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Theorem fourierdlem14 40855
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 40843 . . . . . . . . . . 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 482 . . . . . . . 8 (𝜑𝑉 ∈ (ℝ ↑𝑚 (0...𝑀)))
8 elmapi 8031 . . . . . . . 8 (𝑉 ∈ (ℝ ↑𝑚 (0...𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
97, 8syl 17 . . . . . . 7 (𝜑𝑉:(0...𝑀)⟶ℝ)
109ffvelrnda 6502 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑉𝑖) ∈ ℝ)
11 fourierdlem14.x . . . . . . 7 (𝜑𝑋 ∈ ℝ)
1211adantr 466 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑋 ∈ ℝ)
1310, 12resubcld 10660 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑉𝑖) − 𝑋) ∈ ℝ)
14 fourierdlem14.q . . . . 5 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))
1513, 14fmptd 6527 . . . 4 (𝜑𝑄:(0...𝑀)⟶ℝ)
16 reex 10229 . . . . . 6 ℝ ∈ V
1716a1i 11 . . . . 5 (𝜑 → ℝ ∈ V)
18 ovex 6823 . . . . . 6 (0...𝑀) ∈ V
1918a1i 11 . . . . 5 (𝜑 → (0...𝑀) ∈ V)
2017, 19elmapd 8023 . . . 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 6332 . . . . . . . 8 (𝑖 = 0 → (𝑉𝑖) = (𝑉‘0))
2423oveq1d 6808 . . . . . . 7 (𝑖 = 0 → ((𝑉𝑖) − 𝑋) = ((𝑉‘0) − 𝑋))
2524adantl 467 . . . . . 6 ((𝜑𝑖 = 0) → ((𝑉𝑖) − 𝑋) = ((𝑉‘0) − 𝑋))
26 0zd 11591 . . . . . . . . 9 (𝜑 → 0 ∈ ℤ)
272nnzd 11683 . . . . . . . . 9 (𝜑𝑀 ∈ ℤ)
2826, 27, 263jca 1122 . . . . . . . 8 (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ))
29 0le0 11312 . . . . . . . . 9 0 ≤ 0
3029a1i 11 . . . . . . . 8 (𝜑 → 0 ≤ 0)
31 0red 10243 . . . . . . . . 9 (𝜑 → 0 ∈ ℝ)
322nnred 11237 . . . . . . . . 9 (𝜑𝑀 ∈ ℝ)
332nngt0d 11266 . . . . . . . . 9 (𝜑 → 0 < 𝑀)
3431, 32, 33ltled 10387 . . . . . . . 8 (𝜑 → 0 ≤ 𝑀)
3528, 30, 34jca32 505 . . . . . . 7 (𝜑 → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (0 ≤ 0 ∧ 0 ≤ 𝑀)))
36 elfz2 12540 . . . . . . 7 (0 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 ∈ ℤ) ∧ (0 ≤ 0 ∧ 0 ≤ 𝑀)))
3735, 36sylibr 224 . . . . . 6 (𝜑 → 0 ∈ (0...𝑀))
389, 37ffvelrnd 6503 . . . . . . 7 (𝜑 → (𝑉‘0) ∈ ℝ)
3938, 11resubcld 10660 . . . . . 6 (𝜑 → ((𝑉‘0) − 𝑋) ∈ ℝ)
4022, 25, 37, 39fvmptd 6430 . . . . 5 (𝜑 → (𝑄‘0) = ((𝑉‘0) − 𝑋))
416simprd 483 . . . . . . . 8 (𝜑 → (((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉𝑀) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1))))
4241simpld 482 . . . . . . 7 (𝜑 → ((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉𝑀) = (𝐵 + 𝑋)))
4342simpld 482 . . . . . 6 (𝜑 → (𝑉‘0) = (𝐴 + 𝑋))
4443oveq1d 6808 . . . . 5 (𝜑 → ((𝑉‘0) − 𝑋) = ((𝐴 + 𝑋) − 𝑋))
45 fourierdlem14.1 . . . . . . 7 (𝜑𝐴 ∈ ℝ)
4645recnd 10270 . . . . . 6 (𝜑𝐴 ∈ ℂ)
4711recnd 10270 . . . . . 6 (𝜑𝑋 ∈ ℂ)
4846, 47pncand 10595 . . . . 5 (𝜑 → ((𝐴 + 𝑋) − 𝑋) = 𝐴)
4940, 44, 483eqtrd 2809 . . . 4 (𝜑 → (𝑄‘0) = 𝐴)
50 fveq2 6332 . . . . . . . 8 (𝑖 = 𝑀 → (𝑉𝑖) = (𝑉𝑀))
5150oveq1d 6808 . . . . . . 7 (𝑖 = 𝑀 → ((𝑉𝑖) − 𝑋) = ((𝑉𝑀) − 𝑋))
5251adantl 467 . . . . . 6 ((𝜑𝑖 = 𝑀) → ((𝑉𝑖) − 𝑋) = ((𝑉𝑀) − 𝑋))
5326, 27, 273jca 1122 . . . . . . . 8 (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ))
5432leidd 10796 . . . . . . . 8 (𝜑𝑀𝑀)
5553, 34, 54jca32 505 . . . . . . 7 (𝜑 → ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (0 ≤ 𝑀𝑀𝑀)))
56 elfz2 12540 . . . . . . 7 (𝑀 ∈ (0...𝑀) ↔ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (0 ≤ 𝑀𝑀𝑀)))
5755, 56sylibr 224 . . . . . 6 (𝜑𝑀 ∈ (0...𝑀))
589, 57ffvelrnd 6503 . . . . . . 7 (𝜑 → (𝑉𝑀) ∈ ℝ)
5958, 11resubcld 10660 . . . . . 6 (𝜑 → ((𝑉𝑀) − 𝑋) ∈ ℝ)
6022, 52, 57, 59fvmptd 6430 . . . . 5 (𝜑 → (𝑄𝑀) = ((𝑉𝑀) − 𝑋))
6142simprd 483 . . . . . 6 (𝜑 → (𝑉𝑀) = (𝐵 + 𝑋))
6261oveq1d 6808 . . . . 5 (𝜑 → ((𝑉𝑀) − 𝑋) = ((𝐵 + 𝑋) − 𝑋))
63 fourierdlem14.2 . . . . . . 7 (𝜑𝐵 ∈ ℝ)
6463recnd 10270 . . . . . 6 (𝜑𝐵 ∈ ℂ)
6564, 47pncand 10595 . . . . 5 (𝜑 → ((𝐵 + 𝑋) − 𝑋) = 𝐵)
6660, 62, 653eqtrd 2809 . . . 4 (𝜑 → (𝑄𝑀) = 𝐵)
6749, 66jca 501 . . 3 (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵))
68 elfzofz 12693 . . . . . . 7 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
6968, 10sylan2 580 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉𝑖) ∈ ℝ)
709adantr 466 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
71 fzofzp1 12773 . . . . . . . 8 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
7271adantl 467 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
7370, 72ffvelrnd 6503 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
7411adantr 466 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
7541simprd 483 . . . . . . 7 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1)))
7675r19.21bi 3081 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉𝑖) < (𝑉‘(𝑖 + 1)))
7769, 73, 74, 76ltsub1dd 10841 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉𝑖) − 𝑋) < ((𝑉‘(𝑖 + 1)) − 𝑋))
7868adantl 467 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
7968, 13sylan2 580 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉𝑖) − 𝑋) ∈ ℝ)
8014fvmpt2 6433 . . . . . 6 ((𝑖 ∈ (0...𝑀) ∧ ((𝑉𝑖) − 𝑋) ∈ ℝ) → (𝑄𝑖) = ((𝑉𝑖) − 𝑋))
8178, 79, 80syl2anc 573 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) = ((𝑉𝑖) − 𝑋))
82 fveq2 6332 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑉𝑖) = (𝑉𝑗))
8382oveq1d 6808 . . . . . . . . 9 (𝑖 = 𝑗 → ((𝑉𝑖) − 𝑋) = ((𝑉𝑗) − 𝑋))
8483cbvmptv 4884 . . . . . . . 8 (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑉𝑗) − 𝑋))
8514, 84eqtri 2793 . . . . . . 7 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉𝑗) − 𝑋))
8685a1i 11 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉𝑗) − 𝑋)))
87 fveq2 6332 . . . . . . . 8 (𝑗 = (𝑖 + 1) → (𝑉𝑗) = (𝑉‘(𝑖 + 1)))
8887oveq1d 6808 . . . . . . 7 (𝑗 = (𝑖 + 1) → ((𝑉𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
8988adantl 467 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑉𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
9073, 74resubcld 10660 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
9186, 89, 72, 90fvmptd 6430 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
9277, 81, 913brtr4d 4818 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
9392ralrimiva 3115 . . 3 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
9421, 67, 93jca32 505 . 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 40843 . . 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 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  {crab 3065  Vcvv 3351   class class class wbr 4786  cmpt 4863  wf 6027  cfv 6031  (class class class)co 6793  𝑚 cmap 8009  cr 10137  0cc0 10138  1c1 10139   + caddc 10141   < clt 10276  cle 10277  cmin 10468  cn 11222  cz 11579  ...cfz 12533  ..^cfzo 12673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-n0 11495  df-z 11580  df-uz 11889  df-fz 12534  df-fzo 12674
This theorem is referenced by:  fourierdlem74  40914  fourierdlem75  40915  fourierdlem84  40924  fourierdlem85  40925  fourierdlem88  40928  fourierdlem103  40943  fourierdlem104  40944
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