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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 4568 | . 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 4527 | . . 3 âĒ (ð = ð â if(ð = 0, ð, ð) = if(ð = 0, ð, ð)) | |
3 | eqeq1 2730 | . . . 4 âĒ (ð = ð â (ð = 0 â ð = 0)) | |
4 | id 22 | . . . 4 âĒ (ð = ð â ð = ð) | |
5 | 3, 4 | ifbieq2d 4549 | . . 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 7563 | . 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 395 = wceq 1533 â wcel 2098 ifcif 4523 {ctp 4627 (class class class)co 7405 â cmpo 7407 0cc0 11112 1c1 11113 -cneg 11449 |
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-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: signsw0glem 34094 signswmnd 34098 signswrid 34099 signswlid 34100 signswn0 34101 signswch 34102 |
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