Theorem lediv2 8406

Description: Division of a positive number by both sides of 'less than or equal to'. (Contributed by NM, 10-Jan-2006.)

Assertion

Ref	Expression
lediv2	|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ B <-> ( C / B ) <_ ( C / A ) ) )

Proof of Theorem lediv2

Step	Hyp	Ref	Expression
1		simp2l 970	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B e. RR )
2		simp2r 971	. . . . 5 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> 0 < B )
3	1, 2	gt0ap0d 8159	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B # 0 )
4	1, 3	rerecclapd 8354	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( 1 / B ) e. RR )
5		simp1l 968	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> A e. RR )
6		simp1r 969	. . . . 5 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> 0 < A )
7	5, 6	gt0ap0d 8159	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> A # 0 )
8	5, 7	rerecclapd 8354	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( 1 / A ) e. RR )
9		simp3l 972	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> C e. RR )
10		simp3r 973	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> 0 < C )
11		lemul2 8372	. . 3 |- ( ( ( 1 / B ) e. RR /\ ( 1 / A ) e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( 1 / B ) <_ ( 1 / A ) <-> ( C x. ( 1 / B ) ) <_ ( C x. ( 1 / A ) ) ) )
12	4, 8, 9, 10, 11	syl112anc 1179	. 2 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( 1 / B ) <_ ( 1 / A ) <-> ( C x. ( 1 / B ) ) <_ ( C x. ( 1 / A ) ) ) )
13		lerec 8399	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A <_ B <-> ( 1 / B ) <_ ( 1 / A ) ) )
14	13	3adant3 964	. 2 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ B <-> ( 1 / B ) <_ ( 1 / A ) ) )
15	9	recnd 7570	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> C e. CC )
16	1	recnd 7570	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B e. CC )
17	15, 16, 3	divrecapd 8314	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( C / B ) = ( C x. ( 1 / B ) ) )
18	5	recnd 7570	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> A e. CC )
19	15, 18, 7	divrecapd 8314	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( C / A ) = ( C x. ( 1 / A ) ) )
20	17, 19	breq12d 3864	. 2 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( C / B ) <_ ( C / A ) <-> ( C x. ( 1 / B ) ) <_ ( C x. ( 1 / A ) ) ) )
21	12, 14, 20	3bitr4d 219	1 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ B <-> ( C / B ) <_ ( C / A ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 925   e. wcel 1439   class class class wbr 3851  (class class class)co 5666   RRcr 7403   0cc0 7404   1c1 7405   x. cmul 7409   < clt 7576   <_ cle 7577   / cdiv 8193

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-mulrcl 7498  ax-addcom 7499  ax-mulcom 7500  ax-addass 7501  ax-mulass 7502  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-1rid 7506  ax-0id 7507  ax-rnegex 7508  ax-precex 7509  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-apti 7514  ax-pre-ltadd 7515  ax-pre-mulgt0 7516  ax-pre-mulext 7517

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-po 4132  df-iso 4133  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-iota 4993  df-fun 5030  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-reap 8106  df-ap 8113  df-div 8194

This theorem is referenced by:  lediv2d  9252  nnledivrp  9291  isprm6  11458  divdenle  11507