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Mirrors > Home > MPE Home > Th. List > divdiv1d | Structured version Visualization version GIF version |
Description: Division into a fraction. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
div1d.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
divcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
divmuld.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
divmuld.4 | ⊢ (𝜑 → 𝐵 ≠ 0) |
divdiv23d.5 | ⊢ (𝜑 → 𝐶 ≠ 0) |
Ref | Expression |
---|---|
divdiv1d | ⊢ (𝜑 → ((𝐴 / 𝐵) / 𝐶) = (𝐴 / (𝐵 · 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | div1d.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | divcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | divmuld.4 | . 2 ⊢ (𝜑 → 𝐵 ≠ 0) | |
4 | divmuld.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
5 | divdiv23d.5 | . 2 ⊢ (𝜑 → 𝐶 ≠ 0) | |
6 | divdiv1 11975 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐵) / 𝐶) = (𝐴 / (𝐵 · 𝐶))) | |
7 | 1, 2, 3, 4, 5, 6 | syl122anc 1378 | 1 ⊢ (𝜑 → ((𝐴 / 𝐵) / 𝐶) = (𝐴 / (𝐵 · 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 (class class class)co 7430 ℂcc 11150 0cc0 11152 · cmul 11157 / cdiv 11917 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 |
This theorem is referenced by: discr 14275 hashf1 14492 bcfallfac 16076 eftlub 16141 tanval2 16165 sinhval 16186 sqrt2irrlem 16280 bitsp1 16464 4sqlem7 16977 4sqlem10 16980 uniioombl 25637 dvrec 26007 dvsincos 26033 dvcvx 26073 taylthlem2 26430 taylthlem2OLD 26431 mcubic 26904 cubic2 26905 quart1lem 26912 quart1 26913 log2cnv 27001 log2tlbnd 27002 birthdaylem2 27009 efrlim 27026 efrlimOLD 27027 bcmono 27335 m1lgs 27446 chto1lb 27536 vmalogdivsum2 27596 selberg3lem1 27615 selberg4lem1 27618 selberg4 27619 selberg34r 27629 pntrlog2bndlem2 27636 pntrlog2bndlem4 27638 pntpbnd2 27645 pntibndlem2 27649 pntlemg 27656 quad3d 32760 nnproddivdvdsd 41981 dvrelogpow2b 42049 aks4d1p1p7 42055 bcled 42159 bcle2d 42160 irrapxlem5 42813 divdiv3d 45308 mccllem 45552 clim1fr1 45556 sinaover2ne0 45823 dvnprodlem2 45902 wallispi2lem1 46026 stirlinglem3 46031 stirlinglem4 46032 stirlinglem7 46035 stirlinglem15 46043 dirker2re 46047 dirkerdenne0 46048 dirkertrigeqlem2 46054 dirkertrigeqlem3 46055 dirkertrigeq 46056 dirkercncflem1 46058 dirkercncflem2 46059 dirkercncflem4 46061 fourierdlem56 46117 fourierdlem66 46127 sqwvfourb 46184 fouriersw 46186 itscnhlc0xyqsol 48614 |
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