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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > signstf | Structured version Visualization version GIF version |
Description: The zero skipping sign word is a 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 |
---|---|
signstf | ⊢ (𝐹 ∈ Word ℝ → (𝑇‘𝐹) ∈ Word ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | signsv.p | . . . 4 ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏)) | |
2 | signsv.w | . . . 4 ⊢ 𝑊 = {〈(Base‘ndx), {-1, 0, 1}〉, 〈(+g‘ndx), ⨣ 〉} | |
3 | signsv.t | . . . 4 ⊢ 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(♯‘𝑓)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓‘𝑖)))))) | |
4 | signsv.v | . . . 4 ⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(♯‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0)) | |
5 | 1, 2, 3, 4 | signstfv 34557 | . . 3 ⊢ (𝐹 ∈ Word ℝ → (𝑇‘𝐹) = (𝑛 ∈ (0..^(♯‘𝐹)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝐹‘𝑖)))))) |
6 | neg1rr 12379 | . . . . 5 ⊢ -1 ∈ ℝ | |
7 | 0re 11261 | . . . . 5 ⊢ 0 ∈ ℝ | |
8 | 1re 11259 | . . . . 5 ⊢ 1 ∈ ℝ | |
9 | tpssi 4843 | . . . . 5 ⊢ ((-1 ∈ ℝ ∧ 0 ∈ ℝ ∧ 1 ∈ ℝ) → {-1, 0, 1} ⊆ ℝ) | |
10 | 6, 7, 8, 9 | mp3an 1460 | . . . 4 ⊢ {-1, 0, 1} ⊆ ℝ |
11 | 1, 2 | signswbase 34548 | . . . . 5 ⊢ {-1, 0, 1} = (Base‘𝑊) |
12 | 1, 2 | signswmnd 34551 | . . . . . 6 ⊢ 𝑊 ∈ Mnd |
13 | 12 | a1i 11 | . . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑛 ∈ (0..^(♯‘𝐹))) → 𝑊 ∈ Mnd) |
14 | fzo0ssnn0 13782 | . . . . . . . 8 ⊢ (0..^(♯‘𝐹)) ⊆ ℕ0 | |
15 | nn0uz 12918 | . . . . . . . 8 ⊢ ℕ0 = (ℤ≥‘0) | |
16 | 14, 15 | sseqtri 4032 | . . . . . . 7 ⊢ (0..^(♯‘𝐹)) ⊆ (ℤ≥‘0) |
17 | 16 | a1i 11 | . . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (0..^(♯‘𝐹)) ⊆ (ℤ≥‘0)) |
18 | 17 | sselda 3995 | . . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑛 ∈ (0..^(♯‘𝐹))) → 𝑛 ∈ (ℤ≥‘0)) |
19 | wrdf 14554 | . . . . . . . . 9 ⊢ (𝐹 ∈ Word ℝ → 𝐹:(0..^(♯‘𝐹))⟶ℝ) | |
20 | 19 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑛 ∈ (0..^(♯‘𝐹))) ∧ 𝑖 ∈ (0...𝑛)) → 𝐹:(0..^(♯‘𝐹))⟶ℝ) |
21 | fzssfzo 34533 | . . . . . . . . . 10 ⊢ (𝑛 ∈ (0..^(♯‘𝐹)) → (0...𝑛) ⊆ (0..^(♯‘𝐹))) | |
22 | 21 | adantl 481 | . . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑛 ∈ (0..^(♯‘𝐹))) → (0...𝑛) ⊆ (0..^(♯‘𝐹))) |
23 | 22 | sselda 3995 | . . . . . . . 8 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑛 ∈ (0..^(♯‘𝐹))) ∧ 𝑖 ∈ (0...𝑛)) → 𝑖 ∈ (0..^(♯‘𝐹))) |
24 | 20, 23 | ffvelcdmd 7105 | . . . . . . 7 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑛 ∈ (0..^(♯‘𝐹))) ∧ 𝑖 ∈ (0...𝑛)) → (𝐹‘𝑖) ∈ ℝ) |
25 | 24 | rexrd 11309 | . . . . . 6 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑛 ∈ (0..^(♯‘𝐹))) ∧ 𝑖 ∈ (0...𝑛)) → (𝐹‘𝑖) ∈ ℝ*) |
26 | sgncl 34520 | . . . . . 6 ⊢ ((𝐹‘𝑖) ∈ ℝ* → (sgn‘(𝐹‘𝑖)) ∈ {-1, 0, 1}) | |
27 | 25, 26 | syl 17 | . . . . 5 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑛 ∈ (0..^(♯‘𝐹))) ∧ 𝑖 ∈ (0...𝑛)) → (sgn‘(𝐹‘𝑖)) ∈ {-1, 0, 1}) |
28 | 11, 13, 18, 27 | gsumncl 34534 | . . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑛 ∈ (0..^(♯‘𝐹))) → (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝐹‘𝑖)))) ∈ {-1, 0, 1}) |
29 | 10, 28 | sselid 3993 | . . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑛 ∈ (0..^(♯‘𝐹))) → (𝑊 Σg (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝐹‘𝑖)))) ∈ ℝ) |
30 | 5, 29 | fmpt3d 7136 | . 2 ⊢ (𝐹 ∈ Word ℝ → (𝑇‘𝐹):(0..^(♯‘𝐹))⟶ℝ) |
31 | iswrdi 14553 | . 2 ⊢ ((𝑇‘𝐹):(0..^(♯‘𝐹))⟶ℝ → (𝑇‘𝐹) ∈ Word ℝ) | |
32 | 30, 31 | syl 17 | 1 ⊢ (𝐹 ∈ Word ℝ → (𝑇‘𝐹) ∈ Word ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 ifcif 4531 {cpr 4633 {ctp 4635 〈cop 4637 ↦ cmpt 5231 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ℝcr 11152 0cc0 11153 1c1 11154 ℝ*cxr 11292 − cmin 11490 -cneg 11491 ℕ0cn0 12524 ℤ≥cuz 12876 ...cfz 13544 ..^cfzo 13691 ♯chash 14366 Word cword 14549 sgncsgn 15122 Σcsu 15719 ndxcnx 17227 Basecbs 17245 +gcplusg 17298 Σg cgsu 17487 Mndcmnd 18760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-word 14550 df-sgn 15123 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-0g 17488 df-gsum 17489 df-mgm 18666 df-sgrp 18745 df-mnd 18761 |
This theorem is referenced by: signstres 34569 signsvtp 34577 signsvtn 34578 |
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