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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > signsvf0 | Structured version Visualization version GIF version |
Description: There is no change of sign in the empty word. (Contributed by Thierry Arnoux, 8-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 |
---|---|
signsvf0 | ⊢ (𝑉‘∅) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wrd0 13519 | . . 3 ⊢ ∅ ∈ Word ℝ | |
2 | signsv.p | . . . 4 ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏)) | |
3 | signsv.w | . . . 4 ⊢ 𝑊 = {〈(Base‘ndx), {-1, 0, 1}〉, 〈(+g‘ndx), ⨣ 〉} | |
4 | signsv.t | . . . 4 ⊢ 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(♯‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓‘𝑖)))))) | |
5 | signsv.v | . . . 4 ⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(♯‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0)) | |
6 | 2, 3, 4, 5 | signsvvfval 30988 | . . 3 ⊢ (∅ ∈ Word ℝ → (𝑉‘∅) = Σ𝑗 ∈ (1..^(♯‘∅))if(((𝑇‘∅)‘𝑗) ≠ ((𝑇‘∅)‘(𝑗 − 1)), 1, 0)) |
7 | 1, 6 | ax-mp 5 | . 2 ⊢ (𝑉‘∅) = Σ𝑗 ∈ (1..^(♯‘∅))if(((𝑇‘∅)‘𝑗) ≠ ((𝑇‘∅)‘(𝑗 − 1)), 1, 0) |
8 | hash0 13353 | . . . . 5 ⊢ (♯‘∅) = 0 | |
9 | 8 | oveq2i 6802 | . . . 4 ⊢ (1..^(♯‘∅)) = (1..^0) |
10 | 0le1 10751 | . . . . 5 ⊢ 0 ≤ 1 | |
11 | 1z 11607 | . . . . . 6 ⊢ 1 ∈ ℤ | |
12 | 0z 11588 | . . . . . 6 ⊢ 0 ∈ ℤ | |
13 | fzon 12690 | . . . . . 6 ⊢ ((1 ∈ ℤ ∧ 0 ∈ ℤ) → (0 ≤ 1 ↔ (1..^0) = ∅)) | |
14 | 11, 12, 13 | mp2an 672 | . . . . 5 ⊢ (0 ≤ 1 ↔ (1..^0) = ∅) |
15 | 10, 14 | mpbi 220 | . . . 4 ⊢ (1..^0) = ∅ |
16 | 9, 15 | eqtri 2793 | . . 3 ⊢ (1..^(♯‘∅)) = ∅ |
17 | 16 | sumeq1i 14629 | . 2 ⊢ Σ𝑗 ∈ (1..^(♯‘∅))if(((𝑇‘∅)‘𝑗) ≠ ((𝑇‘∅)‘(𝑗 − 1)), 1, 0) = Σ𝑗 ∈ ∅ if(((𝑇‘∅)‘𝑗) ≠ ((𝑇‘∅)‘(𝑗 − 1)), 1, 0) |
18 | sum0 14653 | . 2 ⊢ Σ𝑗 ∈ ∅ if(((𝑇‘∅)‘𝑗) ≠ ((𝑇‘∅)‘(𝑗 − 1)), 1, 0) = 0 | |
19 | 7, 17, 18 | 3eqtri 2797 | 1 ⊢ (𝑉‘∅) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∅c0 4063 ifcif 4225 {cpr 4318 {ctp 4320 〈cop 4322 class class class wbr 4786 ↦ cmpt 4863 ‘cfv 6029 (class class class)co 6791 ↦ cmpt2 6793 ℝcr 10135 0cc0 10136 1c1 10137 ≤ cle 10275 − cmin 10466 -cneg 10467 ℤcz 11577 ...cfz 12526 ..^cfzo 12666 ♯chash 13314 Word cword 13480 sgncsgn 14027 Σcsu 14617 ndxcnx 16054 Basecbs 16057 +gcplusg 16142 Σg cgsu 16302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-inf2 8700 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 ax-pre-sup 10214 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 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-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-isom 6038 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-om 7211 df-1st 7313 df-2nd 7314 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-1o 7711 df-oadd 7715 df-er 7894 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-sup 8502 df-oi 8569 df-card 8963 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-div 10885 df-nn 11221 df-2 11279 df-3 11280 df-n0 11493 df-z 11578 df-uz 11887 df-rp 12029 df-fz 12527 df-fzo 12667 df-seq 13002 df-exp 13061 df-hash 13315 df-word 13488 df-cj 14040 df-re 14041 df-im 14042 df-sqrt 14176 df-abs 14177 df-clim 14420 df-sum 14618 |
This theorem is referenced by: (None) |
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