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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > signswlid | Structured version Visualization version GIF version |
Description: The zero-skipping operation keeps nonzeros. (Contributed by Thierry Arnoux, 12-Oct-2018.) |
Ref | Expression |
---|---|
signsw.p | ⒠⨣ = (π β {-1, 0, 1}, π β {-1, 0, 1} β¦ if(π = 0, π, π)) |
signsw.w | β’ π = {β¨(Baseβndx), {-1, 0, 1}β©, β¨(+gβndx), ⨣ β©} |
Ref | Expression |
---|---|
signswlid | β’ (((π β {-1, 0, 1} β§ π β {-1, 0, 1}) β§ π β 0) β (π ⨣ π) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | signsw.p | . . . 4 ⒠⨣ = (π β {-1, 0, 1}, π β {-1, 0, 1} β¦ if(π = 0, π, π)) | |
2 | 1 | signspval 34241 | . . 3 β’ ((π β {-1, 0, 1} β§ π β {-1, 0, 1}) β (π ⨣ π) = if(π = 0, π, π)) |
3 | 2 | adantr 479 | . 2 β’ (((π β {-1, 0, 1} β§ π β {-1, 0, 1}) β§ π β 0) β (π ⨣ π) = if(π = 0, π, π)) |
4 | simpr 483 | . . . 4 β’ (((π β {-1, 0, 1} β§ π β {-1, 0, 1}) β§ π β 0) β π β 0) | |
5 | 4 | neneqd 2935 | . . 3 β’ (((π β {-1, 0, 1} β§ π β {-1, 0, 1}) β§ π β 0) β Β¬ π = 0) |
6 | 5 | iffalsed 4535 | . 2 β’ (((π β {-1, 0, 1} β§ π β {-1, 0, 1}) β§ π β 0) β if(π = 0, π, π) = π) |
7 | 3, 6 | eqtrd 2765 | 1 β’ (((π β {-1, 0, 1} β§ π β {-1, 0, 1}) β§ π β 0) β (π ⨣ π) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2930 ifcif 4524 {cpr 4626 {ctp 4628 β¨cop 4630 βcfv 6543 (class class class)co 7416 β cmpo 7418 0cc0 11138 1c1 11139 -cneg 11475 ndxcnx 17161 Basecbs 17179 +gcplusg 17232 |
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-sep 5294 ax-nul 5301 ax-pr 5423 |
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-rab 3420 df-v 3465 df-sbc 3769 df-dif 3942 df-un 3944 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 |
This theorem is referenced by: signsvtn0 34259 |
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