divdir - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > divdir		Structured version Visualization version GIF version

Theorem divdir 11913

Description: Distribution of division over addition. (Contributed by NM, 31-Jul-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.)

**Assertion**
Ref	Expression
divdir	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 / 𝐶) + (𝐵 / 𝐶)))

**Proof of Theorem divdir**
Step	Hyp	Ref	Expression
1		simp1 1134	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → 𝐴 ∈ ℂ)
2		simp2 1135	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → 𝐵 ∈ ℂ)
3		reccl 11895	. . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) → (1 / 𝐶) ∈ ℂ)
4	3	3ad2ant3 1133	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (1 / 𝐶) ∈ ℂ)
5	1, 2, 4	adddird 11255	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 + 𝐵) · (1 / 𝐶)) = ((𝐴 · (1 / 𝐶)) + (𝐵 · (1 / 𝐶))))
6	1, 2	addcld 11249	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (𝐴 + 𝐵) ∈ ℂ)
7		simp3l 1199	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → 𝐶 ∈ ℂ)
8		simp3r 1200	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → 𝐶 ≠ 0)
9		divrec 11904	. . 3 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 + 𝐵) · (1 / 𝐶)))
10	6, 7, 8, 9	syl3anc 1369	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 + 𝐵) / 𝐶) = ((𝐴 + 𝐵) · (1 / 𝐶)))
11		divrec 11904	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) → (𝐴 / 𝐶) = (𝐴 · (1 / 𝐶)))
12	1, 7, 8, 11	syl3anc 1369	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (𝐴 / 𝐶) = (𝐴 · (1 / 𝐶)))
13		divrec 11904	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0) → (𝐵 / 𝐶) = (𝐵 · (1 / 𝐶)))
14	2, 7, 8, 13	syl3anc 1369	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (𝐵 / 𝐶) = (𝐵 · (1 / 𝐶)))
15	12, 14	oveq12d 7432	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 / 𝐶) + (𝐵 / 𝐶)) = ((𝐴 · (1 / 𝐶)) + (𝐵 · (1 / 𝐶))))
16	5, 10, 15	3eqtr4d 2777	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 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 (class class class)co 7414 ℂcc 11122 0cc0 11124 1c1 11125 + caddc 11127 · cmul 11129 / cdiv 11887

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888

This theorem is referenced by: muldivdir 11923 divsubdir 11924 divadddiv 11945 divdirzi 11982 divdird 12044 2halves 12456 halfaddsub 12461 zdivadd 12649 nneo 12662 rpnnen1lem5 12981 2tnp1ge0ge0 13812 fldiv 13843 modcyc 13889 mulsubdivbinom2 14239 crim 15080 efival 16114 flodddiv4 16375 divgcdcoprm0 16621 pythagtriplem17 16785 ptolemy 26405 relogbmul 26683 harmonicbnd4 26917 ppiub 27111 logfacrlim 27131 bposlem9 27199 2lgslem3a 27303 2lgslem3b 27304 2lgslem3c 27305 2lgslem3d 27306 chpchtlim 27386 mudivsum 27437 selberglem2 27453 pntrsumo1 27472 pntibndlem2 27498 pntibndlem3 27499 pntlemb 27504 dpfrac1 32584 heiborlem6 37211 zofldiv2ALTV 46915 zofldiv2 47517 sinhpcosh 48084 onetansqsecsq 48105

W3C validator