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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 11317 | . 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 3018 (class class class)co 7158 ℂcc 10537 0cc0 10539 1c1 10540 · cmul 10544 / cdiv 11299 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 |
This theorem is referenced by: expaddzlem 13475 rediv 14492 imdiv 14499 geo2sum 15231 clim2div 15247 efaddlem 15448 sinhval 15509 cvsmuleqdivd 23740 sca2rab 24115 itg2mulclem 24349 itg2mulc 24350 dvmptdivc 24564 dvexp3 24577 dvlip 24592 dvradcnv 25011 tanregt0 25125 logtayl 25245 cxpeq 25340 chordthmlem2 25413 chordthmlem4 25415 heron 25418 dquartlem1 25431 asinlem3 25451 asinsin 25472 efiatan2 25497 atantayl2 25518 amgmlem 25569 basellem8 25667 chebbnd1lem3 26049 dchrmusum2 26072 dchrvmasumlem3 26077 dchrisum0lem1 26094 selberg2lem 26128 logdivbnd 26134 pntrsumo1 26143 pntrlog2bndlem5 26159 pntibndlem2 26169 pntlemr 26180 pntlemo 26185 nmblolbii 28578 blocnilem 28583 nmbdoplbi 29803 nmcoplbi 29807 nmbdfnlbi 29828 nmcfnlbi 29831 logdivsqrle 31923 knoppndvlem7 33859 dvtan 34944 dvasin 34980 areacirclem1 34984 areacirclem4 34987 areaquad 39830 wallispi2lem1 42363 stirlinglem4 42369 stirlinglem5 42370 stirlinglem15 42380 dirkertrigeqlem2 42391 dirkertrigeq 42393 dirkercncflem2 42396 fourierdlem30 42429 fourierdlem57 42455 fourierdlem58 42456 fourierdlem62 42460 fourierdlem95 42493 nn0digval 44667 eenglngeehlnmlem1 44731 eenglngeehlnmlem2 44732 |
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