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Theorem subdird 11695

Description: Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.)

**Hypotheses**
Ref	Expression
mulm1d.1	⊢ (𝜑 → 𝐴 ∈ ℂ)
mulnegd.2	⊢ (𝜑 → 𝐵 ∈ ℂ)
subdid.3	⊢ (𝜑 → 𝐶 ∈ ℂ)

**Assertion**
Ref	Expression
subdird	⊢ (𝜑 → ((𝐴 − 𝐵) · 𝐶) = ((𝐴 · 𝐶) − (𝐵 · 𝐶)))

**Proof of Theorem subdird**
Step	Hyp	Ref	Expression
1		mulm1d.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℂ)
2		mulnegd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ ℂ)
3		subdid.3	. 2 ⊢ (𝜑 → 𝐶 ∈ ℂ)
4		subdir 11672	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 − 𝐵) · 𝐶) = ((𝐴 · 𝐶) − (𝐵 · 𝐶)))
5	1, 2, 3, 4	syl3anc 1369	1 ⊢ (𝜑 → ((𝐴 − 𝐵) · 𝐶) = ((𝐴 · 𝐶) − (𝐵 · 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 (class class class)co 7414 ℂcc 11130 · cmul 11137 − cmin 11468

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 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208

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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 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 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-ltxr 11277 df-sub 11470

This theorem is referenced by: mulsubfacd 11699 ltmul1a 12087 xp1d2m1eqxm1d2 12490 div4p1lem1div2 12491 lincmb01cmp 13498 iccf1o 13499 modmul1 13915 remullem 15101 mulcn2 15566 fsumparts 15778 pwdif 15840 geo2sum 15845 fallfacfwd 16006 bpoly4 16029 modprm0 16767 mul4sqlem 16915 vdwapun 16936 icopnfcnv 24860 itgconst 25741 itgmulc2lem2 25755 dvmulbr 25862 dvmulbrOLD 25863 dvrec 25880 dvsincos 25906 cmvth 25916 cmvthOLD 25917 dvcvx 25946 dvfsumlem1 25953 dvfsumlem2 25954 dvfsumlem2OLD 25955 coeeulem 26151 abelthlem6 26366 tangtx 26433 tanarg 26546 logdivlti 26547 logcnlem4 26572 affineequiv 26748 affineequiv2 26749 chordthmlem2 26758 chordthmlem4 26760 mcubic 26772 dquartlem2 26777 quart1lem 26780 quart1 26781 quartlem1 26782 dvatan 26860 atantayl 26862 lgamcvg2 26980 wilthlem2 26994 logfaclbnd 27148 logexprlim 27151 perfectlem2 27156 dchrsum2 27194 sumdchr2 27196 bposlem9 27218 lgsquadlem1 27306 2sqmod 27362 chebbnd1lem3 27397 rpvmasumlem 27413 log2sumbnd 27470 chpdifbndlem1 27479 selberg3lem1 27483 selberg4lem1 27486 selbergr 27494 selberg3r 27495 selberg4r 27496 pntrlog2bndlem3 27505 pntrlog2bndlem5 27507 pntibndlem2 27517 pntlemo 27533 ttgcontlem1 28688 brbtwn2 28709 colinearalglem1 28710 axsegconlem9 28729 axcontlem2 28769 axcontlem7 28774 axcontlem8 28775 sinccvglem 35270 bj-bary1lem 36783 bj-bary1lem1 36784 itgmulc2nclem2 37154 bfp 37291 pellexlem6 42248 congmul 42382 areaquad 42638 itgsinexp 45337 stoweidlem13 45395 stoweidlem14 45396 stoweidlem26 45408 fourierdlem6 45495 fourierdlem26 45515 fourierdlem42 45531 fourierdlem65 45553 fourierdlem95 45583 smfmullem1 46173 sigarmf 46236 cevathlem2 46250 perfectALTVlem2 47056 submuladdmuld 47768 affinecomb2 47770 affineid 47771 rrx2linest 47809 itscnhlinecirc02plem2 47850 inlinecirc02p 47854 joinlmulsubmuld 48201

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