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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > signstfval | Structured version Visualization version GIF version |
Description: Value of the zero-skipping sign 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 |
---|---|
signstfval | β’ ((πΉ β Word β β§ π β (0..^(β―βπΉ))) β ((πβπΉ)βπ) = (π Ξ£g (π β (0...π) β¦ (sgnβ(πΉβπ))))) |
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 34248 | . . 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 | 5 | adantr 479 | . 2 β’ ((πΉ β Word β β§ π β (0..^(β―βπΉ))) β (πβπΉ) = (π β (0..^(β―βπΉ)) β¦ (π Ξ£g (π β (0...π) β¦ (sgnβ(πΉβπ)))))) |
7 | simpr 483 | . . . . 5 β’ (((πΉ β Word β β§ π β (0..^(β―βπΉ))) β§ π = π) β π = π) | |
8 | 7 | oveq2d 7429 | . . . 4 β’ (((πΉ β Word β β§ π β (0..^(β―βπΉ))) β§ π = π) β (0...π) = (0...π)) |
9 | 8 | mpteq1d 5239 | . . 3 β’ (((πΉ β Word β β§ π β (0..^(β―βπΉ))) β§ π = π) β (π β (0...π) β¦ (sgnβ(πΉβπ))) = (π β (0...π) β¦ (sgnβ(πΉβπ)))) |
10 | 9 | oveq2d 7429 | . 2 β’ (((πΉ β Word β β§ π β (0..^(β―βπΉ))) β§ π = π) β (π Ξ£g (π β (0...π) β¦ (sgnβ(πΉβπ)))) = (π Ξ£g (π β (0...π) β¦ (sgnβ(πΉβπ))))) |
11 | simpr 483 | . 2 β’ ((πΉ β Word β β§ π β (0..^(β―βπΉ))) β π β (0..^(β―βπΉ))) | |
12 | ovexd 7448 | . 2 β’ ((πΉ β Word β β§ π β (0..^(β―βπΉ))) β (π Ξ£g (π β (0...π) β¦ (sgnβ(πΉβπ)))) β V) | |
13 | 6, 10, 11, 12 | fvmptd 7005 | 1 β’ ((πΉ β Word β β§ π β (0..^(β―βπΉ))) β ((πβπΉ)βπ) = (π Ξ£g (π β (0...π) β¦ (sgnβ(πΉβπ))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2930 Vcvv 3463 ifcif 4525 {cpr 4627 {ctp 4629 β¨cop 4631 β¦ cmpt 5227 βcfv 6543 (class class class)co 7413 β cmpo 7415 βcr 11132 0cc0 11133 1c1 11134 β cmin 11469 -cneg 11470 ...cfz 13511 ..^cfzo 13654 β―chash 14316 Word cword 14491 sgncsgn 15060 Ξ£csu 15659 ndxcnx 17156 Basecbs 17174 +gcplusg 17227 Ξ£g cgsu 17416 |
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 5281 ax-sep 5295 ax-nul 5302 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7416 |
This theorem is referenced by: signstcl 34250 signstfvn 34254 signstfvp 34256 |
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