Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 15005 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵))) | |
5 | 1, 2, 3, 4 | syl3anc 1370 | 1 ⊢ (𝜑 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ‘cfv 6432 (class class class)co 7271 ℂcc 10870 0cc0 10872 / cdiv 11632 abscabs 14943 |
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 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-sup 9179 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 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 12582 df-rp 12730 df-seq 13720 df-exp 13781 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 |
This theorem is referenced by: reccn2 15304 rlimno1 15363 o1fsum 15523 divrcnv 15562 georeclim 15582 eftabs 15783 efcllem 15785 efaddlem 15800 mul4sqlem 16652 gzrngunit 20662 pjthlem1 24599 iblabsr 24992 iblmulc2 24993 c1liplem1 25158 ftc1lem4 25201 ulmdvlem1 25557 dvradcnv 25578 eff1olem 25702 logcnlem4 25798 lawcoslem1 25963 isosctrlem3 25968 cxploglim2 26126 fsumharmonic 26159 lgamgulmlem2 26177 lgamgulmlem5 26180 lgamcvg2 26202 logfacrlim 26370 2sqlem3 26566 dchrmusum2 26640 dchrvmasumlem3 26645 dchrisum0lem1 26662 dchrisum0lem2a 26663 mudivsum 26676 mulogsumlem 26677 2vmadivsumlem 26686 selberg3lem1 26703 selberg3lem2 26704 selberg4lem1 26706 pntrlog2bndlem1 26723 pntrlog2bndlem3 26725 pntrlog2bndlem5 26727 pntrlog2bndlem6 26729 pntpbnd1a 26731 pntpbnd2 26733 pntibndlem2 26737 pntlemo 26753 pjhthlem1 29749 qqhnm 31936 unbdqndv2lem1 34685 unbdqndv2lem2 34686 knoppndvlem10 34697 knoppndvlem14 34701 iblmulc2nc 35838 ftc1cnnclem 35844 pellexlem2 40649 pellexlem6 40653 modabsdifz 40805 cvgdvgrat 41901 binomcxplemnotnn0 41944 0ellimcdiv 43161 dvdivbd 43435 fourierdlem30 43649 fourierdlem39 43658 etransclem23 43769 |
Copyright terms: Public domain | W3C validator |