divdivdiv - Metamath Proof Explorer

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

Theorem divdivdiv 10726

Description: Division of two ratios. Theorem I.15 of [Apostol] p. 18. (Contributed by NM, 2-Aug-2004.)

**Assertion**
Ref	Expression
divdivdiv	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

**Proof of Theorem divdivdiv**
Step	Hyp	Ref	Expression
1		simprrl 804	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
2		simprll 802	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
3		simprlr 803	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
4		divcl 10691	. . . . . . 7 $/\$ $/\$
5	1, 2, 3, 4	syl3anc 1326	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
6		simpll 790	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7		simplrl 800	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simplrr 801	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
9		divcl 10691	. . . . . . 7 $/\$ $/\$
10	6, 7, 8, 9	syl3anc 1326	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
11	5, 10	mulcomd 10061	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
12		simplr 792	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
13		simprl 794	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14		divmuldiv 10725	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
15	6, 1, 12, 13, 14	syl22anc 1327	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16	11, 15	eqtrd 2656	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
17	16	oveq2d 6666	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18		simprr 796	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19		divmuldiv 10725	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
20	2, 1, 18, 13, 19	syl22anc 1327	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21	2, 1	mulcomd 10061	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22	21	oveq1d 6665	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23	1, 2	mulcld 10060	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24		simprrr 805	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25	1, 2, 24, 3	mulne0d 10679	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26		divid 10714	. . . . . . . 8 $/\$
27	23, 25, 26	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	22, 27	eqtrd 2656	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29	20, 28	eqtrd 2656	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	29	oveq1d 6665	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31		divcl 10691	. . . . . 6 $/\$ $/\$
32	2, 1, 24, 31	syl3anc 1326	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	32, 5, 10	mulassd 10063	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	10	mulid2d 10058	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	30, 33, 34	3eqtr3d 2664	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	17, 35	eqtr3d 2658	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	6, 1	mulcld 10060	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38	7, 2	mulcld 10060	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39		mulne0 10669	. . . . 5 $/\$ $/\$ $/\$
40	39	ad2ant2lr 784	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41		divcl 10691	. . . 4 $/\$ $/\$
42	37, 38, 40, 41	syl3anc 1326	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43		divne0 10697	. . . 4 $/\$ $/\$ $/\$
44	43	adantl 482	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45		divmul 10688	. . 3 $/\$ $/\$ $/\$
46	10, 42, 32, 44, 45	syl112anc 1330	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47	36, 46	mpbird 247	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990

wne 2794 (class class class)co 6650

cc 9934

cc0 9936

c1 9937

cmul 9941

cdiv 10684

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685

This theorem is referenced by: recdiv 10731 divcan7 10734 divdiv1 10736 divdiv2 10737 divdivdivi 10788 divdivdivd 10848 qreccl 11808 pnt2 25302

W3C validator