Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fourierdlem14 Structured version   Visualization version   GIF version

Theorem fourierdlem14 40810
Description: Given the partition 𝑉, 𝑄 is the partition shifted to the left by 𝑋. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem14.1 (𝜑𝐴 ∈ ℝ)
fourierdlem14.2 (𝜑𝐵 ∈ ℝ)
fourierdlem14.x (𝜑𝑋 ∈ ℝ)
fourierdlem14.p 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑋) ∧ (𝑝𝑚) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem14.o 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem14.m (𝜑𝑀 ∈ ℕ)
fourierdlem14.v (𝜑𝑉 ∈ (𝑃𝑀))
fourierdlem14.q 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))
Assertion
Ref Expression
fourierdlem14 (𝜑𝑄 ∈ (𝑂𝑀))
Distinct variable groups:   𝐴,𝑚,𝑝   𝐵,𝑚,𝑝   𝑖,𝑀,𝑚,𝑝   𝑄,𝑖,𝑝   𝑖,𝑉,𝑝   𝑖,𝑋,𝑚,𝑝   𝜑,𝑖
Allowed substitution hints:   𝜑(𝑚,𝑝)   𝐴(𝑖)   𝐵(𝑖)   𝑃(𝑖,𝑚,𝑝)   𝑄(𝑚)   𝑂(𝑖,𝑚,𝑝)   𝑉(𝑚)

Proof of Theorem fourierdlem14
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 fourierdlem14.v . . . . . . . . . 10 (𝜑𝑉 ∈ (𝑃𝑀))
2 fourierdlem14.m . . . . . . . . . . 11 (𝜑𝑀 ∈ ℕ)
3 fourierdlem14.p . . . . . . . . . . . 12 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = (𝐴 + 𝑋) ∧ (𝑝𝑚) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
43fourierdlem2 40798 . . . . . . . . . . 11 (𝑀 ∈ ℕ → (𝑉 ∈ (𝑃𝑀) ↔ (𝑉 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉𝑀) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1))))))
52, 4syl 17 . . . . . . . . . 10 (𝜑 → (𝑉 ∈ (𝑃𝑀) ↔ (𝑉 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉𝑀) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1))))))
61, 5mpbid 222 . . . . . . . . 9 (𝜑 → (𝑉 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉𝑀) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1)))))
76simpld 477 . . . . . . . 8 (𝜑𝑉 ∈ (ℝ ↑𝑚 (0...𝑀)))
8 elmapi 8033 . . . . . . . 8 (𝑉 ∈ (ℝ ↑𝑚 (0...𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
97, 8syl 17 . . . . . . 7 (𝜑𝑉:(0...𝑀)⟶ℝ)
109ffvelrnda 6510 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑉𝑖) ∈ ℝ)
11 fourierdlem14.x . . . . . . 7 (𝜑𝑋 ∈ ℝ)
1211adantr 472 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑋 ∈ ℝ)
1310, 12resubcld 10621 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑉𝑖) − 𝑋) ∈ ℝ)
14 fourierdlem14.q . . . . 5 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))
1513, 14fmptd 6536 . . . 4 (𝜑𝑄:(0...𝑀)⟶ℝ)
16 reex 10190 . . . . . 6 ℝ ∈ V
1716a1i 11 . . . . 5 (𝜑 → ℝ ∈ V)
18 ovex 6829 . . . . . 6 (0...𝑀) ∈ V
1918a1i 11 . . . . 5 (𝜑 → (0...𝑀) ∈ V)
2017, 19elmapd 8025 . . . 4 (𝜑 → (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ))
2115, 20mpbird 247 . . 3 (𝜑𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)))
2214a1i 11 . . . . . 6 (𝜑𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋)))
23 fveq2 6340 . . . . . . . 8 (𝑖 = 0 → (𝑉𝑖) = (𝑉‘0))
2423oveq1d 6816 . . . . . . 7 (𝑖 = 0 → ((𝑉𝑖) − 𝑋) = ((𝑉‘0) − 𝑋))
2524adantl 473 . . . . . 6 ((𝜑𝑖 = 0) → ((𝑉𝑖) − 𝑋) = ((𝑉‘0) − 𝑋))
26 0zd 11552 . . . . . . . . 9 (𝜑 → 0 ∈ ℤ)
272nnzd 11644 . . . . . . . . 9 (𝜑𝑀 ∈ ℤ)
2826, 27, 263jca 1403 . . . . . . . 8 (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ))
29 0le0 11273 . . . . . . . . 9 0 ≤ 0
3029a1i 11 . . . . . . . 8 (𝜑 → 0 ≤ 0)
31 0red 10204 . . . . . . . . 9 (𝜑 → 0 ∈ ℝ)
322nnred 11198 . . . . . . . . 9 (𝜑𝑀 ∈ ℝ)
332nngt0d 11227 . . . . . . . . 9 (𝜑 → 0 < 𝑀)
3431, 32, 33ltled 10348 . . . . . . . 8 (𝜑 → 0 ≤ 𝑀)
3528, 30, 34jca32 559 . . . . . . 7 (𝜑 → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (0 ≤ 0 ∧ 0 ≤ 𝑀)))
36 elfz2 12497 . . . . . . 7 (0 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (0 ≤ 0 ∧ 0 ≤ 𝑀)))
3735, 36sylibr 224 . . . . . 6 (𝜑 → 0 ∈ (0...𝑀))
389, 37ffvelrnd 6511 . . . . . . 7 (𝜑 → (𝑉‘0) ∈ ℝ)
3938, 11resubcld 10621 . . . . . 6 (𝜑 → ((𝑉‘0) − 𝑋) ∈ ℝ)
4022, 25, 37, 39fvmptd 6438 . . . . 5 (𝜑 → (𝑄‘0) = ((𝑉‘0) − 𝑋))
416simprd 482 . . . . . . . 8 (𝜑 → (((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉𝑀) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1))))
4241simpld 477 . . . . . . 7 (𝜑 → ((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉𝑀) = (𝐵 + 𝑋)))
4342simpld 477 . . . . . 6 (𝜑 → (𝑉‘0) = (𝐴 + 𝑋))
4443oveq1d 6816 . . . . 5 (𝜑 → ((𝑉‘0) − 𝑋) = ((𝐴 + 𝑋) − 𝑋))
45 fourierdlem14.1 . . . . . . 7 (𝜑𝐴 ∈ ℝ)
4645recnd 10231 . . . . . 6 (𝜑𝐴 ∈ ℂ)
4711recnd 10231 . . . . . 6 (𝜑𝑋 ∈ ℂ)
4846, 47pncand 10556 . . . . 5 (𝜑 → ((𝐴 + 𝑋) − 𝑋) = 𝐴)
4940, 44, 483eqtrd 2786 . . . 4 (𝜑 → (𝑄‘0) = 𝐴)
50 fveq2 6340 . . . . . . . 8 (𝑖 = 𝑀 → (𝑉𝑖) = (𝑉𝑀))
5150oveq1d 6816 . . . . . . 7 (𝑖 = 𝑀 → ((𝑉𝑖) − 𝑋) = ((𝑉𝑀) − 𝑋))
5251adantl 473 . . . . . 6 ((𝜑𝑖 = 𝑀) → ((𝑉𝑖) − 𝑋) = ((𝑉𝑀) − 𝑋))
5326, 27, 273jca 1403 . . . . . . . 8 (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ))
5432leidd 10757 . . . . . . . 8 (𝜑𝑀𝑀)
5553, 34, 54jca32 559 . . . . . . 7 (𝜑 → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (0 ≤ 𝑀𝑀𝑀)))
56 elfz2 12497 . . . . . . 7 (𝑀 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (0 ≤ 𝑀𝑀𝑀)))
5755, 56sylibr 224 . . . . . 6 (𝜑𝑀 ∈ (0...𝑀))
589, 57ffvelrnd 6511 . . . . . . 7 (𝜑 → (𝑉𝑀) ∈ ℝ)
5958, 11resubcld 10621 . . . . . 6 (𝜑 → ((𝑉𝑀) − 𝑋) ∈ ℝ)
6022, 52, 57, 59fvmptd 6438 . . . . 5 (𝜑 → (𝑄𝑀) = ((𝑉𝑀) − 𝑋))
6142simprd 482 . . . . . 6 (𝜑 → (𝑉𝑀) = (𝐵 + 𝑋))
6261oveq1d 6816 . . . . 5 (𝜑 → ((𝑉𝑀) − 𝑋) = ((𝐵 + 𝑋) − 𝑋))
63 fourierdlem14.2 . . . . . . 7 (𝜑𝐵 ∈ ℝ)
6463recnd 10231 . . . . . 6 (𝜑𝐵 ∈ ℂ)
6564, 47pncand 10556 . . . . 5 (𝜑 → ((𝐵 + 𝑋) − 𝑋) = 𝐵)
6660, 62, 653eqtrd 2786 . . . 4 (𝜑 → (𝑄𝑀) = 𝐵)
6749, 66jca 555 . . 3 (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵))
68 elfzofz 12650 . . . . . . 7 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
6968, 10sylan2 492 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉𝑖) ∈ ℝ)
709adantr 472 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
71 fzofzp1 12730 . . . . . . . 8 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
7271adantl 473 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
7370, 72ffvelrnd 6511 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
7411adantr 472 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
7541simprd 482 . . . . . . 7 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1)))
7675r19.21bi 3058 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉𝑖) < (𝑉‘(𝑖 + 1)))
7769, 73, 74, 76ltsub1dd 10802 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉𝑖) − 𝑋) < ((𝑉‘(𝑖 + 1)) − 𝑋))
7868adantl 473 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
7968, 13sylan2 492 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉𝑖) − 𝑋) ∈ ℝ)
8014fvmpt2 6441 . . . . . 6 ((𝑖 ∈ (0...𝑀) ∧ ((𝑉𝑖) − 𝑋) ∈ ℝ) → (𝑄𝑖) = ((𝑉𝑖) − 𝑋))
8178, 79, 80syl2anc 696 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) = ((𝑉𝑖) − 𝑋))
82 fveq2 6340 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑉𝑖) = (𝑉𝑗))
8382oveq1d 6816 . . . . . . . . 9 (𝑖 = 𝑗 → ((𝑉𝑖) − 𝑋) = ((𝑉𝑗) − 𝑋))
8483cbvmptv 4890 . . . . . . . 8 (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑉𝑗) − 𝑋))
8514, 84eqtri 2770 . . . . . . 7 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉𝑗) − 𝑋))
8685a1i 11 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉𝑗) − 𝑋)))
87 fveq2 6340 . . . . . . . 8 (𝑗 = (𝑖 + 1) → (𝑉𝑗) = (𝑉‘(𝑖 + 1)))
8887oveq1d 6816 . . . . . . 7 (𝑗 = (𝑖 + 1) → ((𝑉𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
8988adantl 473 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑉𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
9073, 74resubcld 10621 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
9186, 89, 72, 90fvmptd 6438 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
9277, 81, 913brtr4d 4824 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
9392ralrimiva 3092 . . 3 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
9421, 67, 93jca32 559 . 2 (𝜑 → (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
95 fourierdlem14.o . . . 4 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑𝑚 (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
9695fourierdlem2 40798 . . 3 (𝑀 ∈ ℕ → (𝑄 ∈ (𝑂𝑀) ↔ (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
972, 96syl 17 . 2 (𝜑 → (𝑄 ∈ (𝑂𝑀) ↔ (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
9894, 97mpbird 247 1 (𝜑𝑄 ∈ (𝑂𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1620  wcel 2127  wral 3038  {crab 3042  Vcvv 3328   class class class wbr 4792  cmpt 4869  wf 6033  cfv 6037  (class class class)co 6801  𝑚 cmap 8011  cr 10098  0cc0 10099  1c1 10100   + caddc 10102   < clt 10237  cle 10238  cmin 10429  cn 11183  cz 11540  ...cfz 12490  ..^cfzo 12630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-cnex 10155  ax-resscn 10156  ax-1cn 10157  ax-icn 10158  ax-addcl 10159  ax-addrcl 10160  ax-mulcl 10161  ax-mulrcl 10162  ax-mulcom 10163  ax-addass 10164  ax-mulass 10165  ax-distr 10166  ax-i2m1 10167  ax-1ne0 10168  ax-1rid 10169  ax-rnegex 10170  ax-rrecex 10171  ax-cnre 10172  ax-pre-lttri 10173  ax-pre-lttrn 10174  ax-pre-ltadd 10175  ax-pre-mulgt0 10176
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-nel 3024  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-riota 6762  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-om 7219  df-1st 7321  df-2nd 7322  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-er 7899  df-map 8013  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10239  df-mnf 10240  df-xr 10241  df-ltxr 10242  df-le 10243  df-sub 10431  df-neg 10432  df-nn 11184  df-n0 11456  df-z 11541  df-uz 11851  df-fz 12491  df-fzo 12631
This theorem is referenced by:  fourierdlem74  40869  fourierdlem75  40870  fourierdlem84  40879  fourierdlem85  40880  fourierdlem88  40883  fourierdlem103  40898  fourierdlem104  40899
  Copyright terms: Public domain W3C validator