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Theorem fourierdlem3 40645
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 6698 . . . . . 6 (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
21oveq2d 6706 . . . . 5 (𝑚 = 𝑀 → ((-π[,]π) ↑𝑚 (0...𝑚)) = ((-π[,]π) ↑𝑚 (0...𝑀)))
3 fveq2 6229 . . . . . . . 8 (𝑚 = 𝑀 → (𝑝𝑚) = (𝑝𝑀))
43eqeq1d 2653 . . . . . . 7 (𝑚 = 𝑀 → ((𝑝𝑚) = π ↔ (𝑝𝑀) = π))
54anbi2d 740 . . . . . 6 (𝑚 = 𝑀 → (((𝑝‘0) = -π ∧ (𝑝𝑚) = π) ↔ ((𝑝‘0) = -π ∧ (𝑝𝑀) = π)))
6 oveq2 6698 . . . . . . 7 (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀))
76raleqdv 3174 . . . . . 6 (𝑚 = 𝑀 → (∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))))
85, 7anbi12d 747 . . . . 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)))))
92, 8rabeqbidv 3226 . . . 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)))})
10 fourierdlem3.1 . . . 4 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ ((-π[,]π) ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑚) = π) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
11 ovex 6718 . . . . 5 ((-π[,]π) ↑𝑚 (0...𝑀)) ∈ V
1211rabex 4845 . . . 4 {𝑝 ∈ ((-π[,]π) ↑𝑚 (0...𝑀)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))} ∈ V
139, 10, 12fvmpt 6321 . . 3 (𝑀 ∈ ℕ → (𝑃𝑀) = {𝑝 ∈ ((-π[,]π) ↑𝑚 (0...𝑀)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
1413eleq2d 2716 . 2 (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃𝑀) ↔ 𝑄 ∈ {𝑝 ∈ ((-π[,]π) ↑𝑚 (0...𝑀)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))}))
15 fveq1 6228 . . . . . 6 (𝑝 = 𝑄 → (𝑝‘0) = (𝑄‘0))
1615eqeq1d 2653 . . . . 5 (𝑝 = 𝑄 → ((𝑝‘0) = -π ↔ (𝑄‘0) = -π))
17 fveq1 6228 . . . . . 6 (𝑝 = 𝑄 → (𝑝𝑀) = (𝑄𝑀))
1817eqeq1d 2653 . . . . 5 (𝑝 = 𝑄 → ((𝑝𝑀) = π ↔ (𝑄𝑀) = π))
1916, 18anbi12d 747 . . . 4 (𝑝 = 𝑄 → (((𝑝‘0) = -π ∧ (𝑝𝑀) = π) ↔ ((𝑄‘0) = -π ∧ (𝑄𝑀) = π)))
20 fveq1 6228 . . . . . 6 (𝑝 = 𝑄 → (𝑝𝑖) = (𝑄𝑖))
21 fveq1 6228 . . . . . 6 (𝑝 = 𝑄 → (𝑝‘(𝑖 + 1)) = (𝑄‘(𝑖 + 1)))
2220, 21breq12d 4698 . . . . 5 (𝑝 = 𝑄 → ((𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑄𝑖) < (𝑄‘(𝑖 + 1))))
2322ralbidv 3015 . . . 4 (𝑝 = 𝑄 → (∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))
2419, 23anbi12d 747 . . 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)))))
2524elrab 3396 . 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)))))
2614, 25syl6bb 276 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 196  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945   class class class wbr 4685  cmpt 4762  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  0cc0 9974  1c1 9975   + caddc 9977   < clt 10112  -cneg 10305  cn 11058  [,]cicc 12216  ...cfz 12364  ..^cfzo 12504  πcpi 14841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693
This theorem is referenced by: (None)
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