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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 | ⒠⨣ = (π β {-1, 0, 1}, π β {-1, 0, 1} β¦ if(π = 0, π, π)) |
signsv.w | β’ π = {β¨(Baseβndx), {-1, 0, 1}β©, β¨(+gβndx), ⨣ β©} |
signsv.t | β’ π = (π β Word β β¦ (π β (0..^(β―βπ)) β¦ (π Ξ£g (π β (0...π) β¦ (sgnβ(πβπ)))))) |
signsv.v | β’ π = (π β Word β β¦ Ξ£π β (1..^(β―βπ))if(((πβπ)βπ) β ((πβπ)β(π β 1)), 1, 0)) |
Ref | Expression |
---|---|
signstf | β’ (πΉ β Word β β (πβπΉ) β Word β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | signsv.p | . . . 4 ⒠⨣ = (π β {-1, 0, 1}, π β {-1, 0, 1} β¦ if(π = 0, π, π)) | |
2 | signsv.w | . . . 4 β’ π = {β¨(Baseβndx), {-1, 0, 1}β©, β¨(+gβndx), ⨣ β©} | |
3 | signsv.t | . . . 4 β’ π = (π β Word β β¦ (π β (0..^(β―βπ)) β¦ (π Ξ£g (π β (0...π) β¦ (sgnβ(πβπ)))))) | |
4 | signsv.v | . . . 4 β’ π = (π β Word β β¦ Ξ£π β (1..^(β―βπ))if(((πβπ)βπ) β ((πβπ)β(π β 1)), 1, 0)) | |
5 | 1, 2, 3, 4 | signstfv 34228 | . . 3 β’ (πΉ β Word β β (πβπΉ) = (π β (0..^(β―βπΉ)) β¦ (π Ξ£g (π β (0...π) β¦ (sgnβ(πΉβπ)))))) |
6 | neg1rr 12365 | . . . . 5 β’ -1 β β | |
7 | 0re 11254 | . . . . 5 β’ 0 β β | |
8 | 1re 11252 | . . . . 5 β’ 1 β β | |
9 | tpssi 4844 | . . . . 5 β’ ((-1 β β β§ 0 β β β§ 1 β β) β {-1, 0, 1} β β) | |
10 | 6, 7, 8, 9 | mp3an 1457 | . . . 4 β’ {-1, 0, 1} β β |
11 | 1, 2 | signswbase 34219 | . . . . 5 β’ {-1, 0, 1} = (Baseβπ) |
12 | 1, 2 | signswmnd 34222 | . . . . . 6 β’ π β Mnd |
13 | 12 | a1i 11 | . . . . 5 β’ ((πΉ β Word β β§ π β (0..^(β―βπΉ))) β π β Mnd) |
14 | fzo0ssnn0 13753 | . . . . . . . 8 β’ (0..^(β―βπΉ)) β β0 | |
15 | nn0uz 12902 | . . . . . . . 8 β’ β0 = (β€β₯β0) | |
16 | 14, 15 | sseqtri 4018 | . . . . . . 7 β’ (0..^(β―βπΉ)) β (β€β₯β0) |
17 | 16 | a1i 11 | . . . . . 6 β’ (πΉ β Word β β (0..^(β―βπΉ)) β (β€β₯β0)) |
18 | 17 | sselda 3982 | . . . . 5 β’ ((πΉ β Word β β§ π β (0..^(β―βπΉ))) β π β (β€β₯β0)) |
19 | wrdf 14509 | . . . . . . . . 9 β’ (πΉ β Word β β πΉ:(0..^(β―βπΉ))βΆβ) | |
20 | 19 | ad2antrr 724 | . . . . . . . 8 β’ (((πΉ β Word β β§ π β (0..^(β―βπΉ))) β§ π β (0...π)) β πΉ:(0..^(β―βπΉ))βΆβ) |
21 | fzssfzo 34204 | . . . . . . . . . 10 β’ (π β (0..^(β―βπΉ)) β (0...π) β (0..^(β―βπΉ))) | |
22 | 21 | adantl 480 | . . . . . . . . 9 β’ ((πΉ β Word β β§ π β (0..^(β―βπΉ))) β (0...π) β (0..^(β―βπΉ))) |
23 | 22 | sselda 3982 | . . . . . . . 8 β’ (((πΉ β Word β β§ π β (0..^(β―βπΉ))) β§ π β (0...π)) β π β (0..^(β―βπΉ))) |
24 | 20, 23 | ffvelcdmd 7100 | . . . . . . 7 β’ (((πΉ β Word β β§ π β (0..^(β―βπΉ))) β§ π β (0...π)) β (πΉβπ) β β) |
25 | 24 | rexrd 11302 | . . . . . 6 β’ (((πΉ β Word β β§ π β (0..^(β―βπΉ))) β§ π β (0...π)) β (πΉβπ) β β*) |
26 | sgncl 34191 | . . . . . 6 β’ ((πΉβπ) β β* β (sgnβ(πΉβπ)) β {-1, 0, 1}) | |
27 | 25, 26 | syl 17 | . . . . 5 β’ (((πΉ β Word β β§ π β (0..^(β―βπΉ))) β§ π β (0...π)) β (sgnβ(πΉβπ)) β {-1, 0, 1}) |
28 | 11, 13, 18, 27 | gsumncl 34205 | . . . 4 β’ ((πΉ β Word β β§ π β (0..^(β―βπΉ))) β (π Ξ£g (π β (0...π) β¦ (sgnβ(πΉβπ)))) β {-1, 0, 1}) |
29 | 10, 28 | sselid 3980 | . . 3 β’ ((πΉ β Word β β§ π β (0..^(β―βπΉ))) β (π Ξ£g (π β (0...π) β¦ (sgnβ(πΉβπ)))) β β) |
30 | 5, 29 | fmpt3d 7131 | . 2 β’ (πΉ β Word β β (πβπΉ):(0..^(β―βπΉ))βΆβ) |
31 | iswrdi 14508 | . 2 β’ ((πβπΉ):(0..^(β―βπΉ))βΆβ β (πβπΉ) β Word β) | |
32 | 30, 31 | syl 17 | 1 β’ (πΉ β Word β β (πβπΉ) β Word β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2937 β wss 3949 ifcif 4532 {cpr 4634 {ctp 4636 β¨cop 4638 β¦ cmpt 5235 βΆwf 6549 βcfv 6553 (class class class)co 7426 β cmpo 7428 βcr 11145 0cc0 11146 1c1 11147 β*cxr 11285 β cmin 11482 -cneg 11483 β0cn0 12510 β€β₯cuz 12860 ...cfz 13524 ..^cfzo 13667 β―chash 14329 Word cword 14504 sgncsgn 15073 Ξ£csu 15672 ndxcnx 17169 Basecbs 17187 +gcplusg 17240 Ξ£g cgsu 17429 Mndcmnd 18701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 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 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-seq 14007 df-hash 14330 df-word 14505 df-sgn 15074 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-0g 17430 df-gsum 17431 df-mgm 18607 df-sgrp 18686 df-mnd 18702 |
This theorem is referenced by: signstres 34240 signsvtp 34248 signsvtn 34249 |
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