Theorem fourierdlem14 46159

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 46147	. . . . . . . . . . 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 8768	. . . . . . . 8 ⊢ (𝑉 ∈ (ℝ ↑m (0...𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑉:(0...𝑀)⟶ℝ)
10	9	ffvelcdmda 7012	. . . . . 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 11540	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑉‘𝑖) − 𝑋) ∈ ℝ)
14		fourierdlem14.q	. . . . 5 ⊢ 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋))
15	13, 14	fmptd 7042	. . . 4 ⊢ (𝜑 → 𝑄:(0...𝑀)⟶ℝ)
16		reex 11092	. . . . . 6 ⊢ ℝ ∈ V
17	16	a1i 11	. . . . 5 ⊢ (𝜑 → ℝ ∈ V)
18		ovex 7374	. . . . . 6 ⊢ (0...𝑀) ∈ V
19	18	a1i 11	. . . . 5 ⊢ (𝜑 → (0...𝑀) ∈ V)
20	17, 19	elmapd 8759	. . . 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 6817	. . . . . . . 8 ⊢ (𝑖 = 0 → (𝑉‘𝑖) = (𝑉‘0))
24	23	oveq1d 7356	. . . . . . 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 12475	. . . . . . 7 ⊢ (𝜑 → 0 ∈ ℤ)
27	2	nnzd 12490	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℤ)
28		0le0 12221	. . . . . . . 8 ⊢ 0 ≤ 0
29	28	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 0)
30		0red 11110	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
31	2	nnred 12135	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℝ)
32	2	nngt0d 12169	. . . . . . . 8 ⊢ (𝜑 → 0 < 𝑀)
33	30, 31, 32	ltled 11256	. . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝑀)
34	26, 27, 26, 29, 33	elfzd 13410	. . . . . 6 ⊢ (𝜑 → 0 ∈ (0...𝑀))
35	9, 34	ffvelcdmd 7013	. . . . . . 7 ⊢ (𝜑 → (𝑉‘0) ∈ ℝ)
36	35, 11	resubcld 11540	. . . . . 6 ⊢ (𝜑 → ((𝑉‘0) − 𝑋) ∈ ℝ)
37	22, 25, 34, 36	fvmptd 6931	. . . . 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 7356	. . . . 5 ⊢ (𝜑 → ((𝑉‘0) − 𝑋) = ((𝐴 + 𝑋) − 𝑋))
42		fourierdlem14.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
43	42	recnd 11135	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ)
44	11	recnd 11135	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℂ)
45	43, 44	pncand 11468	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝑋) − 𝑋) = 𝐴)
46	37, 41, 45	3eqtrd 2770	. . . 4 ⊢ (𝜑 → (𝑄‘0) = 𝐴)
47		fveq2 6817	. . . . . . . 8 ⊢ (𝑖 = 𝑀 → (𝑉‘𝑖) = (𝑉‘𝑀))
48	47	oveq1d 7356	. . . . . . 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 11678	. . . . . . 7 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
51	26, 27, 27, 33, 50	elfzd 13410	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
52	9, 51	ffvelcdmd 7013	. . . . . . 7 ⊢ (𝜑 → (𝑉‘𝑀) ∈ ℝ)
53	52, 11	resubcld 11540	. . . . . 6 ⊢ (𝜑 → ((𝑉‘𝑀) − 𝑋) ∈ ℝ)
54	22, 49, 51, 53	fvmptd 6931	. . . . 5 ⊢ (𝜑 → (𝑄‘𝑀) = ((𝑉‘𝑀) − 𝑋))
55	39	simprd 495	. . . . . 6 ⊢ (𝜑 → (𝑉‘𝑀) = (𝐵 + 𝑋))
56	55	oveq1d 7356	. . . . 5 ⊢ (𝜑 → ((𝑉‘𝑀) − 𝑋) = ((𝐵 + 𝑋) − 𝑋))
57		fourierdlem14.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
58	57	recnd 11135	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℂ)
59	58, 44	pncand 11468	. . . . 5 ⊢ (𝜑 → ((𝐵 + 𝑋) − 𝑋) = 𝐵)
60	54, 56, 59	3eqtrd 2770	. . . 4 ⊢ (𝜑 → (𝑄‘𝑀) = 𝐵)
61	46, 60	jca 511	. . 3 ⊢ (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄‘𝑀) = 𝐵))
62		elfzofz 13570	. . . . . . 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 13659	. . . . . . . 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 7013	. . . . . 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 3224	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑉‘𝑖) < (𝑉‘(𝑖 + 1)))
71	63, 67, 68, 70	ltsub1dd 11724	. . . . 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 6935	. . . . . 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 6817	. . . . . . . . . 10 ⊢ (𝑖 = 𝑗 → (𝑉‘𝑖) = (𝑉‘𝑗))
77	76	oveq1d 7356	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → ((𝑉‘𝑖) − 𝑋) = ((𝑉‘𝑗) − 𝑋))
78	77	cbvmptv 5190	. . . . . . . 8 ⊢ (𝑖 ∈ (0...𝑀) ↦ ((𝑉‘𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑉‘𝑗) − 𝑋))
79	14, 78	eqtri 2754	. . . . . . 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 6817	. . . . . . . 8 ⊢ (𝑗 = (𝑖 + 1) → (𝑉‘𝑗) = (𝑉‘(𝑖 + 1)))
82	81	oveq1d 7356	. . . . . . 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 11540	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑉‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
85	80, 83, 66, 84	fvmptd 6931	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
86	71, 75, 85	3brtr4d 5118	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))
87	86	ralrimiva 3124	. . 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 46147	. . 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 1541   ∈ wcel 2111  ∀wral 3047  {crab 3395  Vcvv 3436   class class class wbr 5086   ↦ cmpt 5167  ⟶wf 6472  ‘cfv 6476  (class class class)co 7341   ↑_m cmap 8745  ℝcr 11000  0cc0 11001  1c1 11002   + caddc 11004   < clt 11141   ≤ cle 11142   − cmin 11339  ℕcn 12120  ...cfz 13402  ..^cfzo 13549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-cnex 11057  ax-resscn 11058  ax-1cn 11059  ax-icn 11060  ax-addcl 11061  ax-addrcl 11062  ax-mulcl 11063  ax-mulrcl 11064  ax-mulcom 11065  ax-addass 11066  ax-mulass 11067  ax-distr 11068  ax-i2m1 11069  ax-1ne0 11070  ax-1rid 11071  ax-rnegex 11072  ax-rrecex 11073  ax-cnre 11074  ax-pre-lttri 11075  ax-pre-lttrn 11076  ax-pre-ltadd 11077  ax-pre-mulgt0 11078

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-er 8617  df-map 8747  df-en 8865  df-dom 8866  df-sdom 8867  df-pnf 11143  df-mnf 11144  df-xr 11145  df-ltxr 11146  df-le 11147  df-sub 11341  df-neg 11342  df-nn 12121  df-n0 12377  df-z 12464  df-uz 12728  df-fz 13403  df-fzo 13550

This theorem is referenced by:  fourierdlem74  46218  fourierdlem75  46219  fourierdlem84  46228  fourierdlem85  46229  fourierdlem88  46232  fourierdlem103  46247  fourierdlem104  46248