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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 14489 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵))) | |
5 | 1, 2, 3, 4 | syl3anc 1364 | 1 ⊢ (𝜑 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 ‘cfv 6225 (class class class)co 7016 ℂcc 10381 0cc0 10383 / cdiv 11145 abscabs 14427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-sup 8752 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-n0 11746 df-z 11830 df-uz 12094 df-rp 12240 df-seq 13220 df-exp 13280 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 |
This theorem is referenced by: reccn2 14787 rlimno1 14844 o1fsum 15001 divrcnv 15040 georeclim 15061 eftabs 15262 efcllem 15264 efaddlem 15279 mul4sqlem 16118 gzrngunit 20293 pjthlem1 23723 iblabsr 24113 iblmulc2 24114 c1liplem1 24276 ftc1lem4 24319 ulmdvlem1 24671 dvradcnv 24692 eff1olem 24813 logcnlem4 24909 lawcoslem1 25074 isosctrlem3 25079 cxploglim2 25238 fsumharmonic 25271 lgamgulmlem2 25289 lgamgulmlem5 25292 lgamcvg2 25314 logfacrlim 25482 2sqlem3 25678 dchrmusum2 25752 dchrvmasumlem3 25757 dchrisum0lem1 25774 dchrisum0lem2a 25775 mudivsum 25788 mulogsumlem 25789 2vmadivsumlem 25798 selberg3lem1 25815 selberg3lem2 25816 selberg4lem1 25818 pntrlog2bndlem1 25835 pntrlog2bndlem3 25837 pntrlog2bndlem5 25839 pntrlog2bndlem6 25841 pntpbnd1a 25843 pntpbnd2 25845 pntibndlem2 25849 pntlemo 25865 pjhthlem1 28859 qqhnm 30848 unbdqndv2lem1 33457 unbdqndv2lem2 33458 knoppndvlem10 33469 knoppndvlem14 33473 iblmulc2nc 34488 ftc1cnnclem 34496 pellexlem2 38912 pellexlem6 38916 modabsdifz 39068 cvgdvgrat 40183 binomcxplemnotnn0 40226 0ellimcdiv 41472 dvdivbd 41749 fourierdlem30 41964 fourierdlem39 41973 etransclem23 42084 |
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