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Theorem signstfv 34029
Description: Value of the zero-skipping sign word. (Contributed by Thierry Arnoux, 8-Oct-2018.)
Hypotheses
Ref Expression
signsv.p ⨣ = (π‘Ž ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, π‘Ž, 𝑏))
signsv.w π‘Š = {⟨(Baseβ€˜ndx), {-1, 0, 1}⟩, ⟨(+gβ€˜ndx), ⨣ ⟩}
signsv.t 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(β™―β€˜π‘“)) ↦ (π‘Š Ξ£g (𝑖 ∈ (0...𝑛) ↦ (sgnβ€˜(π‘“β€˜π‘–))))))
signsv.v 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(β™―β€˜π‘“))if(((π‘‡β€˜π‘“)β€˜π‘—) β‰  ((π‘‡β€˜π‘“)β€˜(𝑗 βˆ’ 1)), 1, 0))
Assertion
Ref Expression
signstfv (𝐹 ∈ Word ℝ β†’ (π‘‡β€˜πΉ) = (𝑛 ∈ (0..^(β™―β€˜πΉ)) ↦ (π‘Š Ξ£g (𝑖 ∈ (0...𝑛) ↦ (sgnβ€˜(πΉβ€˜π‘–))))))
Distinct variable groups:   𝑓,𝑖,𝑛,𝐹   𝑓,π‘Š
Allowed substitution hints:   ⨣ (𝑓,𝑖,𝑗,𝑛,π‘Ž,𝑏)   𝑇(𝑓,𝑖,𝑗,𝑛,π‘Ž,𝑏)   𝐹(𝑗,π‘Ž,𝑏)   𝑉(𝑓,𝑖,𝑗,𝑛,π‘Ž,𝑏)   π‘Š(𝑖,𝑗,𝑛,π‘Ž,𝑏)
Proof of Theorem signstfv
StepHypRef Expression
1 fveq2 6881 . . . 4 (𝑓 = 𝐹 β†’ (β™―β€˜π‘“) = (β™―β€˜πΉ))
21oveq2d 7417 . . 3 (𝑓 = 𝐹 β†’ (0..^(β™―β€˜π‘“)) = (0..^(β™―β€˜πΉ)))
3 simpl 482 . . . . . . 7 ((𝑓 = 𝐹 ∧ 𝑖 ∈ (0...𝑛)) β†’ 𝑓 = 𝐹)
43fveq1d 6883 . . . . . 6 ((𝑓 = 𝐹 ∧ 𝑖 ∈ (0...𝑛)) β†’ (π‘“β€˜π‘–) = (πΉβ€˜π‘–))
54fveq2d 6885 . . . . 5 ((𝑓 = 𝐹 ∧ 𝑖 ∈ (0...𝑛)) β†’ (sgnβ€˜(π‘“β€˜π‘–)) = (sgnβ€˜(πΉβ€˜π‘–)))
65mpteq2dva 5238 . . . 4 (𝑓 = 𝐹 β†’ (𝑖 ∈ (0...𝑛) ↦ (sgnβ€˜(π‘“β€˜π‘–))) = (𝑖 ∈ (0...𝑛) ↦ (sgnβ€˜(πΉβ€˜π‘–))))
76oveq2d 7417 . . 3 (𝑓 = 𝐹 β†’ (π‘Š Ξ£g (𝑖 ∈ (0...𝑛) ↦ (sgnβ€˜(π‘“β€˜π‘–)))) = (π‘Š Ξ£g (𝑖 ∈ (0...𝑛) ↦ (sgnβ€˜(πΉβ€˜π‘–)))))
82, 7mpteq12dv 5229 . 2 (𝑓 = 𝐹 β†’ (𝑛 ∈ (0..^(β™―β€˜π‘“)) ↦ (π‘Š Ξ£g (𝑖 ∈ (0...𝑛) ↦ (sgnβ€˜(π‘“β€˜π‘–))))) = (𝑛 ∈ (0..^(β™―β€˜πΉ)) ↦ (π‘Š Ξ£g (𝑖 ∈ (0...𝑛) ↦ (sgnβ€˜(πΉβ€˜π‘–))))))
9 signsv.t . 2 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(β™―β€˜π‘“)) ↦ (π‘Š Ξ£g (𝑖 ∈ (0...𝑛) ↦ (sgnβ€˜(π‘“β€˜π‘–))))))
10 ovex 7434 . . 3 (0..^(β™―β€˜πΉ)) ∈ V
1110mptex 7216 . 2 (𝑛 ∈ (0..^(β™―β€˜πΉ)) ↦ (π‘Š Ξ£g (𝑖 ∈ (0...𝑛) ↦ (sgnβ€˜(πΉβ€˜π‘–))))) ∈ V
128, 9, 11fvmpt 6988 1 (𝐹 ∈ Word ℝ β†’ (π‘‡β€˜πΉ) = (𝑛 ∈ (0..^(β™―β€˜πΉ)) ↦ (π‘Š Ξ£g (𝑖 ∈ (0...𝑛) ↦ (sgnβ€˜(πΉβ€˜π‘–))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   β‰  wne 2932  ifcif 4520  {cpr 4622  {ctp 4624  βŸ¨cop 4626   ↦ cmpt 5221  β€˜cfv 6533  (class class class)co 7401   ∈ cmpo 7403  β„cr 11104  0cc0 11105  1c1 11106   βˆ’ cmin 11440  -cneg 11441  ...cfz 13480  ..^cfzo 13623  β™―chash 14286  Word cword 14460  sgncsgn 15029  Ξ£csu 15628  ndxcnx 17124  Basecbs 17142  +gcplusg 17195   Ξ£g cgsu 17384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404
This theorem is referenced by:  signstfval  34030  signstf  34032  signstlen  34033  signstf0  34034
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