Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > signstcl | Structured version Visualization version GIF version |
Description: Closure of the zero skipping sign word. (Contributed by Thierry Arnoux, 9-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 |
---|---|
signstcl | ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘𝐹)‘𝑁) ∈ {-1, 0, 1}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | signsv.p | . . 3 ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏)) | |
2 | signsv.w | . . 3 ⊢ 𝑊 = {〈(Base‘ndx), {-1, 0, 1}〉, 〈(+g‘ndx), ⨣ 〉} | |
3 | signsv.t | . . 3 ⊢ 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(♯‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓‘𝑖)))))) | |
4 | signsv.v | . . 3 ⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(♯‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0)) | |
5 | 1, 2, 3, 4 | signstfval 32115 | . 2 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘𝐹)‘𝑁) = (𝑊 Σg (𝑖 ∈ (0...𝑁) ↦ (sgn‘(𝐹‘𝑖))))) |
6 | 1, 2 | signswbase 32105 | . . 3 ⊢ {-1, 0, 1} = (Base‘𝑊) |
7 | 1, 2 | signswmnd 32108 | . . . 4 ⊢ 𝑊 ∈ Mnd |
8 | 7 | a1i 11 | . . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → 𝑊 ∈ Mnd) |
9 | fzo0ssnn0 13211 | . . . . . 6 ⊢ (0..^(♯‘𝐹)) ⊆ ℕ0 | |
10 | nn0uz 12364 | . . . . . 6 ⊢ ℕ0 = (ℤ≥‘0) | |
11 | 9, 10 | sseqtri 3913 | . . . . 5 ⊢ (0..^(♯‘𝐹)) ⊆ (ℤ≥‘0) |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝐹 ∈ Word ℝ → (0..^(♯‘𝐹)) ⊆ (ℤ≥‘0)) |
13 | 12 | sselda 3877 | . . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → 𝑁 ∈ (ℤ≥‘0)) |
14 | wrdf 13962 | . . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → 𝐹:(0..^(♯‘𝐹))⟶ℝ) | |
15 | 14 | ad2antrr 726 | . . . . . 6 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ 𝑖 ∈ (0...𝑁)) → 𝐹:(0..^(♯‘𝐹))⟶ℝ) |
16 | fzssfzo 32090 | . . . . . . . 8 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → (0...𝑁) ⊆ (0..^(♯‘𝐹))) | |
17 | 16 | adantl 485 | . . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → (0...𝑁) ⊆ (0..^(♯‘𝐹))) |
18 | 17 | sselda 3877 | . . . . . 6 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ 𝑖 ∈ (0...𝑁)) → 𝑖 ∈ (0..^(♯‘𝐹))) |
19 | 15, 18 | ffvelrnd 6864 | . . . . 5 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ 𝑖 ∈ (0...𝑁)) → (𝐹‘𝑖) ∈ ℝ) |
20 | 19 | rexrd 10771 | . . . 4 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ 𝑖 ∈ (0...𝑁)) → (𝐹‘𝑖) ∈ ℝ*) |
21 | sgncl 32077 | . . . 4 ⊢ ((𝐹‘𝑖) ∈ ℝ* → (sgn‘(𝐹‘𝑖)) ∈ {-1, 0, 1}) | |
22 | 20, 21 | syl 17 | . . 3 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ 𝑖 ∈ (0...𝑁)) → (sgn‘(𝐹‘𝑖)) ∈ {-1, 0, 1}) |
23 | 6, 8, 13, 22 | gsumncl 32091 | . 2 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → (𝑊 Σg (𝑖 ∈ (0...𝑁) ↦ (sgn‘(𝐹‘𝑖)))) ∈ {-1, 0, 1}) |
24 | 5, 23 | eqeltrd 2833 | 1 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘𝐹)‘𝑁) ∈ {-1, 0, 1}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ⊆ wss 3843 ifcif 4414 {cpr 4518 {ctp 4520 〈cop 4522 ↦ cmpt 5110 ⟶wf 6335 ‘cfv 6339 (class class class)co 7172 ∈ cmpo 7174 ℝcr 10616 0cc0 10617 1c1 10618 ℝ*cxr 10754 − cmin 10950 -cneg 10951 ℕ0cn0 11978 ℤ≥cuz 12326 ...cfz 12983 ..^cfzo 13126 ♯chash 13784 Word cword 13957 sgncsgn 14537 Σcsu 15137 ndxcnx 16585 Basecbs 16588 +gcplusg 16670 Σg cgsu 16819 Mndcmnd 18029 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-om 7602 df-1st 7716 df-2nd 7717 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-er 8322 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-card 9443 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-nn 11719 df-2 11781 df-n0 11979 df-z 12065 df-uz 12327 df-fz 12984 df-fzo 13127 df-seq 13463 df-hash 13785 df-word 13958 df-sgn 14538 df-struct 16590 df-ndx 16591 df-slot 16592 df-base 16594 df-plusg 16683 df-0g 16820 df-gsum 16821 df-mgm 17970 df-sgrp 18019 df-mnd 18030 |
This theorem is referenced by: signsvtn0 32121 signstfvneq0 32123 signstfvcl 32124 signstfveq0 32128 |
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