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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 20654 | . . 3 β’ (πΉ β π΄ β π β Ring) |
3 | eqid 2724 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
4 | abv0.z | . . . 4 β’ 0 = (0gβπ ) | |
5 | 3, 4 | ring0cl 20156 | . . 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 2724 | . . 3 β’ 0 = 0 | |
8 | 1, 3, 4 | abveq0 20659 | . . 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 684 | 1 β’ (πΉ β π΄ β (πΉβ 0 ) = 0) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βcfv 6533 0cc0 11106 Basecbs 17143 0gc0g 17384 Ringcrg 20128 AbsValcabv 20649 |
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 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-map 8818 df-0g 17386 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18856 df-ring 20130 df-abv 20650 |
This theorem is referenced by: abvdom 20671 abvres 20672 abvcxp 27464 qabvle 27474 ostthlem1 27476 ostth2lem2 27483 ostth3 27487 |
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