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Mirrors > Home > MPE Home > Th. List > abvne0 | Structured version Visualization version GIF version |
Description: The absolute value of a nonzero number is nonzero. (Contributed by Mario Carneiro, 8-Sep-2014.) |
Ref | Expression |
---|---|
abvf.a | β’ π΄ = (AbsValβπ ) |
abvf.b | β’ π΅ = (Baseβπ ) |
abveq0.z | β’ 0 = (0gβπ ) |
Ref | Expression |
---|---|
abvne0 | β’ ((πΉ β π΄ β§ π β π΅ β§ π β 0 ) β (πΉβπ) β 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abvf.a | . . . 4 β’ π΄ = (AbsValβπ ) | |
2 | abvf.b | . . . 4 β’ π΅ = (Baseβπ ) | |
3 | abveq0.z | . . . 4 β’ 0 = (0gβπ ) | |
4 | 1, 2, 3 | abveq0 20706 | . . 3 β’ ((πΉ β π΄ β§ π β π΅) β ((πΉβπ) = 0 β π = 0 )) |
5 | 4 | necon3bid 2982 | . 2 β’ ((πΉ β π΄ β§ π β π΅) β ((πΉβπ) β 0 β π β 0 )) |
6 | 5 | biimp3ar 1467 | 1 β’ ((πΉ β π΄ β§ π β π΅ β§ π β 0 ) β (πΉβπ) β 0) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1085 = wceq 1534 β wcel 2099 β wne 2937 βcfv 6548 0cc0 11139 Basecbs 17180 0gc0g 17421 AbsValcabv 20696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-map 8847 df-abv 20697 |
This theorem is referenced by: abvgt0 20708 abv1z 20712 abvrec 20716 abvdiv 20717 abvdom 20718 |
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