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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 20831 | . . 3 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
3 | eqid 2735 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | abv0.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
5 | 3, 4 | ring0cl 20281 | . . 3 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅)) |
6 | 2, 5 | syl 17 | . 2 ⊢ (𝐹 ∈ 𝐴 → 0 ∈ (Base‘𝑅)) |
7 | eqid 2735 | . . 3 ⊢ 0 = 0 | |
8 | 1, 3, 4 | abveq0 20836 | . . 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 1537 ∈ wcel 2106 ‘cfv 6563 0cc0 11153 Basecbs 17245 0gc0g 17486 Ringcrg 20251 AbsValcabv 20826 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-ring 20253 df-abv 20827 |
This theorem is referenced by: abvdom 20848 abvres 20849 abvcxp 27674 qabvle 27684 ostthlem1 27686 ostth2lem2 27693 ostth3 27697 fiabv 42523 |
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