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Mirrors > Home > MPE Home > Th. List > abv0 | Structured version Visualization version GIF version |
Description: The absolute value of zero is zero. (Contributed by Mario Carneiro, 8-Sep-2014.) |
Ref | Expression |
---|---|
abv0.a | β’ π΄ = (AbsValβπ ) |
abv0.z | β’ 0 = (0gβπ ) |
Ref | Expression |
---|---|
abv0 | β’ (πΉ β π΄ β (πΉβ 0 ) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abv0.a | . . . 4 β’ π΄ = (AbsValβπ ) | |
2 | 1 | abvrcl 20323 | . . 3 β’ (πΉ β π΄ β π β Ring) |
3 | eqid 2733 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
4 | abv0.z | . . . 4 β’ 0 = (0gβπ ) | |
5 | 3, 4 | ring0cl 19998 | . . 3 β’ (π β Ring β 0 β (Baseβπ )) |
6 | 2, 5 | syl 17 | . 2 β’ (πΉ β π΄ β 0 β (Baseβπ )) |
7 | eqid 2733 | . . 3 β’ 0 = 0 | |
8 | 1, 3, 4 | abveq0 20328 | . . 3 β’ ((πΉ β π΄ β§ 0 β (Baseβπ )) β ((πΉβ 0 ) = 0 β 0 = 0 )) |
9 | 7, 8 | mpbiri 258 | . 2 β’ ((πΉ β π΄ β§ 0 β (Baseβπ )) β (πΉβ 0 ) = 0) |
10 | 6, 9 | mpdan 686 | 1 β’ (πΉ β π΄ β (πΉβ 0 ) = 0) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βcfv 6500 0cc0 11059 Basecbs 17091 0gc0g 17329 Ringcrg 19972 AbsValcabv 20318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-map 8773 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-ring 19974 df-abv 20319 |
This theorem is referenced by: abvdom 20340 abvres 20341 abvcxp 26986 qabvle 26996 ostthlem1 26998 ostth2lem2 27005 ostth3 27009 |
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