Theorem fourierdlem3 39003

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 6557	. . . . . 6 ⊢ (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
2	1	oveq2d 6565	. . . . 5 ⊢ (𝑚 = 𝑀 → ((-π[,]π) ↑𝑚 (0...𝑚)) = ((-π[,]π) ↑𝑚 (0...𝑀)))
3		fveq2 6103	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑝‘𝑚) = (𝑝‘𝑀))
4	3	eqeq1d 2612	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ((𝑝‘𝑚) = π ↔ (𝑝‘𝑀) = π))
5	4	anbi2d 736	. . . . . 6 ⊢ (𝑚 = 𝑀 → (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ↔ ((𝑝‘0) = -π ∧ (𝑝‘𝑀) = π)))
6		oveq2 6557	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀))
7	6	raleqdv 3121	. . . . . 6 ⊢ (𝑚 = 𝑀 → (∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))))
8	5, 7	anbi12d 743	. . . . 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)))))
9	2, 8	rabeqbidv 3168	. . . 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)))})
10		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)))})
11		ovex 6577	. . . . 5 ⊢ ((-π[,]π) ↑𝑚 (0...𝑀)) ∈ V
12	11	rabex 4740	. . . 4 ⊢ {𝑝 ∈ ((-π[,]π) ↑𝑚 (0...𝑀)) ∣ (((𝑝‘0) = -π ∧ (𝑝‘𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)))} ∈ V
13	9, 10, 12	fvmpt 6191	. . 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)))})
14	13	eleq2d 2673	. 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)))}))
15		fveq1 6102	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘0) = (𝑄‘0))
16	15	eqeq1d 2612	. . . . 5 ⊢ (𝑝 = 𝑄 → ((𝑝‘0) = -π ↔ (𝑄‘0) = -π))
17		fveq1 6102	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘𝑀) = (𝑄‘𝑀))
18	17	eqeq1d 2612	. . . . 5 ⊢ (𝑝 = 𝑄 → ((𝑝‘𝑀) = π ↔ (𝑄‘𝑀) = π))
19	16, 18	anbi12d 743	. . . 4 ⊢ (𝑝 = 𝑄 → (((𝑝‘0) = -π ∧ (𝑝‘𝑀) = π) ↔ ((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π)))
20		fveq1 6102	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘𝑖) = (𝑄‘𝑖))
21		fveq1 6102	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘(𝑖 + 1)) = (𝑄‘(𝑖 + 1)))
22	20, 21	breq12d 4596	. . . . 5 ⊢ (𝑝 = 𝑄 → ((𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
23	22	ralbidv 2969	. . . 4 ⊢ (𝑝 = 𝑄 → (∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
24	19, 23	anbi12d 743	. . 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)))))
25	24	elrab 3331	. 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)))))
26	14, 25	syl6bb 275	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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953  -cneg 10146  ℕcn 10897  [,]cicc 12049  ...cfz 12197  ..^cfzo 12334  πcpi 14636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552

This theorem is referenced by: (None)