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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > signsvvfval | Structured version Visualization version GIF version |
Description: The value of 𝑉, which represents the number of times the sign changes in a word. (Contributed by Thierry Arnoux, 7-Oct-2018.) |
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‘(𝑓‘𝑖)))))) |
signsv.v | ⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(♯‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0)) |
Ref | Expression |
---|---|
signsvvfval | ⊢ (𝐹 ∈ Word ℝ → (𝑉‘𝐹) = Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)), 1, 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6433 | . . . 4 ⊢ (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹)) | |
2 | 1 | oveq2d 6921 | . . 3 ⊢ (𝑓 = 𝐹 → (1..^(♯‘𝑓)) = (1..^(♯‘𝐹))) |
3 | fveq2 6433 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑇‘𝑓) = (𝑇‘𝐹)) | |
4 | 3 | fveq1d 6435 | . . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑇‘𝑓)‘𝑗) = ((𝑇‘𝐹)‘𝑗)) |
5 | 3 | fveq1d 6435 | . . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑇‘𝑓)‘(𝑗 − 1)) = ((𝑇‘𝐹)‘(𝑗 − 1))) |
6 | 4, 5 | neeq12d 3060 | . . . . 5 ⊢ (𝑓 = 𝐹 → (((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)) ↔ ((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)))) |
7 | 6 | ifbid 4328 | . . . 4 ⊢ (𝑓 = 𝐹 → if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0) = if(((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)), 1, 0)) |
8 | 7 | adantr 474 | . . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑗 ∈ (1..^(♯‘𝑓))) → if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0) = if(((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)), 1, 0)) |
9 | 2, 8 | sumeq12dv 14814 | . 2 ⊢ (𝑓 = 𝐹 → Σ𝑗 ∈ (1..^(♯‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0) = Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)), 1, 0)) |
10 | signsv.v | . 2 ⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(♯‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0)) | |
11 | sumex 14795 | . 2 ⊢ Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)), 1, 0) ∈ V | |
12 | 9, 10, 11 | fvmpt 6529 | 1 ⊢ (𝐹 ∈ Word ℝ → (𝑉‘𝐹) = Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)), 1, 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ≠ wne 2999 ifcif 4306 {cpr 4399 {ctp 4401 〈cop 4403 ↦ cmpt 4952 ‘cfv 6123 (class class class)co 6905 ↦ cmpt2 6907 ℝcr 10251 0cc0 10252 1c1 10253 − cmin 10585 -cneg 10586 ...cfz 12619 ..^cfzo 12760 ♯chash 13410 Word cword 13574 sgncsgn 14203 Σcsu 14793 ndxcnx 16219 Basecbs 16222 +gcplusg 16305 Σg cgsu 16454 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-seq 13096 df-sum 14794 |
This theorem is referenced by: signsvf0 31206 signsvf1 31207 signsvfn 31208 |
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