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Mirrors > Home > MPE Home > Th. List > divdir | Structured version Visualization version GIF version |
Description: Distribution of division over addition. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
divdir | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1127 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → 𝐴 ∈ ℂ) | |
2 | simp2 1128 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → 𝐵 ∈ ℂ) | |
3 | reccl 11040 | . . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) → (1 / 𝐶) ∈ ℂ) | |
4 | 3 | 3ad2ant3 1126 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (1 / 𝐶) ∈ ℂ) |
5 | 1, 2, 4 | adddird 10402 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 + 𝐵) · (1 / 𝐶)) = ((𝐴 · (1 / 𝐶)) + (𝐵 · (1 / 𝐶)))) |
6 | 1, 2 | addcld 10396 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (𝐴 + 𝐵) ∈ ℂ) |
7 | simp3l 1215 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → 𝐶 ∈ ℂ) | |
8 | simp3r 1216 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → 𝐶 ≠ 0) | |
9 | divrec 11049 | . . 3 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 + 𝐵) · (1 / 𝐶))) | |
10 | 6, 7, 8, 9 | syl3anc 1439 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 + 𝐵) · (1 / 𝐶))) |
11 | divrec 11049 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) → (𝐴 / 𝐶) = (𝐴 · (1 / 𝐶))) | |
12 | 1, 7, 8, 11 | syl3anc 1439 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (𝐴 / 𝐶) = (𝐴 · (1 / 𝐶))) |
13 | divrec 11049 | . . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) → (𝐵 / 𝐶) = (𝐵 · (1 / 𝐶))) | |
14 | 2, 7, 8, 13 | syl3anc 1439 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (𝐵 / 𝐶) = (𝐵 · (1 / 𝐶))) |
15 | 12, 14 | oveq12d 6940 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) + (𝐵 / 𝐶)) = ((𝐴 · (1 / 𝐶)) + (𝐵 · (1 / 𝐶)))) |
16 | 5, 10, 15 | 3eqtr4d 2823 | 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 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 (class class class)co 6922 ℂcc 10270 0cc0 10272 1c1 10273 + caddc 10275 · cmul 10277 / cdiv 11032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 |
This theorem is referenced by: muldivdir 11068 divsubdir 11069 divadddiv 11090 divdirzi 11127 divdird 11189 2halves 11610 halfaddsub 11615 zdivadd 11800 nneo 11813 rpnnen1lem5 12128 2tnp1ge0ge0 12949 fldiv 12978 modcyc 13024 mulsubdivbinom2 13367 crim 14262 efival 15284 flodddiv4 15543 divgcdcoprm0 15784 pythagtriplem17 15940 ptolemy 24686 relogbmul 24955 harmonicbnd4 25189 ppiub 25381 logfacrlim 25401 bposlem9 25469 2lgslem3a 25573 2lgslem3b 25574 2lgslem3c 25575 2lgslem3d 25576 chpchtlim 25620 mudivsum 25671 selberglem2 25687 pntrsumo1 25706 pntibndlem2 25732 pntibndlem3 25733 pntlemb 25738 dpfrac1 30162 heiborlem6 34234 zofldiv2ALTV 42592 zofldiv2 43333 sinhpcosh 43582 onetansqsecsq 43603 |
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