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Theorem fourierdlem3 45124
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 ∧ (𝑝𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
Assertion
Ref Expression
fourierdlem3 (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ ((-π[,]π) ↑m (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (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
StepHypRef Expression
1 oveq2 7419 . . . . . 6 (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
21oveq2d 7427 . . . . 5 (𝑚 = 𝑀 → ((-π[,]π) ↑m (0...𝑚)) = ((-π[,]π) ↑m (0...𝑀)))
3 fveqeq2 6899 . . . . . . 7 (𝑚 = 𝑀 → ((𝑝𝑚) = π ↔ (𝑝𝑀) = π))
43anbi2d 627 . . . . . 6 (𝑚 = 𝑀 → (((𝑝‘0) = -π ∧ (𝑝𝑚) = π) ↔ ((𝑝‘0) = -π ∧ (𝑝𝑀) = π)))
5 oveq2 7419 . . . . . . 7 (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀))
65raleqdv 3323 . . . . . 6 (𝑚 = 𝑀 → (∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))))
74, 6anbi12d 629 . . . . 5 (𝑚 = 𝑀 → ((((𝑝‘0) = -π ∧ (𝑝𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑝‘0) = -π ∧ (𝑝𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))))
82, 7rabeqbidv 3447 . . . 4 (𝑚 = 𝑀 → {𝑝 ∈ ((-π[,]π) ↑m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))} = {𝑝 ∈ ((-π[,]π) ↑m (0...𝑀)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
9 fourierdlem3.1 . . . 4 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ ((-π[,]π) ↑m (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
10 ovex 7444 . . . . 5 ((-π[,]π) ↑m (0...𝑀)) ∈ V
1110rabex 5331 . . . 4 {𝑝 ∈ ((-π[,]π) ↑m (0...𝑀)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))} ∈ V
128, 9, 11fvmpt 6997 . . 3 (𝑀 ∈ ℕ → (𝑃𝑀) = {𝑝 ∈ ((-π[,]π) ↑m (0...𝑀)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
1312eleq2d 2817 . 2 (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃𝑀) ↔ 𝑄 ∈ {𝑝 ∈ ((-π[,]π) ↑m (0...𝑀)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))}))
14 fveq1 6889 . . . . . 6 (𝑝 = 𝑄 → (𝑝‘0) = (𝑄‘0))
1514eqeq1d 2732 . . . . 5 (𝑝 = 𝑄 → ((𝑝‘0) = -π ↔ (𝑄‘0) = -π))
16 fveq1 6889 . . . . . 6 (𝑝 = 𝑄 → (𝑝𝑀) = (𝑄𝑀))
1716eqeq1d 2732 . . . . 5 (𝑝 = 𝑄 → ((𝑝𝑀) = π ↔ (𝑄𝑀) = π))
1815, 17anbi12d 629 . . . 4 (𝑝 = 𝑄 → (((𝑝‘0) = -π ∧ (𝑝𝑀) = π) ↔ ((𝑄‘0) = -π ∧ (𝑄𝑀) = π)))
19 fveq1 6889 . . . . . 6 (𝑝 = 𝑄 → (𝑝𝑖) = (𝑄𝑖))
20 fveq1 6889 . . . . . 6 (𝑝 = 𝑄 → (𝑝‘(𝑖 + 1)) = (𝑄‘(𝑖 + 1)))
2119, 20breq12d 5160 . . . . 5 (𝑝 = 𝑄 → ((𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑄𝑖) < (𝑄‘(𝑖 + 1))))
2221ralbidv 3175 . . . 4 (𝑝 = 𝑄 → (∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))
2318, 22anbi12d 629 . . 3 (𝑝 = 𝑄 → ((((𝑝‘0) = -π ∧ (𝑝𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))) ↔ (((𝑄‘0) = -π ∧ (𝑄𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
2423elrab 3682 . 2 (𝑄 ∈ {𝑝 ∈ ((-π[,]π) ↑m (0...𝑀)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))} ↔ (𝑄 ∈ ((-π[,]π) ↑m (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
2513, 24bitrdi 286 1 (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ ((-π[,]π) ↑m (0...𝑀)) ∧ (((𝑄‘0) = -π ∧ (𝑄𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104  wral 3059  {crab 3430   class class class wbr 5147  cmpt 5230  cfv 6542  (class class class)co 7411  m cmap 8822  0cc0 11112  1c1 11113   + caddc 11115   < clt 11252  -cneg 11449  cn 12216  [,]cicc 13331  ...cfz 13488  ..^cfzo 13631  πcpi 16014
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414
This theorem is referenced by: (None)
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