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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 6901 | . . . 4 ⊢ (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹)) | |
2 | 1 | oveq2d 7440 | . . 3 ⊢ (𝑓 = 𝐹 → (1..^(♯‘𝑓)) = (1..^(♯‘𝐹))) |
3 | fveq2 6901 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑇‘𝑓) = (𝑇‘𝐹)) | |
4 | 3 | fveq1d 6903 | . . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑇‘𝑓)‘𝑗) = ((𝑇‘𝐹)‘𝑗)) |
5 | 3 | fveq1d 6903 | . . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑇‘𝑓)‘(𝑗 − 1)) = ((𝑇‘𝐹)‘(𝑗 − 1))) |
6 | 4, 5 | neeq12d 2992 | . . . . 5 ⊢ (𝑓 = 𝐹 → (((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)) ↔ ((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)))) |
7 | 6 | ifbid 4556 | . . . 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 479 | . . 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 15710 | . 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 15692 | . 2 ⊢ Σ𝑗 ∈ (1..^(♯‘𝐹))if(((𝑇‘𝐹)‘𝑗) ≠ ((𝑇‘𝐹)‘(𝑗 − 1)), 1, 0) ∈ V | |
12 | 9, 10, 11 | fvmpt 7009 | 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 1534 ∈ wcel 2099 ≠ wne 2930 ifcif 4533 {cpr 4635 {ctp 4637 〈cop 4639 ↦ cmpt 5236 ‘cfv 6554 (class class class)co 7424 ∈ cmpo 7426 ℝcr 11157 0cc0 11158 1c1 11159 − cmin 11494 -cneg 11495 ...cfz 13538 ..^cfzo 13681 ♯chash 14347 Word cword 14522 sgncsgn 15091 Σcsu 15690 ndxcnx 17195 Basecbs 17213 +gcplusg 17266 Σg cgsu 17455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-seq 14022 df-sum 15691 |
This theorem is referenced by: signsvf0 34426 signsvf1 34427 signsvfn 34428 |
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