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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 11978 | . 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 1381 | 1 ⊢ (𝜑 → ((𝐴 / 𝐵) / 𝐶) = (𝐴 / (𝐵 · 𝐶))) |
| 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 · 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: discr 14279 hashf1 14496 bcfallfac 16080 eftlub 16145 tanval2 16169 sinhval 16190 sqrt2irrlem 16284 bitsp1 16468 4sqlem7 16982 4sqlem10 16985 uniioombl 25624 dvrec 25993 dvsincos 26019 dvcvx 26059 taylthlem2 26416 taylthlem2OLD 26417 mcubic 26890 cubic2 26891 quart1lem 26898 quart1 26899 log2cnv 26987 log2tlbnd 26988 birthdaylem2 26995 efrlim 27012 efrlimOLD 27013 bcmono 27321 m1lgs 27432 chto1lb 27522 vmalogdivsum2 27582 selberg3lem1 27601 selberg4lem1 27604 selberg4 27605 selberg34r 27615 pntrlog2bndlem2 27622 pntrlog2bndlem4 27624 pntpbnd2 27631 pntibndlem2 27635 pntlemg 27642 quad3d 32754 nnproddivdvdsd 42001 dvrelogpow2b 42069 aks4d1p1p7 42075 bcled 42179 bcle2d 42180 irrapxlem5 42837 divdiv3d 45370 mccllem 45612 clim1fr1 45616 sinaover2ne0 45883 dvnprodlem2 45962 wallispi2lem1 46086 stirlinglem3 46091 stirlinglem4 46092 stirlinglem7 46095 stirlinglem15 46103 dirker2re 46107 dirkerdenne0 46108 dirkertrigeqlem2 46114 dirkertrigeqlem3 46115 dirkertrigeq 46116 dirkercncflem1 46118 dirkercncflem2 46119 dirkercncflem4 46121 fourierdlem56 46177 fourierdlem66 46187 sqwvfourb 46244 fouriersw 46246 itscnhlc0xyqsol 48686 |
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