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Theorem fourierdlem3 42272
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 7153 . . . . . 6 (𝑚 = 𝑀 → (0...𝑚) = (0...𝑀))
21oveq2d 7161 . . . . 5 (𝑚 = 𝑀 → ((-π[,]π) ↑m (0...𝑚)) = ((-π[,]π) ↑m (0...𝑀)))
3 fveqeq2 6672 . . . . . . 7 (𝑚 = 𝑀 → ((𝑝𝑚) = π ↔ (𝑝𝑀) = π))
43anbi2d 628 . . . . . 6 (𝑚 = 𝑀 → (((𝑝‘0) = -π ∧ (𝑝𝑚) = π) ↔ ((𝑝‘0) = -π ∧ (𝑝𝑀) = π)))
5 oveq2 7153 . . . . . . 7 (𝑚 = 𝑀 → (0..^𝑚) = (0..^𝑀))
65raleqdv 3413 . . . . . 6 (𝑚 = 𝑀 → (∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1))))
74, 6anbi12d 630 . . . . 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 3483 . . . 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 7178 . . . . 5 ((-π[,]π) ↑m (0...𝑀)) ∈ V
1110rabex 5226 . . . 4 {𝑝 ∈ ((-π[,]π) ↑m (0...𝑀)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))} ∈ V
128, 9, 11fvmpt 6761 . . 3 (𝑀 ∈ ℕ → (𝑃𝑀) = {𝑝 ∈ ((-π[,]π) ↑m (0...𝑀)) ∣ (((𝑝‘0) = -π ∧ (𝑝𝑀) = π) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
1312eleq2d 2895 . 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 6662 . . . . . 6 (𝑝 = 𝑄 → (𝑝‘0) = (𝑄‘0))
1514eqeq1d 2820 . . . . 5 (𝑝 = 𝑄 → ((𝑝‘0) = -π ↔ (𝑄‘0) = -π))
16 fveq1 6662 . . . . . 6 (𝑝 = 𝑄 → (𝑝𝑀) = (𝑄𝑀))
1716eqeq1d 2820 . . . . 5 (𝑝 = 𝑄 → ((𝑝𝑀) = π ↔ (𝑄𝑀) = π))
1815, 17anbi12d 630 . . . 4 (𝑝 = 𝑄 → (((𝑝‘0) = -π ∧ (𝑝𝑀) = π) ↔ ((𝑄‘0) = -π ∧ (𝑄𝑀) = π)))
19 fveq1 6662 . . . . . 6 (𝑝 = 𝑄 → (𝑝𝑖) = (𝑄𝑖))
20 fveq1 6662 . . . . . 6 (𝑝 = 𝑄 → (𝑝‘(𝑖 + 1)) = (𝑄‘(𝑖 + 1)))
2119, 20breq12d 5070 . . . . 5 (𝑝 = 𝑄 → ((𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ (𝑄𝑖) < (𝑄‘(𝑖 + 1))))
2221ralbidv 3194 . . . 4 (𝑝 = 𝑄 → (∀𝑖 ∈ (0..^𝑀)(𝑝𝑖) < (𝑝‘(𝑖 + 1)) ↔ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))
2318, 22anbi12d 630 . . 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 3677 . 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, 24syl6bb 288 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 207  wa 396   = wceq 1528  wcel 2105  wral 3135  {crab 3139   class class class wbr 5057  cmpt 5137  cfv 6348  (class class class)co 7145  m cmap 8395  0cc0 10525  1c1 10526   + caddc 10528   < clt 10663  -cneg 10859  cn 11626  [,]cicc 12729  ...cfz 12880  ..^cfzo 13021  πcpi 15408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148
This theorem is referenced by: (None)
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