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Mirrors > Home > MPE Home > Th. List > ne0gt0d | Structured version Visualization version GIF version |
Description: A nonzero nonnegative number is positive. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ne0gt0d.2 | ⊢ (𝜑 → 0 ≤ 𝐴) |
ne0gt0d.3 | ⊢ (𝜑 → 𝐴 ≠ 0) |
Ref | Expression |
---|---|
ne0gt0d | ⊢ (𝜑 → 0 < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ne0gt0d.3 | . 2 ⊢ (𝜑 → 𝐴 ≠ 0) | |
2 | ltd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | ne0gt0d.2 | . . 3 ⊢ (𝜑 → 0 ≤ 𝐴) | |
4 | ne0gt0 11369 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (𝐴 ≠ 0 ↔ 0 < 𝐴)) | |
5 | 2, 3, 4 | syl2anc 582 | . 2 ⊢ (𝜑 → (𝐴 ≠ 0 ↔ 0 < 𝐴)) |
6 | 1, 5 | mpbid 231 | 1 ⊢ (𝜑 → 0 < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2099 ≠ wne 2930 class class class wbr 5153 ℝcr 11157 0cc0 11158 < clt 11298 ≤ cle 11299 |
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 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11215 ax-1cn 11216 ax-addrcl 11219 ax-rnegex 11229 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 |
This theorem is referenced by: sqrtgt0 15263 absrpcl 15293 sqreulem 15364 fprodle 15998 efgt0 16105 abvgt0 20799 nmrpcl 24620 lebnumlem1 24978 ipcau2 25253 recxpcl 26702 mulcxp 26712 rlimcnp 26993 lgsdilem 27353 pntleml 27640 ttgcontlem1 28818 axsegconlem6 28856 axpaschlem 28874 axcontlem2 28899 axcontlem4 28901 axcontlem7 28904 xrge0iifhom 33752 cndprobprob 34272 usgrgt2cycl 34958 tan2h 37313 dvasin 37405 radcnvrat 43988 ioodvbdlimc1lem2 45553 ioodvbdlimc2lem 45555 fourierdlem30 45758 fourierdlem48 45775 fourierdlem49 45776 fourierdlem54 45781 fourierdlem102 45829 fourierdlem114 45841 sqwvfoura 45849 |
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