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Theorem lt2mul2div 8788
Description: 'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.)
Assertion
Ref Expression
lt2mul2div  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( ( A  x.  B )  < 
( C  x.  D
)  <->  ( A  /  D )  <  ( C  /  B ) ) )

Proof of Theorem lt2mul2div
StepHypRef Expression
1 simprl 526 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  C  e.  RR )
21recnd 7941 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  C  e.  CC )
3 simprrl 534 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  D  e.  RR )
43recnd 7941 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  D  e.  CC )
52, 4mulcomd 7934 . . . . 5  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( C  x.  D )  =  ( D  x.  C ) )
65oveq1d 5866 . . . 4  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( ( C  x.  D )  /  B )  =  ( ( D  x.  C
)  /  B ) )
7 simplrl 530 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  B  e.  RR )
87recnd 7941 . . . . 5  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  B  e.  CC )
9 simplrr 531 . . . . . 6  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  0  <  B
)
107, 9gt0ap0d 8541 . . . . 5  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  B #  0 )
114, 2, 8, 10divassapd 8736 . . . 4  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( ( D  x.  C )  /  B )  =  ( D  x.  ( C  /  B ) ) )
126, 11eqtrd 2203 . . 3  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( ( C  x.  D )  /  B )  =  ( D  x.  ( C  /  B ) ) )
1312breq2d 3999 . 2  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( A  < 
( ( C  x.  D )  /  B
)  <->  A  <  ( D  x.  ( C  /  B ) ) ) )
14 simpll 524 . . 3  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  A  e.  RR )
151, 3remulcld 7943 . . 3  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( C  x.  D )  e.  RR )
16 simplr 525 . . 3  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( B  e.  RR  /\  0  < 
B ) )
17 ltmuldiv 8783 . . 3  |-  ( ( A  e.  RR  /\  ( C  x.  D
)  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  x.  B )  <  ( C  x.  D
)  <->  A  <  ( ( C  x.  D )  /  B ) ) )
1814, 15, 16, 17syl3anc 1233 . 2  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( ( A  x.  B )  < 
( C  x.  D
)  <->  A  <  ( ( C  x.  D )  /  B ) ) )
191, 7, 10redivclapd 8745 . . 3  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( C  /  B )  e.  RR )
20 simprr 527 . . 3  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( D  e.  RR  /\  0  < 
D ) )
21 ltdivmul 8785 . . 3  |-  ( ( A  e.  RR  /\  ( C  /  B
)  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) )  ->  ( ( A  /  D )  < 
( C  /  B
)  <->  A  <  ( D  x.  ( C  /  B ) ) ) )
2214, 19, 20, 21syl3anc 1233 . 2  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( ( A  /  D )  < 
( C  /  B
)  <->  A  <  ( D  x.  ( C  /  B ) ) ) )
2313, 18, 223bitr4d 219 1  |-  ( ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  /\  ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) ) )  ->  ( ( A  x.  B )  < 
( C  x.  D
)  <->  ( A  /  D )  <  ( C  /  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 2141   class class class wbr 3987  (class class class)co 5851   RRcr 7766   0cc0 7767    x. cmul 7772    < clt 7947    / cdiv 8582
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 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7858  ax-resscn 7859  ax-1cn 7860  ax-1re 7861  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-mulrcl 7866  ax-addcom 7867  ax-mulcom 7868  ax-addass 7869  ax-mulass 7870  ax-distr 7871  ax-i2m1 7872  ax-0lt1 7873  ax-1rid 7874  ax-0id 7875  ax-rnegex 7876  ax-precex 7877  ax-cnre 7878  ax-pre-ltirr 7879  ax-pre-ltwlin 7880  ax-pre-lttrn 7881  ax-pre-apti 7882  ax-pre-ltadd 7883  ax-pre-mulgt0 7884  ax-pre-mulext 7885
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 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-po 4279  df-iso 4280  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-pnf 7949  df-mnf 7950  df-xr 7951  df-ltxr 7952  df-le 7953  df-sub 8085  df-neg 8086  df-reap 8487  df-ap 8494  df-div 8583
This theorem is referenced by:  lt2mul2divd  9715
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