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 19600 | . . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ((𝐹‘𝑋) = 0 ↔ 𝑋 = 0 )) |
5 | 4 | necon3bid 3063 | . 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ((𝐹‘𝑋) ≠ 0 ↔ 𝑋 ≠ 0 )) |
6 | 5 | biimp3ar 1466 | 1 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐹‘𝑋) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ‘cfv 6358 0cc0 10540 Basecbs 16486 0gc0g 16716 AbsValcabv 19590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-map 8411 df-abv 19591 |
This theorem is referenced by: abvgt0 19602 abv1z 19606 abvrec 19610 abvdiv 19611 abvdom 19612 |
Copyright terms: Public domain | W3C validator |