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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 18644 | . . 3 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
3 | eqid 2610 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | abv0.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
5 | 3, 4 | ring0cl 18392 | . . 3 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅)) |
6 | 2, 5 | syl 17 | . 2 ⊢ (𝐹 ∈ 𝐴 → 0 ∈ (Base‘𝑅)) |
7 | eqid 2610 | . . 3 ⊢ 0 = 0 | |
8 | 1, 3, 4 | abveq0 18649 | . . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 0 ∈ (Base‘𝑅)) → ((𝐹‘ 0 ) = 0 ↔ 0 = 0 )) |
9 | 7, 8 | mpbiri 247 | . 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 0 ∈ (Base‘𝑅)) → (𝐹‘ 0 ) = 0) |
10 | 6, 9 | mpdan 699 | 1 ⊢ (𝐹 ∈ 𝐴 → (𝐹‘ 0 ) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 0cc0 9815 Basecbs 15695 0gc0g 15923 Ringcrg 18370 AbsValcabv 18639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-ring 18372 df-abv 18640 |
This theorem is referenced by: abvdom 18661 abvres 18662 abvcxp 25104 qabvle 25114 ostthlem1 25116 ostth2lem2 25123 ostth3 25127 |
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