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Mirrors > Home > MPE Home > Th. List > absrpcld | Structured version Visualization version GIF version |
Description: The absolute value of a nonzero number is a positive real. (Contributed by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
abscld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
absne0d.2 | ⊢ (𝜑 → 𝐴 ≠ 0) |
Ref | Expression |
---|---|
absrpcld | ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abscld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | absne0d.2 | . 2 ⊢ (𝜑 → 𝐴 ≠ 0) | |
3 | absrpcl 15000 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℝ+) | |
4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ≠ wne 2943 ‘cfv 6433 ℂcc 10869 0cc0 10871 ℝ+crp 12730 abscabs 14945 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 |
This theorem is referenced by: rlimuni 15259 climuni 15261 rlimrecl 15289 reccn2 15306 rlimno1 15365 georeclim 15584 4sqlem11 16656 recld2 23977 c1liplem1 25160 aalioulem2 25493 aaliou2b 25501 ulmdvlem1 25559 abelthlem7 25597 tanregt0 25695 eff1olem 25704 logcnlem2 25798 logcnlem4 25800 logcn 25802 asinlem3 26021 lgamgulmlem2 26179 lgamgulmlem5 26182 lgambdd 26186 lgamucov 26187 ftalem2 26223 ftalem4 26225 dchrabs 26408 sinccvglem 33630 unbdqndv2lem2 34690 rencldnfilem 40642 pellexlem6 40656 modabsdifz 40808 jm2.19 40815 imo72b2lem1 41780 cvgdvgrat 41931 binomcxplemnotnn0 41974 abssubrp 42814 dstregt0 42820 absimnre 43017 lptre2pt 43181 0ellimcdiv 43190 limclner 43192 climxrre 43291 cnrefiisplem 43370 cncficcgt0 43429 |
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