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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 20700 | . . 3 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
3 | eqid 2728 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | abv0.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
5 | 3, 4 | ring0cl 20202 | . . 3 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅)) |
6 | 2, 5 | syl 17 | . 2 ⊢ (𝐹 ∈ 𝐴 → 0 ∈ (Base‘𝑅)) |
7 | eqid 2728 | . . 3 ⊢ 0 = 0 | |
8 | 1, 3, 4 | abveq0 20705 | . . 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 686 | 1 ⊢ (𝐹 ∈ 𝐴 → (𝐹‘ 0 ) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ‘cfv 6548 0cc0 11138 Basecbs 17179 0gc0g 17420 Ringcrg 20172 AbsValcabv 20695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-map 8846 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18892 df-ring 20174 df-abv 20696 |
This theorem is referenced by: abvdom 20717 abvres 20718 abvcxp 27547 qabvle 27557 ostthlem1 27559 ostth2lem2 27566 ostth3 27570 |
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