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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 11350 | . 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 1375 | 1 ⊢ (𝜑 → ((𝐴 / 𝐵) / 𝐶) = (𝐴 / (𝐵 · 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 (class class class)co 7155 ℂcc 10534 0cc0 10536 · cmul 10541 / cdiv 11296 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 |
This theorem is referenced by: discr 13600 hashf1 13814 bcfallfac 15397 eftlub 15461 tanval2 15485 sinhval 15506 sqrt2irrlem 15600 bitsp1 15779 4sqlem7 16279 4sqlem10 16282 uniioombl 24189 dvrec 24551 dvsincos 24577 dvcvx 24616 taylthlem2 24961 mcubic 25424 cubic2 25425 quart1lem 25432 quart1 25433 log2cnv 25521 log2tlbnd 25522 birthdaylem2 25529 efrlim 25546 bcmono 25852 m1lgs 25963 chto1lb 26053 vmalogdivsum2 26113 selberg3lem1 26132 selberg4lem1 26135 selberg4 26136 selberg34r 26146 pntrlog2bndlem2 26153 pntrlog2bndlem4 26155 pntpbnd2 26162 pntibndlem2 26166 pntlemg 26173 irrapxlem5 39421 divdiv3d 41625 mccllem 41876 clim1fr1 41880 sinaover2ne0 42147 dvnprodlem2 42230 wallispi2lem1 42355 stirlinglem3 42360 stirlinglem4 42361 stirlinglem7 42364 stirlinglem15 42372 dirker2re 42376 dirkerdenne0 42377 dirkertrigeqlem2 42383 dirkertrigeqlem3 42384 dirkertrigeq 42385 dirkercncflem1 42387 dirkercncflem2 42388 dirkercncflem4 42390 fourierdlem56 42446 fourierdlem66 42456 sqwvfourb 42513 fouriersw 42515 itscnhlc0xyqsol 44751 |
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