑉‘(𝑖 + 1)))
71	63, 67, 68, 70	ltsub1dd 11844	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘𝑖) − 𝑋) < ((𝑉‘(𝑖 + 1)) − 𝑋))
72	62	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
73	62, 13	sylan2 605	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘𝑖) − 𝑋) ∈ ℝ)
74	14	fvmpt2 7008	. . . . . 6 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ((𝑉‘𝑖) − 𝑋) ∈ ℝ) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋))
75	72, 73, 74	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋))
76		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑉‘𝑖) = (𝑉‘𝑗))
77	76	oveq1d 7438	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝑉‘𝑖) − 𝑋) = ((𝑉‘𝑗) − 𝑋))
78	77	cbvmptv 5220	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋))
79	14, 78	eqtri 2789	. . . . . . 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 6888	. . . . . . . 8 ⊢ (𝑗 = (𝑖 + 1) → (𝑉‘𝑗) = (𝑉‘(𝑖 + 1)))
82	81	oveq1d 7438	. . . . . . 7 ⊢ (𝑗 = (𝑖 + 1) → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
83	82	adantl 487	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑉‘𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
84	67, 68	resubcld 11660	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
85	80, 83, 66, 84	fvmptd 7004	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
86	71, 75, 85	3brtr4d 5148	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
87	86	ralrimiva 3160	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
88	21, 61, 87	jca32 525	. 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 46863	. . 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 18	. 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 260	1 ⊢ (𝜑 → 𝑄 ∈ (𝑂‘𝑀))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3082  {crab 3419  Vcvv 3458   class class class wbr 5114   ↦ cmpt 5197  ⟶wf 6539  ‘cfv 6543  (class class class)co 7423   ↑_m cmap 8833  ℝcr 11117  0cc0 11118  1c1 11119   + caddc 11121   < clt 11261   ≤ cle 11262   − cmin 11459  ℕcn 12251  ...cfz 13553  ..^cfzo 13701

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-er 8703  df-map 8835  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-n0 12523  df-z 12610  df-uz 12881  df-fz 13554  df-fzo 13702

This theorem is used by:  fourierdlem74  46934  fourierdlem75  46935  fourierdlem84  46944  fourierdlem85  46945  fourierdlem88  46948  fourierdlem103  46963  fourierdlem104  46964