![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > divsubdir | Structured version Visualization version GIF version |
Description: Distribution of division over subtraction. (Contributed by NM, 4-Mar-2005.) |
Ref | Expression |
---|---|
divsubdir | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 − 𝐵) / 𝐶) = ((𝐴 / 𝐶) − (𝐵 / 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negcl 11492 | . . . 4 ⊢ (𝐵 ∈ ℂ → -𝐵 ∈ ℂ) | |
2 | divdir 11930 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ -𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 + -𝐵) / 𝐶) = ((𝐴 / 𝐶) + (-𝐵 / 𝐶))) | |
3 | 1, 2 | syl3an2 1161 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 + -𝐵) / 𝐶) = ((𝐴 / 𝐶) + (-𝐵 / 𝐶))) |
4 | negsub 11540 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) | |
5 | 4 | oveq1d 7434 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + -𝐵) / 𝐶) = ((𝐴 − 𝐵) / 𝐶)) |
6 | 5 | 3adant3 1129 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 + -𝐵) / 𝐶) = ((𝐴 − 𝐵) / 𝐶)) |
7 | 3, 6 | eqtr3d 2767 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) + (-𝐵 / 𝐶)) = ((𝐴 − 𝐵) / 𝐶)) |
8 | divneg 11939 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) → -(𝐵 / 𝐶) = (-𝐵 / 𝐶)) | |
9 | 8 | 3expb 1117 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → -(𝐵 / 𝐶) = (-𝐵 / 𝐶)) |
10 | 9 | 3adant1 1127 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → -(𝐵 / 𝐶) = (-𝐵 / 𝐶)) |
11 | 10 | oveq2d 7435 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) + -(𝐵 / 𝐶)) = ((𝐴 / 𝐶) + (-𝐵 / 𝐶))) |
12 | divcl 11911 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) → (𝐴 / 𝐶) ∈ ℂ) | |
13 | 12 | 3expb 1117 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (𝐴 / 𝐶) ∈ ℂ) |
14 | 13 | 3adant2 1128 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (𝐴 / 𝐶) ∈ ℂ) |
15 | divcl 11911 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) → (𝐵 / 𝐶) ∈ ℂ) | |
16 | 15 | 3expb 1117 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (𝐵 / 𝐶) ∈ ℂ) |
17 | 16 | 3adant1 1127 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (𝐵 / 𝐶) ∈ ℂ) |
18 | 14, 17 | negsubd 11609 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) + -(𝐵 / 𝐶)) = ((𝐴 / 𝐶) − (𝐵 / 𝐶))) |
19 | 11, 18 | eqtr3d 2767 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) + (-𝐵 / 𝐶)) = ((𝐴 / 𝐶) − (𝐵 / 𝐶))) |
20 | 7, 19 | eqtr3d 2767 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 − 𝐵) / 𝐶) = ((𝐴 / 𝐶) − (𝐵 / 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 (class class class)co 7419 ℂcc 11138 0cc0 11140 + caddc 11143 − cmin 11476 -cneg 11477 / cdiv 11903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 |
This theorem is referenced by: subdivcomb1 11942 subdivcomb2 11943 divsubdird 12062 1mhlfehlf 12464 halfpm6th 12466 halfaddsub 12478 zeo 12681 quoremz 13856 quoremnn0ALT 13858 mulsubdivbinom2 14257 facndiv 14283 bpoly3 16038 cos2bnd 16168 rpnnen2lem3 16196 rpnnen2lem11 16204 pythagtriplem15 16801 ovolscalem1 25486 sinq12gt0 26487 sincos6thpi 26495 ang180lem2 26787 log2cnv 26921 log2tlbnd 26922 basellem3 27060 ppiub 27182 logfacrlim 27202 logexprlim 27203 bposlem8 27269 gausslemma2dlem1a 27343 chtppilimlem1 27451 vmadivsum 27460 rplogsumlem2 27463 rpvmasumlem 27465 rplogsum 27505 mulog2sumlem1 27512 selberg2lem 27528 selberg2 27529 selbergr 27546 pntlemr 27580 pntlemj 27581 ballotth 34288 nndivsub 36072 heiborlem6 37420 areaquad 42786 lhe4.4ex1a 43908 stirlinglem10 45609 divsub1dir 47771 line2 48011 |
Copyright terms: Public domain | W3C validator |