Theorem lediv2 8900

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 1025	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B e. RR )
2		simp2r 1026	. . . . 5 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> 0 < B )
3	1, 2	gt0ap0d 8638	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B # 0 )
4	1, 3	rerecclapd 8843	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( 1 / B ) e. RR )
5		simp1l 1023	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> A e. RR )
6		simp1r 1024	. . . . 5 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> 0 < A )
7	5, 6	gt0ap0d 8638	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> A # 0 )
8	5, 7	rerecclapd 8843	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( 1 / A ) e. RR )
9		simp3l 1027	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> C e. RR )
10		simp3r 1028	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> 0 < C )
11		lemul2 8866	. . 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 1253	. 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 8893	. . 3 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A <_ B <-> ( 1 / B ) <_ ( 1 / A ) ) )
14	13	3adant3 1019	. 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 8038	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> C e. CC )
16	1	recnd 8038	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B e. CC )
17	15, 16, 3	divrecapd 8802	. . 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 8038	. . . 4 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> A e. CC )
19	15, 18, 7	divrecapd 8802	. . 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 4042	. 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 220	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 104   <-> wb 105   /\ w3a 980   e. wcel 2164   class class class wbr 4029  (class class class)co 5910   RRcr 7861   0cc0 7862   1c1 7863   x. cmul 7867   < clt 8044   <_ cle 8045   / cdiv 8681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-setind 4565  ax-cnex 7953  ax-resscn 7954  ax-1cn 7955  ax-1re 7956  ax-icn 7957  ax-addcl 7958  ax-addrcl 7959  ax-mulcl 7960  ax-mulrcl 7961  ax-addcom 7962  ax-mulcom 7963  ax-addass 7964  ax-mulass 7965  ax-distr 7966  ax-i2m1 7967  ax-0lt1 7968  ax-1rid 7969  ax-0id 7970  ax-rnegex 7971  ax-precex 7972  ax-cnre 7973  ax-pre-ltirr 7974  ax-pre-ltwlin 7975  ax-pre-lttrn 7976  ax-pre-apti 7977  ax-pre-ltadd 7978  ax-pre-mulgt0 7979  ax-pre-mulext 7980

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4322  df-po 4325  df-iso 4326  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-iota 5207  df-fun 5248  df-fv 5254  df-riota 5865  df-ov 5913  df-oprab 5914  df-mpo 5915  df-pnf 8046  df-mnf 8047  df-xr 8048  df-ltxr 8049  df-le 8050  df-sub 8182  df-neg 8183  df-reap 8584  df-ap 8591  df-div 8682

This theorem is referenced by:  lediv2d  9777  nnledivrp  9822  isprm6  12275  divdenle  12325  znidomb  14123