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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 | ⒠⨣ = (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 |
---|---|
signstcl | β’ ((πΉ β Word β β§ π β (0..^(β―βπΉ))) β ((πβπΉ)βπ) β {-1, 0, 1}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | signsv.p | . . 3 ⒠⨣ = (π β {-1, 0, 1}, π β {-1, 0, 1} β¦ if(π = 0, π, π)) | |
2 | signsv.w | . . 3 β’ π = {β¨(Baseβndx), {-1, 0, 1}β©, β¨(+gβndx), ⨣ β©} | |
3 | signsv.t | . . 3 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 | . . 3 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 | signstfval 34225 | . 2 β’ ((πΉ β Word β β§ π β (0..^(β―βπΉ))) β ((πβπΉ)βπ) = (π Ξ£g (π β (0...π) β¦ (sgnβ(πΉβπ))))) |
6 | 1, 2 | signswbase 34215 | . . 3 β’ {-1, 0, 1} = (Baseβπ) |
7 | 1, 2 | signswmnd 34218 | . . . 4 β’ π β Mnd |
8 | 7 | a1i 11 | . . 3 β’ ((πΉ β Word β β§ π β (0..^(β―βπΉ))) β π β Mnd) |
9 | fzo0ssnn0 13743 | . . . . . 6 β’ (0..^(β―βπΉ)) β β0 | |
10 | nn0uz 12892 | . . . . . 6 β’ β0 = (β€β₯β0) | |
11 | 9, 10 | sseqtri 4008 | . . . . 5 β’ (0..^(β―βπΉ)) β (β€β₯β0) |
12 | 11 | a1i 11 | . . . 4 β’ (πΉ β Word β β (0..^(β―βπΉ)) β (β€β₯β0)) |
13 | 12 | sselda 3972 | . . 3 β’ ((πΉ β Word β β§ π β (0..^(β―βπΉ))) β π β (β€β₯β0)) |
14 | wrdf 14499 | . . . . . . 7 β’ (πΉ β Word β β πΉ:(0..^(β―βπΉ))βΆβ) | |
15 | 14 | ad2antrr 724 | . . . . . 6 β’ (((πΉ β Word β β§ π β (0..^(β―βπΉ))) β§ π β (0...π)) β πΉ:(0..^(β―βπΉ))βΆβ) |
16 | fzssfzo 34200 | . . . . . . . 8 β’ (π β (0..^(β―βπΉ)) β (0...π) β (0..^(β―βπΉ))) | |
17 | 16 | adantl 480 | . . . . . . 7 β’ ((πΉ β Word β β§ π β (0..^(β―βπΉ))) β (0...π) β (0..^(β―βπΉ))) |
18 | 17 | sselda 3972 | . . . . . 6 β’ (((πΉ β Word β β§ π β (0..^(β―βπΉ))) β§ π β (0...π)) β π β (0..^(β―βπΉ))) |
19 | 15, 18 | ffvelcdmd 7088 | . . . . 5 β’ (((πΉ β Word β β§ π β (0..^(β―βπΉ))) β§ π β (0...π)) β (πΉβπ) β β) |
20 | 19 | rexrd 11292 | . . . 4 β’ (((πΉ β Word β β§ π β (0..^(β―βπΉ))) β§ π β (0...π)) β (πΉβπ) β β*) |
21 | sgncl 34187 | . . . 4 β’ ((πΉβπ) β β* β (sgnβ(πΉβπ)) β {-1, 0, 1}) | |
22 | 20, 21 | syl 17 | . . 3 β’ (((πΉ β Word β β§ π β (0..^(β―βπΉ))) β§ π β (0...π)) β (sgnβ(πΉβπ)) β {-1, 0, 1}) |
23 | 6, 8, 13, 22 | gsumncl 34201 | . 2 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}) |
24 | 5, 23 | eqeltrd 2825 | 1 β’ ((πΉ β Word β β§ π β (0..^(β―βπΉ))) β ((πβπΉ)βπ) β {-1, 0, 1}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2930 β wss 3939 ifcif 4522 {cpr 4624 {ctp 4626 β¨cop 4628 β¦ cmpt 5224 βΆwf 6537 βcfv 6541 (class class class)co 7414 β cmpo 7416 βcr 11135 0cc0 11136 1c1 11137 β*cxr 11275 β cmin 11472 -cneg 11473 β0cn0 12500 β€β₯cuz 12850 ...cfz 13514 ..^cfzo 13657 β―chash 14319 Word cword 14494 sgncsgn 15063 Ξ£csu 15662 ndxcnx 17159 Basecbs 17177 +gcplusg 17230 Ξ£g cgsu 17419 Mndcmnd 18691 |
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 2696 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4943 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-fzo 13658 df-seq 13997 df-hash 14320 df-word 14495 df-sgn 15064 df-struct 17113 df-slot 17148 df-ndx 17160 df-base 17178 df-plusg 17243 df-0g 17420 df-gsum 17421 df-mgm 18597 df-sgrp 18676 df-mnd 18692 |
This theorem is referenced by: signsvtn0 34231 signstfvneq0 34233 signstfvcl 34234 signstfveq0 34238 |
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