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Mirrors > Home > MPE Home > Th. List > absdivd | Structured version Visualization version GIF version |
Description: Absolute value distributes over division. (Contributed by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
abscld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
abssubd.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
absdivd.2 | ⊢ (𝜑 → 𝐵 ≠ 0) |
Ref | Expression |
---|---|
absdivd | ⊢ (𝜑 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abscld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | abssubd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | absdivd.2 | . 2 ⊢ (𝜑 → 𝐵 ≠ 0) | |
4 | absdiv 14079 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵))) | |
5 | 1, 2, 3, 4 | syl3anc 1366 | 1 ⊢ (𝜑 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ‘cfv 5926 (class class class)co 6690 ℂcc 9972 0cc0 9974 / cdiv 10722 abscabs 14018 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-sup 8389 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-n0 11331 df-z 11416 df-uz 11726 df-rp 11871 df-seq 12842 df-exp 12901 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 |
This theorem is referenced by: reccn2 14371 rlimno1 14428 o1fsum 14589 divrcnv 14628 georeclim 14647 eftabs 14850 efcllem 14852 efaddlem 14867 mul4sqlem 15704 gzrngunit 19860 pjthlem1 23254 iblabsr 23641 iblmulc2 23642 c1liplem1 23804 ftc1lem4 23847 ulmdvlem1 24199 dvradcnv 24220 eff1olem 24339 logcnlem4 24436 lawcoslem1 24590 isosctrlem3 24595 cxploglim2 24750 fsumharmonic 24783 lgamgulmlem2 24801 lgamgulmlem5 24804 lgamcvg2 24826 logfacrlim 24994 2sqlem3 25190 dchrmusum2 25228 dchrvmasumlem3 25233 dchrisum0lem1 25250 dchrisum0lem2a 25251 mudivsum 25264 mulogsumlem 25265 2vmadivsumlem 25274 selberg3lem1 25291 selberg3lem2 25292 selberg4lem1 25294 pntrlog2bndlem1 25311 pntrlog2bndlem3 25313 pntrlog2bndlem5 25315 pntrlog2bndlem6 25317 pntpbnd1a 25319 pntpbnd2 25321 pntibndlem2 25325 pntlemo 25341 pjhthlem1 28378 qqhnm 30162 unbdqndv2lem1 32625 unbdqndv2lem2 32626 knoppndvlem10 32637 knoppndvlem14 32641 iblmulc2nc 33605 ftc1cnnclem 33613 pellexlem2 37711 pellexlem6 37715 modabsdifz 37870 cvgdvgrat 38829 binomcxplemnotnn0 38872 0ellimcdiv 40199 dvdivbd 40456 fourierdlem30 40672 fourierdlem39 40681 etransclem23 40792 |
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