Theorem fourierdlem3 47393

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 7459	. . . . . 6 ⊢ (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
2	1	oveq2d 7467	. . . . 5 ⊢ (𝑚 = 𝑀 → (( -π[,]π) ↑m (0...𝑚)) = (( -π[,]π) ↑m (0...𝑀)))
3		fveqeq2 6925	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ((𝑝‘𝑚) = π ↔ (𝑝‘𝑀) = π))
4	3	anbi2d 642	. . . . . 6 ⊢ (𝑚 = 𝑀 → (((𝑝‘0) = -π ∧ (𝑝‘𝑚) = π) ↔ ((𝑝‘0) = -π ∧ (𝑝‘𝑀) = π)))
5		oveq2 7459	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀))
6	5	raleqdv 3323	. . . . . 6 ⊢ (𝑚 = 𝑀 → (∀𝑖 ∈ (0..^𝑚)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1))))
7	4, 6	anbi12d 644	. . . . 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 3433	. . . 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 7484	. . . . 5 ⊢ (( -π[,]π) ↑m (0...𝑀)) ∈ V
11	10	rabex 5307	. . . 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 7024	. . 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 2850	. 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 6915	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘0) = (𝑄‘0))
15	14	eqeq1d 2765	. . . . 5 ⊢ (𝑝 = 𝑄 → ((𝑝‘0) = -π ↔ (𝑄‘0) = -π))
16		fveq1 6915	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘𝑀) = (𝑄‘𝑀))
17	16	eqeq1d 2765	. . . . 5 ⊢ (𝑝 = 𝑄 → ((𝑝‘𝑀) = π ↔ (𝑄‘𝑀) = π))
18	15, 17	anbi12d 644	. . . 4 ⊢ (𝑝 = 𝑄 → (((𝑝‘0) = -π ∧ (𝑝‘𝑀) = π) ↔ ((𝑄‘0) = -π ∧ (𝑄‘𝑀) = π)))
19		fveq1 6915	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘𝑖) = (𝑄‘𝑖))
20		fveq1 6915	. . . . . 6 ⊢ (𝑝 = 𝑄 → (𝑝‘(𝑖 + 1)) = (𝑄‘(𝑖 + 1)))
21	19, 20	breq12d 5121	. . . . 5 ⊢ (𝑝 = 𝑄 → ((𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
22	21	ralbidv 3189	. . . 4 ⊢ (𝑝 = 𝑄 → (∀𝑖 ∈ (0..^𝑀)(𝑝‘𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑄‘𝑖) < (𝑄‘(𝑖 + 1))))
23	18, 22	anbi12d 644	. . 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 3648	. 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 290	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))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416   class class class wbr 5108   ↦ cmpt 5192  ‘cfv 6566  (class class class)co 7451   ↑_m cmap 8885  0cc0 11239  1c1 11240   + caddc 11242   < clt 11382   -cneg 11581  ℕcn 12375  [,]cicc 13519  ...cfz 13679  ..^cfzo 13828  πcpi 16276

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454

This theorem is used by: (None)