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Theorem lediv23 8809
Description: Swap denominator with other side of 'less than or equal to'. (Contributed by NM, 30-May-2005.)
Assertion
Ref Expression
lediv23 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) <_ C <-> ( A / C ) <_ B ) )
Proof of Theorem lediv23
StepHypRef Expression
1 simp1 992 . . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> A e. RR )
2 simp2l 1018 . . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B e. RR )
3 simp2r 1019 . . . . 5 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> 0 < B )
42, 3gt0ap0d 8548 . . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B # 0 )
51, 2, 4redivclapd 8752 . . 3 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A / B ) e. RR )
6 simp3l 1020 . . 3 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> C e. RR )
7 lemul1 8512 . . 3 |- ( ( ( A / B ) e. RR /\ C e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) <_ C <-> ( ( A / B ) x. B ) <_ ( C x. B ) ) )
85, 6, 2, 3, 7syl112anc 1237 . 2 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) <_ C <-> ( ( A / B ) x. B ) <_ ( C x. B ) ) )
91recnd 7948 . . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> A e. CC )
102recnd 7948 . . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> B e. CC )
119, 10, 4divcanap1d 8708 . . 3 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) x. B ) = A )
1211breq1d 3999 . 2 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( ( A / B ) x. B ) <_ ( C x. B ) <-> A <_ ( C x. B ) ) )
136, 2remulcld 7950 . . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( C x. B ) e. RR )
14 lediv1 8785 . . . 4 |- ( ( A e. RR /\ ( C x. B ) e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ ( C x. B ) <-> ( A / C ) <_ ( ( C x. B ) / C ) ) )
1513, 14syld3an2 1280 . . 3 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ ( C x. B ) <-> ( A / C ) <_ ( ( C x. B ) / C ) ) )
166recnd 7948 . . . . 5 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> C e. CC )
17 simp3r 1021 . . . . . 6 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> 0 < C )
186, 17gt0ap0d 8548 . . . . 5 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> C # 0 )
1910, 16, 18divcanap3d 8712 . . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( C x. B ) / C ) = B )
2019breq2d 4001 . . 3 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / C ) <_ ( ( C x. B ) / C ) <-> ( A / C ) <_ B ) )
2115, 20bitrd 187 . 2 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ ( C x. B ) <-> ( A / C ) <_ B ) )
228, 12, 213bitrd 213 1 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) <_ C <-> ( A / C ) <_ B ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   RRcr 7773   0cc0 7774   x. cmul 7779   < clt 7954   <_ cle 7955   / cdiv 8589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590
This theorem is referenced by:  divle1le  9682  ledivge1le  9683  lediv23d  9715
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