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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 11939 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴)) | |
| 5 | 1, 2, 3, 4 | syl3anc 1373 | 1 ⊢ (𝜑 → (𝐴 / 𝐵) = ((1 / 𝐵) · 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 · cmul 11160 / cdiv 11920 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 |
| This theorem is referenced by: expaddzlem 14146 rediv 15170 imdiv 15177 geo2sum 15909 clim2div 15925 efaddlem 16129 sinhval 16190 cvsmuleqdivd 25167 sca2rab 25547 itg2mulclem 25781 itg2mulc 25782 dvmptdivc 26003 dvexp3 26016 dvlip 26032 dvradcnv 26464 tanregt0 26581 logtayl 26702 cxpeq 26800 chordthmlem2 26876 chordthmlem4 26878 heron 26881 dquartlem1 26894 asinlem3 26914 asinsin 26935 efiatan2 26960 atantayl2 26981 amgmlem 27033 basellem8 27131 chebbnd1lem3 27515 dchrmusum2 27538 dchrvmasumlem3 27543 dchrisum0lem1 27560 selberg2lem 27594 logdivbnd 27600 pntrsumo1 27609 pntrlog2bndlem5 27625 pntibndlem2 27635 pntlemr 27646 pntlemo 27651 nmblolbii 30818 blocnilem 30823 nmbdoplbi 32043 nmcoplbi 32047 nmbdfnlbi 32068 nmcfnlbi 32071 logdivsqrle 34665 knoppndvlem7 36519 dvtan 37677 dvasin 37711 areacirclem1 37715 areacirclem4 37718 readvcot 42394 areaquad 43228 wallispi2lem1 46086 stirlinglem4 46092 stirlinglem5 46093 stirlinglem15 46103 dirkertrigeqlem2 46114 dirkertrigeq 46116 dirkercncflem2 46119 fourierdlem30 46152 fourierdlem57 46178 fourierdlem58 46179 fourierdlem62 46183 fourierdlem95 46216 nn0digval 48521 eenglngeehlnmlem1 48658 eenglngeehlnmlem2 48659 |
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