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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 11922 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴)) | |
5 | 1, 2, 3, 4 | syl3anc 1368 | 1 ⊢ (𝜑 → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 (class class class)co 7419 ℂcc 11138 0cc0 11140 1c1 11141 · cmul 11145 / cdiv 11903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 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-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 |
This theorem is referenced by: expaddzlem 14106 rediv 15114 imdiv 15121 geo2sum 15855 clim2div 15871 efaddlem 16073 sinhval 16134 cvsmuleqdivd 25105 sca2rab 25485 itg2mulclem 25720 itg2mulc 25721 dvmptdivc 25941 dvexp3 25954 dvlip 25970 dvradcnv 26402 tanregt0 26518 logtayl 26639 cxpeq 26737 chordthmlem2 26810 chordthmlem4 26812 heron 26815 dquartlem1 26828 asinlem3 26848 asinsin 26869 efiatan2 26894 atantayl2 26915 amgmlem 26967 basellem8 27065 chebbnd1lem3 27449 dchrmusum2 27472 dchrvmasumlem3 27477 dchrisum0lem1 27494 selberg2lem 27528 logdivbnd 27534 pntrsumo1 27543 pntrlog2bndlem5 27559 pntibndlem2 27569 pntlemr 27580 pntlemo 27585 nmblolbii 30681 blocnilem 30686 nmbdoplbi 31906 nmcoplbi 31910 nmbdfnlbi 31931 nmcfnlbi 31934 logdivsqrle 34413 knoppndvlem7 36124 dvtan 37274 dvasin 37308 areacirclem1 37312 areacirclem4 37315 areaquad 42786 wallispi2lem1 45597 stirlinglem4 45603 stirlinglem5 45604 stirlinglem15 45614 dirkertrigeqlem2 45625 dirkertrigeq 45627 dirkercncflem2 45630 fourierdlem30 45663 fourierdlem57 45689 fourierdlem58 45690 fourierdlem62 45694 fourierdlem95 45727 nn0digval 47859 eenglngeehlnmlem1 47996 eenglngeehlnmlem2 47997 |
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