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Mirrors > Home > ILE Home > Th. List > absgt0ap | GIF version |
Description: The absolute value of a number apart from zero is positive. (Contributed by Jim Kingdon, 13-Aug-2021.) |
Ref | Expression |
---|---|
absgt0ap | ⊢ (𝐴 ∈ ℂ → (𝐴 # 0 ↔ 0 < (abs‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0red 7685 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℝ) | |
2 | abscl 10709 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | |
3 | absge0 10718 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴)) | |
4 | 1, 2, 3 | leltapd 8312 | . 2 ⊢ (𝐴 ∈ ℂ → (0 < (abs‘𝐴) ↔ (abs‘𝐴) # 0)) |
5 | abs00ap 10720 | . 2 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) # 0 ↔ 𝐴 # 0)) | |
6 | 4, 5 | bitr2d 188 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 # 0 ↔ 0 < (abs‘𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∈ wcel 1461 class class class wbr 3893 ‘cfv 5079 ℂcc 7539 0cc0 7541 < clt 7718 # cap 8255 abscabs 10655 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-coll 4001 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-iinf 4460 ax-cnex 7630 ax-resscn 7631 ax-1cn 7632 ax-1re 7633 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-mulrcl 7638 ax-addcom 7639 ax-mulcom 7640 ax-addass 7641 ax-mulass 7642 ax-distr 7643 ax-i2m1 7644 ax-0lt1 7645 ax-1rid 7646 ax-0id 7647 ax-rnegex 7648 ax-precex 7649 ax-cnre 7650 ax-pre-ltirr 7651 ax-pre-ltwlin 7652 ax-pre-lttrn 7653 ax-pre-apti 7654 ax-pre-ltadd 7655 ax-pre-mulgt0 7656 ax-pre-mulext 7657 ax-arch 7658 ax-caucvg 7659 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rmo 2396 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-if 3439 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-tr 3985 df-id 4173 df-po 4176 df-iso 4177 df-iord 4246 df-on 4248 df-ilim 4249 df-suc 4251 df-iom 4463 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-1st 5990 df-2nd 5991 df-recs 6154 df-frec 6240 df-pnf 7720 df-mnf 7721 df-xr 7722 df-ltxr 7723 df-le 7724 df-sub 7852 df-neg 7853 df-reap 8249 df-ap 8256 df-div 8340 df-inn 8625 df-2 8683 df-3 8684 df-4 8685 df-n0 8876 df-z 8953 df-uz 9223 df-rp 9338 df-seqfrec 10106 df-exp 10180 df-cj 10501 df-re 10502 df-im 10503 df-rsqrt 10656 df-abs 10657 |
This theorem is referenced by: nnabscl 10758 absgt0api 10804 mul0inf 10898 |
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