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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 19592 | . . 3 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
3 | eqid 2821 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | abv0.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
5 | 3, 4 | ring0cl 19319 | . . 3 ⊢ (𝑅 ∈ Ring → 0 ∈ (Base‘𝑅)) |
6 | 2, 5 | syl 17 | . 2 ⊢ (𝐹 ∈ 𝐴 → 0 ∈ (Base‘𝑅)) |
7 | eqid 2821 | . . 3 ⊢ 0 = 0 | |
8 | 1, 3, 4 | abveq0 19597 | . . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 0 ∈ (Base‘𝑅)) → ((𝐹‘ 0 ) = 0 ↔ 0 = 0 )) |
9 | 7, 8 | mpbiri 260 | . 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 0 ∈ (Base‘𝑅)) → (𝐹‘ 0 ) = 0) |
10 | 6, 9 | mpdan 685 | 1 ⊢ (𝐹 ∈ 𝐴 → (𝐹‘ 0 ) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 0cc0 10537 Basecbs 16483 0gc0g 16713 Ringcrg 19297 AbsValcabv 19587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-ring 19299 df-abv 19588 |
This theorem is referenced by: abvdom 19609 abvres 19610 abvcxp 26191 qabvle 26201 ostthlem1 26203 ostth2lem2 26210 ostth3 26214 |
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