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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 34093 | . . 3 β’ ((π β {-1, 0, 1} β§ π β {-1, 0, 1}) β (π ⨣ π) = if(π = 0, π, π)) |
3 | 2 | adantr 480 | . 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 484 | . . . 4 β’ (((π β {-1, 0, 1} β§ π β {-1, 0, 1}) β§ π β 0) β π β 0) | |
5 | 4 | neneqd 2939 | . . 3 β’ (((π β {-1, 0, 1} β§ π β {-1, 0, 1}) β§ π β 0) β Β¬ π = 0) |
6 | 5 | iffalsed 4534 | . 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 2766 | 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 395 = wceq 1533 β wcel 2098 β wne 2934 ifcif 4523 {cpr 4625 {ctp 4627 β¨cop 4629 βcfv 6537 (class class class)co 7405 β cmpo 7407 0cc0 11112 1c1 11113 -cneg 11449 ndxcnx 17135 Basecbs 17153 +gcplusg 17206 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6489 df-fun 6539 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 |
This theorem is referenced by: signsvtn0 34111 |
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