![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > abvne0 | Structured version Visualization version GIF version |
Description: The absolute value of a nonzero number is nonzero. (Contributed by Mario Carneiro, 8-Sep-2014.) |
Ref | Expression |
---|---|
abvf.a | ⊢ 𝐴 = (AbsVal‘𝑅) |
abvf.b | ⊢ 𝐵 = (Base‘𝑅) |
abveq0.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
abvne0 | ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐹‘𝑋) ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abvf.a | . . . 4 ⊢ 𝐴 = (AbsVal‘𝑅) | |
2 | abvf.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
3 | abveq0.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
4 | 1, 2, 3 | abveq0 19036 | . . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ((𝐹‘𝑋) = 0 ↔ 𝑋 = 0 )) |
5 | 4 | necon3bid 2987 | . 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ((𝐹‘𝑋) ≠ 0 ↔ 𝑋 ≠ 0 )) |
6 | 5 | biimp3ar 1581 | 1 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐹‘𝑋) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ‘cfv 6030 0cc0 10142 Basecbs 16064 0gc0g 16308 AbsValcabv 19026 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-fv 6038 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-map 8015 df-abv 19027 |
This theorem is referenced by: abvgt0 19038 abv1z 19042 abvrec 19046 abvdiv 19047 abvdom 19048 |
Copyright terms: Public domain | W3C validator |