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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 | ⊢ 𝐴 = (AbsVal‘𝑅) |
| abv0.z | ⊢ 0 = (0g‘𝑅) |
| Ref | Expression |
|---|---|
| abv0 | ⊢ (𝐹 ∈ 𝐴 → (𝐹‘ 0 ) = 0) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | abv0.a | . . . 4 ⊢ 𝐴 = (AbsVal‘𝑅) | |
| 2 | 1 | abvrcl 20814 | . . 3 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
| 3 | eqid 2737 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 4 | abv0.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
| 5 | 3, 4 | ring0cl 20264 | . . 3 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅)) |
| 6 | 2, 5 | syl 17 | . 2 ⊢ (𝐹 ∈ 𝐴 → 0 ∈ (Base‘𝑅)) |
| 7 | eqid 2737 | . . 3 ⊢ 0 = 0 | |
| 8 | 1, 3, 4 | abveq0 20819 | . . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 0 ∈ (Base‘𝑅)) → ((𝐹‘ 0 ) = 0 ↔ 0 = 0 )) |
| 9 | 7, 8 | mpbiri 258 | . 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 0 ∈ (Base‘𝑅)) → (𝐹‘ 0 ) = 0) |
| 10 | 6, 9 | mpdan 687 | 1 ⊢ (𝐹 ∈ 𝐴 → (𝐹‘ 0 ) = 0) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 0cc0 11155 Basecbs 17247 0gc0g 17484 Ringcrg 20230 AbsValcabv 20809 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-ring 20232 df-abv 20810 |
| This theorem is referenced by: abvdom 20831 abvres 20832 abvcxp 27659 qabvle 27669 ostthlem1 27671 ostth2lem2 27678 ostth3 27682 fiabv 42546 |
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