Theorem fourierdlem3 45336

Description: Membership in a partition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypothesis

Ref	Expression
fourierdlem3.1	⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ ((-π[,]π) ↑m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})

Assertion

Ref	Expression
fourierdlem3	⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ ((-π[,]π) ↑m (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))

Distinct variable groups:   𝑖,𝑀,𝑚,𝑝   𝑄,𝑖,𝑝

Allowed substitution hints:   𝑃(𝑖,𝑚,𝑝)   𝑄(𝑚)

Proof of Theorem fourierdlem3

Step	Hyp	Ref	Expression
1		oveq2 7410	. . . . . 6 ⊢ (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
2	1	oveq2d 7418	. . . . 5 ⊢ (𝑚 = 𝑀 → ((-π[,]π) ↑m (0...𝑚)) = ((-π[,]π) ↑m (0...𝑀)))
3		fveqeq2 6891	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ((𝑝‘𝑚) = π ↔ (𝑝‘𝑀) = π))
4	3	anbi2d 628	. . . . . 6 ⊢ (𝑚 = 𝑀 → (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ↔ ((𝑝‘0) = -π ∧ (𝑝‘𝑀) = π)))
5		oveq2 7410	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀))
6	5	raleqdv 3317	. . . . . 6 ⊢ (𝑚 = 𝑀 → (∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))))
7	4, 6	anbi12d 630	. . . . 5 ⊢ (𝑚 = 𝑀 → ((((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑝‘0) = -π ∧ (𝑝‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))))
8	2, 7	rabeqbidv 3441	. . . 4 ⊢ (𝑚 = 𝑀 → {𝑝 ∈ ((-π[,]π) ↑m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))} = {𝑝 ∈ ((-π[,]π) ↑m (0...𝑀)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
9		fourierdlem3.1	. . . 4 ⊢ 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ ((-π[,]π) ↑m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
10		ovex 7435	. . . . 5 ⊢ ((-π[,]π) ↑m (0...𝑀)) ∈ V
11	10	rabex 5323	. . . 4 ⊢ {𝑝 ∈ ((-π[,]π) ↑m (0...𝑀)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))} ∈ V
12	8, 9, 11	fvmpt 6989	. . 3 ⊢ (𝑀 ∈ ℕ → (𝑃‘𝑀) = {𝑝 ∈ ((-π[,]π) ↑m (0...𝑀)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))})
13	12	eleq2d 2811	. 2 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ 𝑄 ∈ {𝑝 ∈ ((-π[,]π) ↑m (0...𝑀)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))}))
14		fveq1 6881	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘0) = (𝑄‘0))
15	14	eqeq1d 2726	. . . . 5 ⊢ (𝑝 = 𝑄 → ((𝑝‘0) = -π ↔ (𝑄‘0) = -π))
16		fveq1 6881	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘𝑀) = (𝑄‘𝑀))
17	16	eqeq1d 2726	. . . . 5 ⊢ (𝑝 = 𝑄 → ((𝑝‘𝑀) = π ↔ (𝑄‘𝑀) = π))
18	15, 17	anbi12d 630	. . . 4 ⊢ (𝑝 = 𝑄 → (((𝑝‘0) = -π ∧ (𝑝‘𝑀) = π) ↔ ((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π)))
19		fveq1 6881	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘𝑖) = (𝑄‘𝑖))
20		fveq1 6881	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘(𝑖 + 1)) = (𝑄‘(𝑖 + 1)))
21	19, 20	breq12d 5152	. . . . 5 ⊢ (𝑝 = 𝑄 → ((𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
22	21	ralbidv 3169	. . . 4 ⊢ (𝑝 = 𝑄 → (∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
23	18, 22	anbi12d 630	. . 3 ⊢ (𝑝 = 𝑄 → ((((𝑝‘0) = -π ∧ (𝑝‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
24	23	elrab 3676	. 2 ⊢ (𝑄 ∈ {𝑝 ∈ ((-π[,]π) ↑m (0...𝑀)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))} ↔ (𝑄 ∈ ((-π[,]π) ↑m (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1)))))
25	13, 24	bitrdi 287	1 ⊢ (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃‘𝑀) ↔ (𝑄 ∈ ((-π[,]π) ↑m (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3053  {crab 3424   class class class wbr 5139   ↦ cmpt 5222  ‘cfv 6534  (class class class)co 7402   ↑_m cmap 8817  0cc0 11107  1c1 11108   + caddc 11110   < clt 11246  -cneg 11443  ℕcn 12210  [,]cicc 13325  ...cfz 13482  ..^cfzo 13625  πcpi 16008

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-ov 7405

This theorem is referenced by: (None)