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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 11315 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴)) | |
5 | 1, 2, 3, 4 | syl3anc 1367 | 1 ⊢ (𝜑 → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 (class class class)co 7156 ℂcc 10535 0cc0 10537 1c1 10538 · cmul 10542 / cdiv 11297 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 |
This theorem is referenced by: expaddzlem 13473 rediv 14490 imdiv 14497 geo2sum 15229 clim2div 15245 efaddlem 15446 sinhval 15507 cvsmuleqdivd 23738 sca2rab 24113 itg2mulclem 24347 itg2mulc 24348 dvmptdivc 24562 dvexp3 24575 dvlip 24590 dvradcnv 25009 tanregt0 25123 logtayl 25243 cxpeq 25338 chordthmlem2 25411 chordthmlem4 25413 heron 25416 dquartlem1 25429 asinlem3 25449 asinsin 25470 efiatan2 25495 atantayl2 25516 amgmlem 25567 basellem8 25665 chebbnd1lem3 26047 dchrmusum2 26070 dchrvmasumlem3 26075 dchrisum0lem1 26092 selberg2lem 26126 logdivbnd 26132 pntrsumo1 26141 pntrlog2bndlem5 26157 pntibndlem2 26167 pntlemr 26178 pntlemo 26183 nmblolbii 28576 blocnilem 28581 nmbdoplbi 29801 nmcoplbi 29805 nmbdfnlbi 29826 nmcfnlbi 29829 logdivsqrle 31921 knoppndvlem7 33857 dvtan 34957 dvasin 34993 areacirclem1 34997 areacirclem4 35000 areaquad 39872 wallispi2lem1 42405 stirlinglem4 42411 stirlinglem5 42412 stirlinglem15 42422 dirkertrigeqlem2 42433 dirkertrigeq 42435 dirkercncflem2 42438 fourierdlem30 42471 fourierdlem57 42497 fourierdlem58 42498 fourierdlem62 42502 fourierdlem95 42535 nn0digval 44709 eenglngeehlnmlem1 44773 eenglngeehlnmlem2 44774 |
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