lediv12a - Metamath Proof Explorer

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Theorem lediv12a 10916

Description: Comparison of ratio of two nonnegative numbers. (Contributed by NM, 31-Dec-2005.)

**Assertion**
Ref	Expression
lediv12a	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

**Proof of Theorem lediv12a**
Step	Hyp	Ref	Expression
1		simplr 792	. . . . 5 $/\$ $/\$ $/\$
2		0re 10040	. . . . . . . 8
3		ltletr 10129	. . . . . . . 8 $/\$ $/\$ $/\$
4	2, 3	mp3an1 1411	. . . . . . 7 $/\$ $/\$
5	4	imp 445	. . . . . 6 $/\$ $/\$ $/\$
6	5	gt0ne0d 10592	. . . . 5 $/\$ $/\$ $/\$
7	1, 6	rereccld 10852	. . . 4 $/\$ $/\$ $/\$
8		gt0ne0 10493	. . . . . 6 $/\$
9		rereccl 10743	. . . . . 6 $/\$
10	8, 9	syldan 487	. . . . 5 $/\$
11	10	ad2ant2r 783	. . . 4 $/\$ $/\$ $/\$
12		recgt0 10867	. . . . . . 7 $/\$
13	1, 5, 12	syl2anc 693	. . . . . 6 $/\$ $/\$ $/\$
14		ltle 10126	. . . . . . 7 $/\$
15	2, 7, 14	sylancr 695	. . . . . 6 $/\$ $/\$ $/\$
16	13, 15	mpd 15	. . . . 5 $/\$ $/\$ $/\$
17		simprr 796	. . . . . 6 $/\$ $/\$ $/\$
18		id 22	. . . . . . . 8 $/\$ $/\$
19	18	ad2ant2r 783	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
20		lerec 10906	. . . . . . 7 $/\$ $/\$ $/\$
21	19, 1, 5, 20	syl12anc 1324	. . . . . 6 $/\$ $/\$ $/\$
22	17, 21	mpbid 222	. . . . 5 $/\$ $/\$ $/\$
23	16, 22	jca 554	. . . 4 $/\$ $/\$ $/\$ $/\$
24	7, 11, 23	jca31 557	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25		simplll 798	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26		simplrl 800	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27		simpllr 799	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	25, 26, 27	jca31 557	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29		simprll 802	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30		simprrl 804	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31	29, 30	jca 554	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		simprlr 803	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	28, 31, 32	jca32 558	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34		simplrr 801	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		simprrr 805	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36	34, 35	jca 554	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		lemul12a 10881	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38	33, 36, 37	sylc 65	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	24, 38	sylan2 491	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40		recn 10026	. . . . . 6
41	40	adantr 481	. . . . 5 $/\$ $/\$ $/\$ $/\$
42		recn 10026	. . . . . . 7
43	42	ad2antlr 763	. . . . . 6 $/\$ $/\$ $/\$
44	43	adantl 482	. . . . 5 $/\$ $/\$ $/\$ $/\$
45	6	adantl 482	. . . . 5 $/\$ $/\$ $/\$ $/\$
46	41, 44, 45	divrecd 10804	. . . 4 $/\$ $/\$ $/\$ $/\$
47	46	adantlr 751	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
48	47	adantlr 751	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49		recn 10026	. . . . . . . 8
50	49	adantr 481	. . . . . . 7 $/\$ $/\$
51		recn 10026	. . . . . . . 8
52	51	ad2antrl 764	. . . . . . 7 $/\$ $/\$
53	8	adantl 482	. . . . . . 7 $/\$ $/\$
54	50, 52, 53	divrecd 10804	. . . . . 6 $/\$ $/\$
55	54	adantrrr 761	. . . . 5 $/\$ $/\$ $/\$
56	55	adantrlr 759	. . . 4 $/\$ $/\$ $/\$ $/\$
57	56	adantll 750	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
58	57	adantlr 751	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
59	39, 48, 58	3brtr4d 4685	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990

wne 2794 class class class wbr 4653 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

c1 9937

cmul 9941

clt 10074

cle 10075

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: lediv2a 10917 lediv12ad 11931 stoweidlem1 40218

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