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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 19719 | . . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ((𝐹‘𝑋) = 0 ↔ 𝑋 = 0 )) |
5 | 4 | necon3bid 2979 | . 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ((𝐹‘𝑋) ≠ 0 ↔ 𝑋 ≠ 0 )) |
6 | 5 | biimp3ar 1471 | 1 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐹‘𝑋) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 ‘cfv 6340 0cc0 10618 Basecbs 16589 0gc0g 16819 AbsValcabv 19709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3401 df-sbc 3682 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-fv 6348 df-ov 7176 df-oprab 7177 df-mpo 7178 df-map 8442 df-abv 19710 |
This theorem is referenced by: abvgt0 19721 abv1z 19725 abvrec 19729 abvdiv 19730 abvdom 19731 |
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