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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 11050 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴)) | |
5 | 1, 2, 3, 4 | syl3anc 1439 | 1 ⊢ (𝜑 → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 (class class class)co 6922 ℂcc 10270 0cc0 10272 1c1 10273 · cmul 10277 / cdiv 11032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 |
This theorem is referenced by: expaddzlem 13221 rediv 14278 imdiv 14285 geo2sum 15008 clim2div 15024 efaddlem 15225 sinhval 15286 cvsmuleqdivd 23341 sca2rab 23716 itg2mulclem 23950 itg2mulc 23951 dvmptdivc 24165 dvexp3 24178 dvlip 24193 dvradcnv 24612 tanregt0 24723 logtayl 24843 cxpeq 24938 chordthmlem2 25011 chordthmlem4 25013 heron 25016 dquartlem1 25029 asinlem3 25049 asinsin 25070 efiatan2 25095 atantayl2 25116 amgmlem 25168 basellem8 25266 chebbnd1lem3 25612 dchrmusum2 25635 dchrvmasumlem3 25640 dchrisum0lem1 25657 selberg2lem 25691 logdivbnd 25697 pntrsumo1 25706 pntrlog2bndlem5 25722 pntibndlem2 25732 pntlemr 25743 pntlemo 25748 nmblolbii 28226 blocnilem 28231 nmbdoplbi 29455 nmcoplbi 29459 nmbdfnlbi 29480 nmcfnlbi 29483 logdivsqrle 31330 knoppndvlem7 33091 dvtan 34069 dvasin 34105 areacirclem1 34109 areacirclem4 34112 areaquad 38742 wallispi2lem1 41197 stirlinglem4 41203 stirlinglem5 41204 stirlinglem15 41214 dirkertrigeqlem2 41225 dirkertrigeq 41227 dirkercncflem2 41230 fourierdlem30 41263 fourierdlem57 41289 fourierdlem58 41290 fourierdlem62 41294 fourierdlem95 41327 nn0digval 43391 eenglngeehlnmlem1 43455 eenglngeehlnmlem2 43456 |
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