![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > absgt0 | Structured version Visualization version GIF version |
Description: The absolute value of a nonzero number is positive. (Contributed by NM, 1-Oct-1999.) (Proof shortened by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
absgt0 | ⊢ (𝐴 ∈ ℂ → (𝐴 ≠ 0 ↔ 0 < (abs‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0red 11159 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℝ) | |
2 | abscl 15164 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | |
3 | absge0 15173 | . . 3 ⊢ (𝐴 ∈ ℂ → 0 ≤ (abs‘𝐴)) | |
4 | 1, 2, 3 | leltned 11309 | . 2 ⊢ (𝐴 ∈ ℂ → (0 < (abs‘𝐴) ↔ (abs‘𝐴) ≠ 0)) |
5 | abs00 15175 | . . 3 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) = 0 ↔ 𝐴 = 0)) | |
6 | 5 | necon3bid 2989 | . 2 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0)) |
7 | 4, 6 | bitr2d 280 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 ≠ 0 ↔ 0 < (abs‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2107 ≠ wne 2944 class class class wbr 5106 ‘cfv 6497 ℂcc 11050 0cc0 11052 < clt 11190 abscabs 15120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-sup 9379 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-n0 12415 df-z 12501 df-uz 12765 df-rp 12917 df-seq 13908 df-exp 13969 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 |
This theorem is referenced by: nnabscl 15211 absgt0i 15285 absabv 20857 dveflem 25346 abelthlem2 25794 nmophmi 30976 unbdqndv2lem1 34975 ftc1anclem7 36160 ftc1anc 36162 pellexlem2 41156 dvgrat 42599 fourierdlem42 44397 |
Copyright terms: Public domain | W3C validator |