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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > signsvf1 | Structured version Visualization version GIF version |
Description: In a single-letter word, which represents a constant polynomial, there is no change of sign. (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 |
---|---|
signsvf1 | ⊢ (𝐾 ∈ ℝ → (𝑉‘〈“𝐾”〉) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s1cl 14482 | . . 3 ⊢ (𝐾 ∈ ℝ → 〈“𝐾”〉 ∈ 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 33059 | . . 3 ⊢ (〈“𝐾”〉 ∈ Word ℝ → (𝑉‘〈“𝐾”〉) = Σ𝑗 ∈ (1..^(♯‘〈“𝐾”〉))if(((𝑇‘〈“𝐾”〉)‘𝑗) ≠ ((𝑇‘〈“𝐾”〉)‘(𝑗 − 1)), 1, 0)) |
7 | 1, 6 | syl 17 | . 2 ⊢ (𝐾 ∈ ℝ → (𝑉‘〈“𝐾”〉) = Σ𝑗 ∈ (1..^(♯‘〈“𝐾”〉))if(((𝑇‘〈“𝐾”〉)‘𝑗) ≠ ((𝑇‘〈“𝐾”〉)‘(𝑗 − 1)), 1, 0)) |
8 | s1len 14486 | . . . . . 6 ⊢ (♯‘〈“𝐾”〉) = 1 | |
9 | 8 | oveq2i 7364 | . . . . 5 ⊢ (1..^(♯‘〈“𝐾”〉)) = (1..^1) |
10 | fzo0 13588 | . . . . 5 ⊢ (1..^1) = ∅ | |
11 | 9, 10 | eqtri 2764 | . . . 4 ⊢ (1..^(♯‘〈“𝐾”〉)) = ∅ |
12 | 11 | sumeq1i 15575 | . . 3 ⊢ Σ𝑗 ∈ (1..^(♯‘〈“𝐾”〉))if(((𝑇‘〈“𝐾”〉)‘𝑗) ≠ ((𝑇‘〈“𝐾”〉)‘(𝑗 − 1)), 1, 0) = Σ𝑗 ∈ ∅ if(((𝑇‘〈“𝐾”〉)‘𝑗) ≠ ((𝑇‘〈“𝐾”〉)‘(𝑗 − 1)), 1, 0) |
13 | sum0 15598 | . . 3 ⊢ Σ𝑗 ∈ ∅ if(((𝑇‘〈“𝐾”〉)‘𝑗) ≠ ((𝑇‘〈“𝐾”〉)‘(𝑗 − 1)), 1, 0) = 0 | |
14 | 12, 13 | eqtri 2764 | . 2 ⊢ Σ𝑗 ∈ (1..^(♯‘〈“𝐾”〉))if(((𝑇‘〈“𝐾”〉)‘𝑗) ≠ ((𝑇‘〈“𝐾”〉)‘(𝑗 − 1)), 1, 0) = 0 |
15 | 7, 14 | eqtrdi 2792 | 1 ⊢ (𝐾 ∈ ℝ → (𝑉‘〈“𝐾”〉) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∅c0 4280 ifcif 4484 {cpr 4586 {ctp 4588 〈cop 4590 ↦ cmpt 5186 ‘cfv 6493 (class class class)co 7353 ∈ cmpo 7355 ℝcr 11046 0cc0 11047 1c1 11048 − cmin 11381 -cneg 11382 ...cfz 13416 ..^cfzo 13559 ♯chash 14222 Word cword 14394 〈“cs1 14475 sgncsgn 14963 Σcsu 15562 ndxcnx 17057 Basecbs 17075 +gcplusg 17125 Σg cgsu 17314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-inf2 9573 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-pre-sup 11125 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-sup 9374 df-oi 9442 df-card 9871 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-div 11809 df-nn 12150 df-2 12212 df-3 12213 df-n0 12410 df-z 12496 df-uz 12760 df-rp 12908 df-fz 13417 df-fzo 13560 df-seq 13899 df-exp 13960 df-hash 14223 df-word 14395 df-s1 14476 df-cj 14976 df-re 14977 df-im 14978 df-sqrt 15112 df-abs 15113 df-clim 15362 df-sum 15563 |
This theorem is referenced by: (None) |
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