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Mirrors > Home > MPE Home > Th. List > divrec2d | Structured version Visualization version GIF version |
Description: Relationship between division and reciprocal. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
div1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
divcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
divcld.3 | ⊢ (𝜑 → 𝐵 ≠ 0) |
Ref | Expression |
---|---|
divrec2d | ⊢ (𝜑 → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | div1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | divcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | divcld.3 | . 2 ⊢ (𝜑 → 𝐵 ≠ 0) | |
4 | divrec2 10702 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴)) | |
5 | 1, 2, 3, 4 | syl3anc 1326 | 1 ⊢ (𝜑 → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 · cmul 9941 / cdiv 10684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 |
This theorem is referenced by: expaddzlem 12903 rediv 13871 imdiv 13878 geo2sum 14604 clim2div 14621 efaddlem 14823 sinhval 14884 cvsmuleqdivd 22934 sca2rab 23280 itg2mulclem 23513 itg2mulc 23514 dvmptdivc 23728 dvexp3 23741 dvlip 23756 dvradcnv 24175 tanregt0 24285 logtayl 24406 cxpeq 24498 chordthmlem2 24560 chordthmlem4 24562 heron 24565 dquartlem1 24578 asinlem3 24598 asinsin 24619 efiatan2 24644 atantayl2 24665 amgmlem 24716 basellem8 24814 chebbnd1lem3 25160 dchrmusum2 25183 dchrvmasumlem3 25188 dchrisum0lem1 25205 selberg2lem 25239 logdivbnd 25245 pntrsumo1 25254 pntrlog2bndlem5 25270 pntibndlem2 25280 pntlemr 25291 pntlemo 25296 nmblolbii 27654 blocnilem 27659 nmbdoplbi 28883 nmcoplbi 28887 nmbdfnlbi 28908 nmcfnlbi 28911 logdivsqrle 30728 knoppndvlem7 32509 dvtan 33460 dvasin 33496 areacirclem1 33500 areacirclem4 33503 areaquad 37802 wallispi2lem1 40288 stirlinglem4 40294 stirlinglem5 40295 stirlinglem15 40305 dirkertrigeqlem2 40316 dirkertrigeq 40318 dirkercncflem2 40321 fourierdlem30 40354 fourierdlem57 40380 fourierdlem58 40381 fourierdlem62 40385 fourierdlem95 40418 nn0digval 42394 |
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