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Theorem fourierdlem14 39645
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 39633 . . . . . . . . . . 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 475 . . . . . . . 8 (𝜑𝑉 ∈ (ℝ ↑𝑚 (0...𝑀)))
8 elmapi 7823 . . . . . . . 8 (𝑉 ∈ (ℝ ↑𝑚 (0...𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
97, 8syl 17 . . . . . . 7 (𝜑𝑉:(0...𝑀)⟶ℝ)
109ffvelrnda 6315 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑉𝑖) ∈ ℝ)
11 fourierdlem14.x . . . . . . 7 (𝜑𝑋 ∈ ℝ)
1211adantr 481 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑋 ∈ ℝ)
1310, 12resubcld 10402 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑉𝑖) − 𝑋) ∈ ℝ)
14 fourierdlem14.q . . . . 5 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))
1513, 14fmptd 6340 . . . 4 (𝜑𝑄:(0...𝑀)⟶ℝ)
16 reex 9971 . . . . . 6 ℝ ∈ V
1716a1i 11 . . . . 5 (𝜑 → ℝ ∈ V)
18 ovex 6632 . . . . . 6 (0...𝑀) ∈ V
1918a1i 11 . . . . 5 (𝜑 → (0...𝑀) ∈ V)
2017, 19elmapd 7816 . . . 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 6148 . . . . . . . 8 (𝑖 = 0 → (𝑉𝑖) = (𝑉‘0))
2423oveq1d 6619 . . . . . . 7 (𝑖 = 0 → ((𝑉𝑖) − 𝑋) = ((𝑉‘0) − 𝑋))
2524adantl 482 . . . . . 6 ((𝜑𝑖 = 0) → ((𝑉𝑖) − 𝑋) = ((𝑉‘0) − 𝑋))
26 0zd 11333 . . . . . . . . 9 (𝜑 → 0 ∈ ℤ)
272nnzd 11425 . . . . . . . . 9 (𝜑𝑀 ∈ ℤ)
2826, 27, 263jca 1240 . . . . . . . 8 (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ))
29 0le0 11054 . . . . . . . . 9 0 ≤ 0
3029a1i 11 . . . . . . . 8 (𝜑 → 0 ≤ 0)
31 0red 9985 . . . . . . . . 9 (𝜑 → 0 ∈ ℝ)
322nnred 10979 . . . . . . . . 9 (𝜑𝑀 ∈ ℝ)
332nngt0d 11008 . . . . . . . . 9 (𝜑 → 0 < 𝑀)
3431, 32, 33ltled 10129 . . . . . . . 8 (𝜑 → 0 ≤ 𝑀)
3528, 30, 34jca32 557 . . . . . . 7 (𝜑 → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (0 ≤ 0 ∧ 0 ≤ 𝑀)))
36 elfz2 12275 . . . . . . 7 (0 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (0 ≤ 0 ∧ 0 ≤ 𝑀)))
3735, 36sylibr 224 . . . . . 6 (𝜑 → 0 ∈ (0...𝑀))
389, 37ffvelrnd 6316 . . . . . . 7 (𝜑 → (𝑉‘0) ∈ ℝ)
3938, 11resubcld 10402 . . . . . 6 (𝜑 → ((𝑉‘0) − 𝑋) ∈ ℝ)
4022, 25, 37, 39fvmptd 6245 . . . . 5 (𝜑 → (𝑄‘0) = ((𝑉‘0) − 𝑋))
416simprd 479 . . . . . . . 8 (𝜑 → (((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉𝑀) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1))))
4241simpld 475 . . . . . . 7 (𝜑 → ((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉𝑀) = (𝐵 + 𝑋)))
4342simpld 475 . . . . . 6 (𝜑 → (𝑉‘0) = (𝐴 + 𝑋))
4443oveq1d 6619 . . . . 5 (𝜑 → ((𝑉‘0) − 𝑋) = ((𝐴 + 𝑋) − 𝑋))
45 fourierdlem14.1 . . . . . . 7 (𝜑𝐴 ∈ ℝ)
4645recnd 10012 . . . . . 6 (𝜑𝐴 ∈ ℂ)
4711recnd 10012 . . . . . 6 (𝜑𝑋 ∈ ℂ)
4846, 47pncand 10337 . . . . 5 (𝜑 → ((𝐴 + 𝑋) − 𝑋) = 𝐴)
4940, 44, 483eqtrd 2659 . . . 4 (𝜑 → (𝑄‘0) = 𝐴)
50 fveq2 6148 . . . . . . . 8 (𝑖 = 𝑀 → (𝑉𝑖) = (𝑉𝑀))
5150oveq1d 6619 . . . . . . 7 (𝑖 = 𝑀 → ((𝑉𝑖) − 𝑋) = ((𝑉𝑀) − 𝑋))
5251adantl 482 . . . . . 6 ((𝜑𝑖 = 𝑀) → ((𝑉𝑖) − 𝑋) = ((𝑉𝑀) − 𝑋))
5326, 27, 273jca 1240 . . . . . . . 8 (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ))
5432leidd 10538 . . . . . . . 8 (𝜑𝑀𝑀)
5553, 34, 54jca32 557 . . . . . . 7 (𝜑 → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (0 ≤ 𝑀𝑀𝑀)))
56 elfz2 12275 . . . . . . 7 (𝑀 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (0 ≤ 𝑀𝑀𝑀)))
5755, 56sylibr 224 . . . . . 6 (𝜑𝑀 ∈ (0...𝑀))
589, 57ffvelrnd 6316 . . . . . . 7 (𝜑 → (𝑉𝑀) ∈ ℝ)
5958, 11resubcld 10402 . . . . . 6 (𝜑 → ((𝑉𝑀) − 𝑋) ∈ ℝ)
6022, 52, 57, 59fvmptd 6245 . . . . 5 (𝜑 → (𝑄𝑀) = ((𝑉𝑀) − 𝑋))
6142simprd 479 . . . . . 6 (𝜑 → (𝑉𝑀) = (𝐵 + 𝑋))
6261oveq1d 6619 . . . . 5 (𝜑 → ((𝑉𝑀) − 𝑋) = ((𝐵 + 𝑋) − 𝑋))
63 fourierdlem14.2 . . . . . . 7 (𝜑𝐵 ∈ ℝ)
6463recnd 10012 . . . . . 6 (𝜑𝐵 ∈ ℂ)
6564, 47pncand 10337 . . . . 5 (𝜑 → ((𝐵 + 𝑋) − 𝑋) = 𝐵)
6660, 62, 653eqtrd 2659 . . . 4 (𝜑 → (𝑄𝑀) = 𝐵)
6749, 66jca 554 . . 3 (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵))
68 elfzofz 12426 . . . . . . 7 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
6968, 10sylan2 491 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉𝑖) ∈ ℝ)
709adantr 481 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
71 fzofzp1 12506 . . . . . . . 8 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
7271adantl 482 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
7370, 72ffvelrnd 6316 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
7411adantr 481 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
7541simprd 479 . . . . . . 7 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1)))
7675r19.21bi 2927 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉𝑖) < (𝑉‘(𝑖 + 1)))
7769, 73, 74, 76ltsub1dd 10583 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉𝑖) − 𝑋) < ((𝑉‘(𝑖 + 1)) − 𝑋))
7868adantl 482 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
7968, 13sylan2 491 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉𝑖) − 𝑋) ∈ ℝ)
8014fvmpt2 6248 . . . . . 6 ((𝑖 ∈ (0...𝑀) ∧ ((𝑉𝑖) − 𝑋) ∈ ℝ) → (𝑄𝑖) = ((𝑉𝑖) − 𝑋))
8178, 79, 80syl2anc 692 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) = ((𝑉𝑖) − 𝑋))
82 fveq2 6148 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑉𝑖) = (𝑉𝑗))
8382oveq1d 6619 . . . . . . . . 9 (𝑖 = 𝑗 → ((𝑉𝑖) − 𝑋) = ((𝑉𝑗) − 𝑋))
8483cbvmptv 4710 . . . . . . . 8 (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑉𝑗) − 𝑋))
8514, 84eqtri 2643 . . . . . . 7 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉𝑗) − 𝑋))
8685a1i 11 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉𝑗) − 𝑋)))
87 fveq2 6148 . . . . . . . 8 (𝑗 = (𝑖 + 1) → (𝑉𝑗) = (𝑉‘(𝑖 + 1)))
8887oveq1d 6619 . . . . . . 7 (𝑗 = (𝑖 + 1) → ((𝑉𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
8988adantl 482 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑉𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
9073, 74resubcld 10402 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
9186, 89, 72, 90fvmptd 6245 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
9277, 81, 913brtr4d 4645 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
9392ralrimiva 2960 . . 3 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
9421, 67, 93jca32 557 . 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 39633 . . 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3186   class class class wbr 4613  cmpt 4673  wf 5843  cfv 5847  (class class class)co 6604  𝑚 cmap 7802  cr 9879  0cc0 9880  1c1 9881   + caddc 9883   < clt 10018  cle 10019  cmin 10210  cn 10964  cz 11321  ...cfz 12268  ..^cfzo 12406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407
This theorem is referenced by:  fourierdlem74  39704  fourierdlem75  39705  fourierdlem84  39714  fourierdlem85  39715  fourierdlem88  39718  fourierdlem103  39733  fourierdlem104  39734
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