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Mirrors > Home > MPE Home > Th. List > Mathboxes > signswrid | Structured version Visualization version GIF version |
Description: The zero-skipping operation propagages nonzeros. (Contributed by Thierry Arnoux, 11-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 |
---|---|
signswrid | β’ (π β {-1, 0, 1} β (π ⨣ 0) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0ex 11070 | . . . 4 β’ 0 β V | |
2 | 1 | tpid2 4718 | . . 3 β’ 0 β {-1, 0, 1} |
3 | signsw.p | . . . 4 ⒠⨣ = (π β {-1, 0, 1}, π β {-1, 0, 1} β¦ if(π = 0, π, π)) | |
4 | 3 | signspval 32831 | . . 3 β’ ((π β {-1, 0, 1} β§ 0 β {-1, 0, 1}) β (π ⨣ 0) = if(0 = 0, π, 0)) |
5 | 2, 4 | mpan2 688 | . 2 β’ (π β {-1, 0, 1} β (π ⨣ 0) = if(0 = 0, π, 0)) |
6 | eqid 2736 | . . 3 β’ 0 = 0 | |
7 | iftrue 4479 | . . 3 β’ (0 = 0 β if(0 = 0, π, 0) = π) | |
8 | 6, 7 | mp1i 13 | . 2 β’ (π β {-1, 0, 1} β if(0 = 0, π, 0) = π) |
9 | 5, 8 | eqtrd 2776 | 1 β’ (π β {-1, 0, 1} β (π ⨣ 0) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1540 β wcel 2105 ifcif 4473 {cpr 4575 {ctp 4577 β¨cop 4579 βcfv 6479 (class class class)co 7337 β cmpo 7339 0cc0 10972 1c1 10973 -cneg 11307 ndxcnx 16991 Basecbs 17009 +gcplusg 17059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pr 5372 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-mulcl 11034 ax-i2m1 11040 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-sbc 3728 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4853 df-br 5093 df-opab 5155 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-iota 6431 df-fun 6481 df-fv 6487 df-ov 7340 df-oprab 7341 df-mpo 7342 |
This theorem is referenced by: signstfveq0 32856 |
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