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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > signspval | Structured version Visualization version GIF version |
Description: The value of the skipping 0 sign operation. (Contributed by Thierry Arnoux, 9-Sep-2018.) |
Ref | Expression |
---|---|
signsw.p | âĒ âĻĢ = (ð â {-1, 0, 1}, ð â {-1, 0, 1} âĶ if(ð = 0, ð, ð)) |
Ref | Expression |
---|---|
signspval | âĒ ((ð â {-1, 0, 1} â§ ð â {-1, 0, 1}) â (ð âĻĢ ð) = if(ð = 0, ð, ð)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifcl 4569 | . 2 âĒ ((ð â {-1, 0, 1} â§ ð â {-1, 0, 1}) â if(ð = 0, ð, ð) â {-1, 0, 1}) | |
2 | ifeq1 4528 | . . 3 âĒ (ð = ð â if(ð = 0, ð, ð) = if(ð = 0, ð, ð)) | |
3 | eqeq1 2729 | . . . 4 âĒ (ð = ð â (ð = 0 â ð = 0)) | |
4 | id 22 | . . . 4 âĒ (ð = ð â ð = ð) | |
5 | 3, 4 | ifbieq2d 4550 | . . 3 âĒ (ð = ð â if(ð = 0, ð, ð) = if(ð = 0, ð, ð)) |
6 | signsw.p | . . 3 âĒ âĻĢ = (ð â {-1, 0, 1}, ð â {-1, 0, 1} âĶ if(ð = 0, ð, ð)) | |
7 | 2, 5, 6 | ovmpog 7577 | . 2 âĒ ((ð â {-1, 0, 1} â§ ð â {-1, 0, 1} â§ if(ð = 0, ð, ð) â {-1, 0, 1}) â (ð âĻĢ ð) = if(ð = 0, ð, ð)) |
8 | 1, 7 | mpd3an3 1458 | 1 âĒ ((ð â {-1, 0, 1} â§ ð â {-1, 0, 1}) â (ð âĻĢ ð) = if(ð = 0, ð, ð)) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 â§ wa 394 = wceq 1533 â wcel 2098 ifcif 4524 {ctp 4628 (class class class)co 7416 â cmpo 7418 0cc0 11138 1c1 11139 -cneg 11475 |
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-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: signsw0glem 34242 signswmnd 34246 signswrid 34247 signswlid 34248 signswn0 34249 signswch 34250 |
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