divcan1d - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > divcan1d		Structured version Visualization version GIF version

Theorem divcan1d 12021

Description: A cancellation law for division. (Contributed by Mario Carneiro, 27-May-2016.)

**Hypotheses**
Ref	Expression
div1d.1	⊢ (𝜑 → 𝐴 ∈ ℂ)
divcld.2	⊢ (𝜑 → 𝐵 ∈ ℂ)
divcld.3	⊢ (𝜑 → 𝐵 ≠ 0)

**Assertion**
Ref	Expression
divcan1d	⊢ (𝜑 → ((𝐴 / 𝐵) · 𝐵) = 𝐴)

**Proof of Theorem divcan1d**
Step	Hyp	Ref	Expression
1		div1d.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℂ)
2		divcld.2	. 2 ⊢ (𝜑 → 𝐵 ∈ ℂ)
3		divcld.3	. 2 ⊢ (𝜑 → 𝐵 ≠ 0)
4		divcan1 11911	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → ((𝐴 / 𝐵) · 𝐵) = 𝐴)
5	1, 2, 3, 4	syl3anc 1369	1 ⊢ (𝜑 → ((𝐴 / 𝐵) · 𝐵) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 (class class class)co 7420 ℂcc 11136 0cc0 11138 · cmul 11143 / cdiv 11901

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 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

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 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 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 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902

This theorem is referenced by: ldiv 12078 ltdiv23 12135 lediv23 12136 recp1lt1 12142 ledivp1 12146 subhalfhalf 12476 xp1d2m1eqxm1d2 12496 div4p1lem1div2 12497 qmulz 12965 iccf1o 13505 ltdifltdiv 13831 bcpasc 14312 sqrtdiv 15244 geo2sum 15851 sqrt2irrlem 16224 dvdsval2 16233 flodddiv4t2lthalf 16392 bitsres 16447 bitsuz 16448 dvdsgcdidd 16512 mulgcddvds 16625 qredeq 16627 isprm6 16684 qmuldeneqnum 16718 hashgcdlem 16756 pcqdiv 16825 pockthlem 16873 prmreclem3 16886 4sqlem5 16910 4sqlem12 16924 4sqlem15 16927 sylow3lem4 19584 odadd1 19802 odadd2 19803 gexexlem 19806 pgpfac1lem3a 20032 pgpfac1lem3 20033 znidomb 21494 znrrg 21498 nmoleub2lem 25040 nmoleub3 25045 i1fmullem 25622 mbfi1fseqlem3 25646 mbfi1fseqlem4 25647 mbfi1fseqlem5 25648 dvcnp2 25848 dvcnp2OLD 25849 dvlip 25925 plydivlem4 26230 cosne0 26462 advlogexp 26588 root1id 26688 cxplogb 26717 ang180lem1 26740 ang180lem3 26742 angpieqvd 26762 chordthmlem 26763 dcubic2 26775 dcubic 26777 dquartlem2 26783 cxploglim2 26910 fsumdvdsdiaglem 27114 logexprlim 27157 bposlem3 27218 lgslem1 27229 gausslemma2dlem1a 27297 lgsquadlem1 27312 2lgslem1a1 27321 log2sumbnd 27476 chpdifbndlem1 27485 selberg4lem1 27492 pntrlog2bndlem3 27511 pntibndlem2 27523 pntlemr 27534 ostth2lem3 27567 ostth2 27569 ostth3 27570 axcontlem7 28780 blocnilem 30613 zringfrac 32996 qqhval2lem 33582 cndprobin 34054 itgexpif 34238 faclimlem1 35337 faclimlem3 35339 nn0prpwlem 35806 itg2addnclem3 37146 bfplem1 37295 rrncmslem 37305 rrnequiv 37308 nnproddivdvdsd 41471 lcmineqlem12 41511 3lexlogpow5ineq2 41526 3lexlogpow2ineq1 41529 aks4d1p8 41558 pellexlem6 42254 jm2.19 42414 jm2.27c 42428 binomcxplemnotnn0 43793 sineq0ALT 44376 xralrple2 44736 ltdiv23neg 44776 stoweidlem42 45430 stirlinglem3 45464 dirkertrigeq 45489 dirkercncflem2 45492 dirkercncflem4 45494 fourierdlem4 45499 fourierdlem63 45557 fourierdlem65 45559 fourierdlem83 45577 fourierdlem89 45583 fourierdlem90 45584 fourierdlem91 45585 etransclem38 45660 smfmullem1 46179 sigarcol 46252 sharhght 46253 proththd 46954 mod0mul 47592 nn0sumshdiglemA 47692 rrx2vlinest 47814

W3C validator

	
		OSZAR »