absdivd - Metamath Proof Explorer

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Theorem absdivd 15434

Description: Absolute value distributes over division. (Contributed by Mario Carneiro, 29-May-2016.)

**Assertion**
Ref	Expression
absdivd	⊢ (𝜑 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵)))

**Proof of Theorem absdivd**
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 15274	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵)))
5	1, 2, 3, 4	syl3anc 1369	1 ⊢ (𝜑 → (abs‘(𝐴 / 𝐵)) = ((abs‘𝐴) / (abs‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ‘cfv 6548 (class class class)co 7420 ℂcc 11136 0cc0 11138 / cdiv 11901 abscabs 15213

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-sup 9465 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215

This theorem is referenced by: reccn2 15573 rlimno1 15632 o1fsum 15791 divrcnv 15830 georeclim 15850 eftabs 16051 efcllem 16053 efaddlem 16069 mul4sqlem 16921 gzrngunit 21365 pjthlem1 25364 iblabsr 25758 iblmulc2 25759 c1liplem1 25928 ftc1lem4 25973 ulmdvlem1 26335 dvradcnv 26356 eff1olem 26481 logcnlem4 26578 lawcoslem1 26746 isosctrlem3 26751 cxploglim2 26910 fsumharmonic 26943 lgamgulmlem2 26961 lgamgulmlem5 26964 lgamcvg2 26986 logfacrlim 27156 2sqlem3 27352 dchrmusum2 27426 dchrvmasumlem3 27431 dchrisum0lem1 27448 dchrisum0lem2a 27449 mudivsum 27462 mulogsumlem 27463 2vmadivsumlem 27472 selberg3lem1 27489 selberg3lem2 27490 selberg4lem1 27492 pntrlog2bndlem1 27509 pntrlog2bndlem3 27511 pntrlog2bndlem5 27513 pntrlog2bndlem6 27515 pntpbnd1a 27517 pntpbnd2 27519 pntibndlem2 27523 pntlemo 27539 pjhthlem1 31200 qqhnm 33591 unbdqndv2lem1 35984 unbdqndv2lem2 35985 knoppndvlem10 35996 knoppndvlem14 36000 iblmulc2nc 37158 ftc1cnnclem 37164 pellexlem2 42250 pellexlem6 42254 modabsdifz 42407 cvgdvgrat 43750 binomcxplemnotnn0 43793 0ellimcdiv 45037 dvdivbd 45311 fourierdlem30 45525 fourierdlem39 45534 etransclem23 45645

	
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