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Theorem fourierdlem14 44352
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 44340 . . . . . . . . . . 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 231 . . . . . . . . 9 (𝜑 → (𝑉 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉𝑀) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1)))))
76simpld 495 . . . . . . . 8 (𝜑𝑉 ∈ (ℝ ↑m (0...𝑀)))
8 elmapi 8787 . . . . . . . 8 (𝑉 ∈ (ℝ ↑m (0...𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
97, 8syl 17 . . . . . . 7 (𝜑𝑉:(0...𝑀)⟶ℝ)
109ffvelcdmda 7035 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑉𝑖) ∈ ℝ)
11 fourierdlem14.x . . . . . . 7 (𝜑𝑋 ∈ ℝ)
1211adantr 481 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑋 ∈ ℝ)
1310, 12resubcld 11583 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑉𝑖) − 𝑋) ∈ ℝ)
14 fourierdlem14.q . . . . 5 𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋))
1513, 14fmptd 7062 . . . 4 (𝜑𝑄:(0...𝑀)⟶ℝ)
16 reex 11142 . . . . . 6 ℝ ∈ V
1716a1i 11 . . . . 5 (𝜑 → ℝ ∈ V)
18 ovex 7390 . . . . . 6 (0...𝑀) ∈ V
1918a1i 11 . . . . 5 (𝜑 → (0...𝑀) ∈ V)
2017, 19elmapd 8779 . . . 4 (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ↔ 𝑄:(0...𝑀)⟶ℝ))
2115, 20mpbird 256 . . 3 (𝜑𝑄 ∈ (ℝ ↑m (0...𝑀)))
2214a1i 11 . . . . . 6 (𝜑𝑄 = (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋)))
23 fveq2 6842 . . . . . . . 8 (𝑖 = 0 → (𝑉𝑖) = (𝑉‘0))
2423oveq1d 7372 . . . . . . 7 (𝑖 = 0 → ((𝑉𝑖) − 𝑋) = ((𝑉‘0) − 𝑋))
2524adantl 482 . . . . . 6 ((𝜑𝑖 = 0) → ((𝑉𝑖) − 𝑋) = ((𝑉‘0) − 𝑋))
26 0zd 12511 . . . . . . 7 (𝜑 → 0 ∈ ℤ)
272nnzd 12526 . . . . . . 7 (𝜑𝑀 ∈ ℤ)
28 0le0 12254 . . . . . . . 8 0 ≤ 0
2928a1i 11 . . . . . . 7 (𝜑 → 0 ≤ 0)
30 0red 11158 . . . . . . . 8 (𝜑 → 0 ∈ ℝ)
312nnred 12168 . . . . . . . 8 (𝜑𝑀 ∈ ℝ)
322nngt0d 12202 . . . . . . . 8 (𝜑 → 0 < 𝑀)
3330, 31, 32ltled 11303 . . . . . . 7 (𝜑 → 0 ≤ 𝑀)
3426, 27, 26, 29, 33elfzd 13432 . . . . . 6 (𝜑 → 0 ∈ (0...𝑀))
359, 34ffvelcdmd 7036 . . . . . . 7 (𝜑 → (𝑉‘0) ∈ ℝ)
3635, 11resubcld 11583 . . . . . 6 (𝜑 → ((𝑉‘0) − 𝑋) ∈ ℝ)
3722, 25, 34, 36fvmptd 6955 . . . . 5 (𝜑 → (𝑄‘0) = ((𝑉‘0) − 𝑋))
386simprd 496 . . . . . . . 8 (𝜑 → (((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉𝑀) = (𝐵 + 𝑋)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1))))
3938simpld 495 . . . . . . 7 (𝜑 → ((𝑉‘0) = (𝐴 + 𝑋) ∧ (𝑉𝑀) = (𝐵 + 𝑋)))
4039simpld 495 . . . . . 6 (𝜑 → (𝑉‘0) = (𝐴 + 𝑋))
4140oveq1d 7372 . . . . 5 (𝜑 → ((𝑉‘0) − 𝑋) = ((𝐴 + 𝑋) − 𝑋))
42 fourierdlem14.1 . . . . . . 7 (𝜑𝐴 ∈ ℝ)
4342recnd 11183 . . . . . 6 (𝜑𝐴 ∈ ℂ)
4411recnd 11183 . . . . . 6 (𝜑𝑋 ∈ ℂ)
4543, 44pncand 11513 . . . . 5 (𝜑 → ((𝐴 + 𝑋) − 𝑋) = 𝐴)
4637, 41, 453eqtrd 2780 . . . 4 (𝜑 → (𝑄‘0) = 𝐴)
47 fveq2 6842 . . . . . . . 8 (𝑖 = 𝑀 → (𝑉𝑖) = (𝑉𝑀))
4847oveq1d 7372 . . . . . . 7 (𝑖 = 𝑀 → ((𝑉𝑖) − 𝑋) = ((𝑉𝑀) − 𝑋))
4948adantl 482 . . . . . 6 ((𝜑𝑖 = 𝑀) → ((𝑉𝑖) − 𝑋) = ((𝑉𝑀) − 𝑋))
5031leidd 11721 . . . . . . 7 (𝜑𝑀𝑀)
5126, 27, 27, 33, 50elfzd 13432 . . . . . 6 (𝜑𝑀 ∈ (0...𝑀))
529, 51ffvelcdmd 7036 . . . . . . 7 (𝜑 → (𝑉𝑀) ∈ ℝ)
5352, 11resubcld 11583 . . . . . 6 (𝜑 → ((𝑉𝑀) − 𝑋) ∈ ℝ)
5422, 49, 51, 53fvmptd 6955 . . . . 5 (𝜑 → (𝑄𝑀) = ((𝑉𝑀) − 𝑋))
5539simprd 496 . . . . . 6 (𝜑 → (𝑉𝑀) = (𝐵 + 𝑋))
5655oveq1d 7372 . . . . 5 (𝜑 → ((𝑉𝑀) − 𝑋) = ((𝐵 + 𝑋) − 𝑋))
57 fourierdlem14.2 . . . . . . 7 (𝜑𝐵 ∈ ℝ)
5857recnd 11183 . . . . . 6 (𝜑𝐵 ∈ ℂ)
5958, 44pncand 11513 . . . . 5 (𝜑 → ((𝐵 + 𝑋) − 𝑋) = 𝐵)
6054, 56, 593eqtrd 2780 . . . 4 (𝜑 → (𝑄𝑀) = 𝐵)
6146, 60jca 512 . . 3 (𝜑 → ((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵))
62 elfzofz 13588 . . . . . . 7 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
6362, 10sylan2 593 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉𝑖) ∈ ℝ)
649adantr 481 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑉:(0...𝑀)⟶ℝ)
65 fzofzp1 13669 . . . . . . . 8 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
6665adantl 482 . . . . . . 7 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
6764, 66ffvelcdmd 7036 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉‘(𝑖 + 1)) ∈ ℝ)
6811adantr 481 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑋 ∈ ℝ)
6938simprd 496 . . . . . . 7 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑉𝑖) < (𝑉‘(𝑖 + 1)))
7069r19.21bi 3234 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑉𝑖) < (𝑉‘(𝑖 + 1)))
7163, 67, 68, 70ltsub1dd 11767 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉𝑖) − 𝑋) < ((𝑉‘(𝑖 + 1)) − 𝑋))
7262adantl 482 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
7362, 13sylan2 593 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉𝑖) − 𝑋) ∈ ℝ)
7414fvmpt2 6959 . . . . . 6 ((𝑖 ∈ (0...𝑀) ∧ ((𝑉𝑖) − 𝑋) ∈ ℝ) → (𝑄𝑖) = ((𝑉𝑖) − 𝑋))
7572, 73, 74syl2anc 584 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) = ((𝑉𝑖) − 𝑋))
76 fveq2 6842 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑉𝑖) = (𝑉𝑗))
7776oveq1d 7372 . . . . . . . . 9 (𝑖 = 𝑗 → ((𝑉𝑖) − 𝑋) = ((𝑉𝑗) − 𝑋))
7877cbvmptv 5218 . . . . . . . 8 (𝑖 ∈ (0...𝑀) ↦ ((𝑉𝑖) − 𝑋)) = (𝑗 ∈ (0...𝑀) ↦ ((𝑉𝑗) − 𝑋))
7914, 78eqtri 2764 . . . . . . 7 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉𝑗) − 𝑋))
8079a1i 11 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑄 = (𝑗 ∈ (0...𝑀) ↦ ((𝑉𝑗) − 𝑋)))
81 fveq2 6842 . . . . . . . 8 (𝑗 = (𝑖 + 1) → (𝑉𝑗) = (𝑉‘(𝑖 + 1)))
8281oveq1d 7372 . . . . . . 7 (𝑗 = (𝑖 + 1) → ((𝑉𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
8382adantl 482 . . . . . 6 (((𝜑𝑖 ∈ (0..^𝑀)) ∧ 𝑗 = (𝑖 + 1)) → ((𝑉𝑗) − 𝑋) = ((𝑉‘(𝑖 + 1)) − 𝑋))
8467, 68resubcld 11583 . . . . . 6 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑉‘(𝑖 + 1)) − 𝑋) ∈ ℝ)
8580, 83, 66, 84fvmptd 6955 . . . . 5 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄‘(𝑖 + 1)) = ((𝑉‘(𝑖 + 1)) − 𝑋))
8671, 75, 853brtr4d 5137 . . . 4 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
8786ralrimiva 3143 . . 3 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
8821, 61, 87jca32 516 . 2 (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
89 fourierdlem14.o . . . 4 𝑂 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
9089fourierdlem2 44340 . . 3 (𝑀 ∈ ℕ → (𝑄 ∈ (𝑂𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
912, 90syl 17 . 2 (𝜑 → (𝑄 ∈ (𝑂𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
9288, 91mpbird 256 1 (𝜑𝑄 ∈ (𝑂𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  {crab 3407  Vcvv 3445   class class class wbr 5105  cmpt 5188  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   < clt 11189  cle 11190  cmin 11385  cn 12153  ...cfz 13424  ..^cfzo 13567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568
This theorem is referenced by:  fourierdlem74  44411  fourierdlem75  44412  fourierdlem84  44421  fourierdlem85  44422  fourierdlem88  44425  fourierdlem103  44440  fourierdlem104  44441
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