Theorem fourierdlem14 46042

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 46030	. . . . . . . . . . 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 8907	. . . . . . . 8 ⊢ (𝑉 ∈ (ℝ ↑m (0...𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑉:(0...𝑀)⟶ℝ)
10	9	ffvelcdmda 7118	. . . . . 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 11718	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑉‘𝑖) − 𝑋) ∈ ℝ)
14		fourierdlem14.q	. . . . 5 ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋))
15	13, 14	fmptd 7148	. . . 4 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
16		reex 11275	. . . . . 6 ⊢ ℝ ∈ V
17	16	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ V)
18		ovex 7481	. . . . . 6 ⊢ (0...𝑀) ∈ V
19	18	a1i 11	. . . . 5 ⊢ (𝜑 → (0...𝑀) ∈ V)
20	17, 19	elmapd 8898	. . . 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 6920	. . . . . . . 8 ⊢ (𝑖 = 0 → (𝑉‘𝑖) = (𝑉‘0))
24	23	oveq1d 7463	. . . . . . 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 12651	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℤ)
27	2	nnzd 12666	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
28		0le0 12394	. . . . . . . 8 ⊢ 0 ≤ 0
29	28	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 0)
30		0red 11293	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
31	2	nnred 12308	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ)
32	2	nngt0d 12342	. . . . . . . 8 ⊢ (𝜑 → 0 < 𝑀)
33	30, 31, 32	ltled 11438	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑀)
34	26, 27, 26, 29, 33	elfzd 13575	. . . . . 6 ⊢ (𝜑 → 0 ∈ (0...𝑀))
35	9, 34	ffvelcdmd 7119	. . . . . . 7 ⊢ (𝜑 → (𝑉‘0) ∈ ℝ)
36	35, 11	resubcld 11718	. . . . . 6 ⊢ (𝜑 → ((𝑉‘0) − 𝑋) ∈ ℝ)
37	22, 25, 34, 36	fvmptd 7036	. . . . 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 7463	. . . . 5 ⊢ (𝜑 → ((𝑉‘0) − 𝑋) = ((𝐴 + 𝑋) − 𝑋))
42		fourierdlem14.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
43	42	recnd 11318	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ)
44	11	recnd 11318	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℂ)
45	43, 44	pncand 11648	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝑋) − 𝑋) = 𝐴)
46	37, 41, 45	3eqtrd 2784	. . . 4 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
47		fveq2 6920	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → (𝑉‘𝑖) = (𝑉‘𝑀))
48	47	oveq1d 7463	. . . . . . 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 11856	. . . . . . 7 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
51	26, 27, 27, 33, 50	elfzd 13575	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
52	9, 51	ffvelcdmd 7119	. . . . . . 7 ⊢ (𝜑 → (𝑉‘𝑀) ∈ ℝ)
53	52, 11	resubcld 11718	. . . . . 6 ⊢ (𝜑 → ((𝑉‘𝑀) − 𝑋) ∈ ℝ)
54	22, 49, 51, 53	fvmptd 7036	. . . . 5 ⊢ (𝜑 → (𝑄‘𝑀) = ((𝑉‘𝑀) − 𝑋))
55	39	simprd 495	. . . . . 6 ⊢ (𝜑 → (𝑉‘𝑀) = (𝐵 + 𝑋))
56	55	oveq1d 7463	. . . . 5 ⊢ (𝜑 → ((𝑉‘𝑀) − 𝑋) = ((𝐵 + 𝑋) − 𝑋))
57		fourierdlem14.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
58	57	recnd 11318	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℂ)
59	58, 44	pncand 11648	. . . . 5 ⊢ (𝜑 → ((𝐵 + 𝑋) − 𝑋) = 𝐵)
60	54, 56, 59	3eqtrd 2784	. . . 4 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
61	46, 60	jca 511	. . 3 ⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵))
62		elfzofz 13732	. . . . . . 7 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
63	62, 10	sylan2 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) ∈ ℝ)
64	9	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
65		fzofzp1 13814	. . . . . . . 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 7119	. . . . . 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 3257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) < (𝑉‘(𝑖 + 1)))
71	63, 67, 68, 70	ltsub1dd 11902	. . . . 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 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘𝑖) − 𝑋) ∈ ℝ)
74	14	fvmpt2 7040	. . . . . 6 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ((𝑉‘𝑖) − 𝑋) ∈ ℝ) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋))
75	72, 73, 74	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) = ((𝑉‘𝑖) − 𝑋))
76		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑉‘𝑖) = (𝑉‘𝑗))
77	76	oveq1d 7463	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝑉‘𝑖) − 𝑋) = ((𝑉‘𝑗) − 𝑋))
78	77	cbvmptv 5279	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋))
79	14, 78	eqtri 2768	. . . . . . 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 6920	. . . . . . . 8 ⊢ (𝑗 = (𝑖 + 1) → (𝑉‘𝑗) = (𝑉‘(𝑖 + 1)))
82	81	oveq1d 7463	. . . . . . 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 11718	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
85	80, 83, 66, 84	fvmptd 7036	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
86	71, 75, 85	3brtr4d 5198	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
87	86	ralrimiva 3152	. . 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 46030	. . 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 1537   ∈ wcel 2108  ∀wral 3067  {crab 3443  Vcvv 3488   class class class wbr 5166   ↦ cmpt 5249  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ↑_m cmap 8884  ℝcr 11183  0cc0 11184  1c1 11185   + caddc 11187   < clt 11324   ≤ cle 11325   − cmin 11520  ℕcn 12293  ...cfz 13567  ..^cfzo 13711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712

This theorem is referenced by:  fourierdlem74  46101  fourierdlem75  46102  fourierdlem84  46111  fourierdlem85  46112  fourierdlem88  46115  fourierdlem103  46130  fourierdlem104  46131