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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 14731 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℝ+) | |
4 | 1, 2, 3 | syl2anc 587 | 1 ⊢ (𝜑 → (abs‘𝐴) ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 ≠ wne 2934 ‘cfv 6333 ℂcc 10606 0cc0 10608 ℝ+crp 12465 abscabs 14676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-pre-sup 10686 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-sup 8972 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-3 11773 df-n0 11970 df-z 12056 df-uz 12318 df-rp 12466 df-seq 13454 df-exp 13515 df-cj 14541 df-re 14542 df-im 14543 df-sqrt 14677 df-abs 14678 |
This theorem is referenced by: rlimuni 14990 climuni 14992 rlimrecl 15020 reccn2 15037 rlimno1 15096 georeclim 15313 4sqlem11 16384 recld2 23559 c1liplem1 24740 aalioulem2 25073 aaliou2b 25081 ulmdvlem1 25139 abelthlem7 25177 tanregt0 25275 eff1olem 25284 logcnlem2 25378 logcnlem4 25380 logcn 25382 asinlem3 25601 lgamgulmlem2 25759 lgamgulmlem5 25762 lgambdd 25766 lgamucov 25767 ftalem2 25803 ftalem4 25805 dchrabs 25988 sinccvglem 33193 unbdqndv2lem2 34320 rencldnfilem 40198 pellexlem6 40212 modabsdifz 40364 jm2.19 40371 imo72b2lem1 41310 cvgdvgrat 41453 binomcxplemnotnn0 41496 abssubrp 42335 dstregt0 42341 absimnre 42541 lptre2pt 42707 0ellimcdiv 42716 limclner 42718 climxrre 42817 cnrefiisplem 42896 cncficcgt0 42955 |
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