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

Theorem fourierdlem14 39014
 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 39002 . . . . . . . . . . 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 221 . . . . . . . . 9 (𝜑 → (𝑉 ∈ (ℝ ↑𝑚 (0...𝑀)) ∧ (((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉𝑀) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1)))))
76simpld 474 . . . . . . . 8 (𝜑𝑉 ∈ (ℝ ↑𝑚 (0...𝑀)))
8 elmapi 7765 . . . . . . . 8 (𝑉 ∈ (ℝ ↑𝑚 (0...𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
97, 8syl 17 . . . . . . 7 (𝜑𝑉:(0...𝑀)⟶ℝ)
109fnvinran 38196 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑉𝑖) ∈ ℝ)
11 fourierdlem14.x . . . . . . 7 (𝜑𝑋 ∈ ℝ)
1211adantr 480 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑋 ∈ ℝ)
1310, 12resubcld 10337 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑉𝑖) − 𝑋) ∈ ℝ)
14 fourierdlem14.q . . . . 5 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))
1513, 14fmptd 6292 . . . 4 (𝜑𝑄:(0...𝑀)⟶ℝ)
16 reex 9906 . . . . . 6 ℝ ∈ V
1716a1i 11 . . . . 5 (𝜑 → ℝ ∈ V)
18 ovex 6577 . . . . . 6 (0...𝑀) ∈ V
1918a1i 11 . . . . 5 (𝜑 → (0...𝑀) ∈ V)
2017, 19elmapd 7758 . . . 4 (𝜑 → (𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ))
2115, 20mpbird 246 . . 3 (𝜑𝑄 ∈ (ℝ ↑𝑚 (0...𝑀)))
2214a1i 11 . . . . . 6 (𝜑𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋)))
23 fveq2 6103 . . . . . . . 8 (𝑖 = 0 → (𝑉𝑖) = (𝑉‘0))
2423oveq1d 6564 . . . . . . 7 (𝑖 = 0 → ((𝑉𝑖) − 𝑋) = ((𝑉‘0) − 𝑋))
2524adantl 481 . . . . . 6 ((𝜑𝑖 = 0) → ((𝑉𝑖) − 𝑋) = ((𝑉‘0) − 𝑋))
26 0zd 11266 . . . . . . . . 9 (𝜑 → 0 ∈ ℤ)
272nnzd 11357 . . . . . . . . 9 (𝜑𝑀 ∈ ℤ)
2826, 27, 263jca 1235 . . . . . . . 8 (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ))
29 0le0 10987 . . . . . . . . 9 0 ≤ 0
3029a1i 11 . . . . . . . 8 (𝜑 → 0 ≤ 0)
31 0red 9920 . . . . . . . . 9 (𝜑 → 0 ∈ ℝ)
322nnred 10912 . . . . . . . . 9 (𝜑𝑀 ∈ ℝ)
332nngt0d 10941 . . . . . . . . 9 (𝜑 → 0 < 𝑀)
3431, 32, 33ltled 10064 . . . . . . . 8 (𝜑 → 0 ≤ 𝑀)
3528, 30, 34jca32 556 . . . . . . 7 (𝜑 → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (0 ≤ 0 ∧ 0 ≤ 𝑀)))
36 elfz2 12204 . . . . . . 7 (0 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (0 ≤ 0 ∧ 0 ≤ 𝑀)))
3735, 36sylibr 223 . . . . . 6 (𝜑 → 0 ∈ (0...𝑀))
389, 37ffvelrnd 6268 . . . . . . 7 (𝜑 → (𝑉‘0) ∈ ℝ)
3938, 11resubcld 10337 . . . . . 6 (𝜑 → ((𝑉‘0) − 𝑋) ∈ ℝ)
4022, 25, 37, 39fvmptd 6197 . . . . 5 (𝜑 → (𝑄‘0) = ((𝑉‘0) − 𝑋))
416simprd 478 . . . . . . . 8 (𝜑 → (((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉𝑀) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1))))
4241simpld 474 . . . . . . 7 (𝜑 → ((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉𝑀) = (𝐵 + 𝑋)))
4342simpld 474 . . . . . 6 (𝜑 → (𝑉‘0) = (𝐴 + 𝑋))
4443oveq1d 6564 . . . . 5 (𝜑 → ((𝑉‘0) − 𝑋) = ((𝐴 + 𝑋) − 𝑋))
45 fourierdlem14.1 . . . . . . 7 (𝜑𝐴 ∈ ℝ)
4645recnd 9947 . . . . . 6 (𝜑𝐴 ∈ ℂ)
4711recnd 9947 . . . . . 6 (𝜑𝑋 ∈ ℂ)
4846, 47pncand 10272 . . . . 5 (𝜑 → ((𝐴 + 𝑋) − 𝑋) = 𝐴)
4940, 44, 483eqtrd 2648 . . . 4 (𝜑 → (𝑄‘0) = 𝐴)
50 fveq2 6103 . . . . . . . 8 (𝑖 = 𝑀 → (𝑉𝑖) = (𝑉𝑀))
5150oveq1d 6564 . . . . . . 7 (𝑖 = 𝑀 → ((𝑉𝑖) − 𝑋) = ((𝑉𝑀) − 𝑋))
5251adantl 481 . . . . . 6 ((𝜑𝑖 = 𝑀) → ((𝑉𝑖) − 𝑋) = ((𝑉𝑀) − 𝑋))
5326, 27, 273jca 1235 . . . . . . . 8 (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ))
5432leidd 10473 . . . . . . . 8 (𝜑𝑀𝑀)
5553, 34, 54jca32 556 . . . . . . 7 (𝜑 → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (0 ≤ 𝑀𝑀𝑀)))
56 elfz2 12204 . . . . . . 7 (𝑀 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (0 ≤ 𝑀𝑀𝑀)))
5755, 56sylibr 223 . . . . . 6 (𝜑𝑀 ∈ (0...𝑀))
589, 57ffvelrnd 6268 . . . . . . 7 (𝜑 → (𝑉𝑀) ∈ ℝ)
5958, 11resubcld 10337 . . . . . 6 (𝜑 → ((𝑉𝑀) − 𝑋) ∈ ℝ)
6022, 52, 57, 59fvmptd 6197 . . . . 5 (𝜑 → (𝑄𝑀) = ((𝑉𝑀) − 𝑋))
6142simprd 478 . . . . . 6 (𝜑 → (𝑉𝑀) = (𝐵 + 𝑋))
6261oveq1d 6564 . . . . 5 (𝜑 → ((𝑉𝑀) − 𝑋) = ((𝐵 + 𝑋) − 𝑋))
63 fourierdlem14.2 . . . . . . 7 (𝜑𝐵 ∈ ℝ)
6463recnd 9947 . . . . . 6 (𝜑𝐵 ∈ ℂ)
6564, 47pncand 10272 . . . . 5 (𝜑 → ((𝐵 + 𝑋) − 𝑋) = 𝐵)
6660, 62, 653eqtrd 2648 . . . 4 (𝜑 → (𝑄𝑀) = 𝐵)
6749, 66jca 553 . . 3 (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵))
68 elfzofz 12354 . . . . . . 7 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
6968, 10sylan2 490 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉𝑖) ∈ ℝ)
709adantr 480 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
71 fzofzp1 12431 . . . . . . . 8 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
7271adantl 481 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
7370, 72ffvelrnd 6268 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
7411adantr 480 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
7541simprd 478 . . . . . . 7 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1)))
7675r19.21bi 2916 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉𝑖) < (𝑉‘(𝑖 + 1)))
7769, 73, 74, 76ltsub1dd 10518 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉𝑖) − 𝑋) < ((𝑉‘(𝑖 + 1)) − 𝑋))
7868adantl 481 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
7968, 13sylan2 490 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉𝑖) − 𝑋) ∈ ℝ)
8014fvmpt2 6200 . . . . . 6 ((𝑖 ∈ (0...𝑀) ∧ ((𝑉𝑖) − 𝑋) ∈ ℝ) → (𝑄𝑖) = ((𝑉𝑖) − 𝑋))
8178, 79, 80syl2anc 691 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) = ((𝑉𝑖) − 𝑋))
82 fveq2 6103 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑉𝑖) = (𝑉𝑗))
8382oveq1d 6564 . . . . . . . . 9 (𝑖 = 𝑗 → ((𝑉𝑖) − 𝑋) = ((𝑉𝑗) − 𝑋))
8483cbvmptv 4678 . . . . . . . 8 (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑉𝑗) − 𝑋))
8514, 84eqtri 2632 . . . . . . 7 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉𝑗) − 𝑋))
8685a1i 11 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉𝑗) − 𝑋)))
87 fveq2 6103 . . . . . . . 8 (𝑗 = (𝑖 + 1) → (𝑉𝑗) = (𝑉‘(𝑖 + 1)))
8887oveq1d 6564 . . . . . . 7 (𝑗 = (𝑖 + 1) → ((𝑉𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
8988adantl 481 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑉𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
9073, 74resubcld 10337 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
9186, 89, 72, 90fvmptd 6197 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
9277, 81, 913brtr4d 4615 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
9392ralrimiva 2949 . . 3 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
9421, 67, 93jca32 556 . 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 39002 . . 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 246 1 (𝜑𝑄 ∈ (𝑂𝑀))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   class class class wbr 4583   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℤcz 11254  ...cfz 12197  ..^cfzo 12334 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335 This theorem is referenced by:  fourierdlem74  39073  fourierdlem75  39074  fourierdlem84  39083  fourierdlem85  39084  fourierdlem88  39087  fourierdlem103  39102  fourierdlem104  39103
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