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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 20428 | . . 3 β’ (πΉ β π΄ β π β Ring) |
3 | eqid 2732 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
4 | abv0.z | . . . 4 β’ 0 = (0gβπ ) | |
5 | 3, 4 | ring0cl 20083 | . . 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 2732 | . . 3 β’ 0 = 0 | |
8 | 1, 3, 4 | abveq0 20433 | . . 3 β’ ((πΉ β π΄ β§ 0 β (Baseβπ )) β ((πΉβ 0 ) = 0 β 0 = 0 )) |
9 | 7, 8 | mpbiri 257 | . 2 β’ ((πΉ β π΄ β§ 0 β (Baseβπ )) β (πΉβ 0 ) = 0) |
10 | 6, 9 | mpdan 685 | 1 β’ (πΉ β π΄ β (πΉβ 0 ) = 0) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βcfv 6543 0cc0 11109 Basecbs 17143 0gc0g 17384 Ringcrg 20055 AbsValcabv 20423 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-map 8821 df-0g 17386 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-grp 18821 df-ring 20057 df-abv 20424 |
This theorem is referenced by: abvdom 20445 abvres 20446 abvcxp 27115 qabvle 27125 ostthlem1 27127 ostth2lem2 27134 ostth3 27138 |
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