Theorem fourierdlem3 46095

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 7361	. . . . . 6 ⊢ (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
2	1	oveq2d 7369	. . . . 5 ⊢ (𝑚 = 𝑀 → ((-π[,]π) ↑m (0...𝑚)) = ((-π[,]π) ↑m (0...𝑀)))
3		fveqeq2 6835	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ((𝑝‘𝑚) = π ↔ (𝑝‘𝑀) = π))
4	3	anbi2d 630	. . . . . 6 ⊢ (𝑚 = 𝑀 → (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ↔ ((𝑝‘0) = -π ∧ (𝑝‘𝑀) = π)))
5		oveq2 7361	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀))
6	5	raleqdv 3290	. . . . . 6 ⊢ (𝑚 = 𝑀 → (∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))))
7	4, 6	anbi12d 632	. . . . 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 3415	. . . 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 7386	. . . . 5 ⊢ ((-π[,]π) ↑m (0...𝑀)) ∈ V
11	10	rabex 5281	. . . 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 6934	. . 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 2814	. 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 6825	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘0) = (𝑄‘0))
15	14	eqeq1d 2731	. . . . 5 ⊢ (𝑝 = 𝑄 → ((𝑝‘0) = -π ↔ (𝑄‘0) = -π))
16		fveq1 6825	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘𝑀) = (𝑄‘𝑀))
17	16	eqeq1d 2731	. . . . 5 ⊢ (𝑝 = 𝑄 → ((𝑝‘𝑀) = π ↔ (𝑄‘𝑀) = π))
18	15, 17	anbi12d 632	. . . 4 ⊢ (𝑝 = 𝑄 → (((𝑝‘0) = -π ∧ (𝑝‘𝑀) = π) ↔ ((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π)))
19		fveq1 6825	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘𝑖) = (𝑄‘𝑖))
20		fveq1 6825	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘(𝑖 + 1)) = (𝑄‘(𝑖 + 1)))
21	19, 20	breq12d 5108	. . . . 5 ⊢ (𝑝 = 𝑄 → ((𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
22	21	ralbidv 3152	. . . 4 ⊢ (𝑝 = 𝑄 → (∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
23	18, 22	anbi12d 632	. . 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 3650	. 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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3396   class class class wbr 5095   ↦ cmpt 5176  ‘cfv 6486  (class class class)co 7353   ↑_m cmap 8760  0cc0 11028  1c1 11029   + caddc 11031   < clt 11168  -cneg 11366  ℕcn 12146  [,]cicc 13269  ...cfz 13428  ..^cfzo 13575  πcpi 15991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356

This theorem is referenced by: (None)