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Theorem signstfv 30974
Description: Value of the zero-skipping sign word. (Contributed by Thierry Arnoux, 8-Oct-2018.)
Hypotheses
Ref Expression
signsv.p = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
signsv.w 𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+g‘ndx), ⟩}
signsv.t 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(♯‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))
signsv.v 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(♯‘𝑓))if(((𝑇𝑓)‘𝑗) ≠ ((𝑇𝑓)‘(𝑗 − 1)), 1, 0))
Assertion
Ref Expression
signstfv (𝐹 ∈ Word ℝ → (𝑇𝐹) = (𝑛 ∈ (0..^(♯‘𝐹)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝐹𝑖))))))
Distinct variable groups:   𝑓,𝑖,𝑛,𝐹   𝑓,𝑊
Allowed substitution hints:   (𝑓,𝑖,𝑗,𝑛,𝑎,𝑏)   𝑇(𝑓,𝑖,𝑗,𝑛,𝑎,𝑏)   𝐹(𝑗,𝑎,𝑏)   𝑉(𝑓,𝑖,𝑗,𝑛,𝑎,𝑏)   𝑊(𝑖,𝑗,𝑛,𝑎,𝑏)

Proof of Theorem signstfv
StepHypRef Expression
1 fveq2 6332 . . . 4 (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹))
21oveq2d 6808 . . 3 (𝑓 = 𝐹 → (0..^(♯‘𝑓)) = (0..^(♯‘𝐹)))
3 simpl 468 . . . . . . 7 ((𝑓 = 𝐹𝑖 ∈ (0...𝑛)) → 𝑓 = 𝐹)
43fveq1d 6334 . . . . . 6 ((𝑓 = 𝐹𝑖 ∈ (0...𝑛)) → (𝑓𝑖) = (𝐹𝑖))
54fveq2d 6336 . . . . 5 ((𝑓 = 𝐹𝑖 ∈ (0...𝑛)) → (sgn‘(𝑓𝑖)) = (sgn‘(𝐹𝑖)))
65mpteq2dva 4876 . . . 4 (𝑓 = 𝐹 → (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))) = (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝐹𝑖))))
76oveq2d 6808 . . 3 (𝑓 = 𝐹 → (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖)))) = (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝐹𝑖)))))
82, 7mpteq12dv 4865 . 2 (𝑓 = 𝐹 → (𝑛 ∈ (0..^(♯‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))) = (𝑛 ∈ (0..^(♯‘𝐹)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝐹𝑖))))))
9 signsv.t . 2 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(♯‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓𝑖))))))
10 ovex 6822 . . 3 (0..^(♯‘𝐹)) ∈ V
1110mptex 6629 . 2 (𝑛 ∈ (0..^(♯‘𝐹)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝐹𝑖))))) ∈ V
128, 9, 11fvmpt 6424 1 (𝐹 ∈ Word ℝ → (𝑇𝐹) = (𝑛 ∈ (0..^(♯‘𝐹)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝐹𝑖))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  wne 2942  ifcif 4223  {cpr 4316  {ctp 4318  cop 4320  cmpt 4861  cfv 6031  (class class class)co 6792  cmpt2 6794  cr 10136  0cc0 10137  1c1 10138  cmin 10467  -cneg 10468  ...cfz 12532  ..^cfzo 12672  chash 13320  Word cword 13486  sgncsgn 14033  Σcsu 14623  ndxcnx 16060  Basecbs 16063  +gcplusg 16148   Σg cgsu 16308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  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-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795
This theorem is referenced by:  signstfval  30975  signstf  30977  signstlen  30978  signstf0  30979
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