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Theorem ltmul12a 9628
Description: Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.)
Assertion
Ref Expression
ltmul12a  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( ( C  e.  RR  /\  D  e.  RR )  /\  (
0  <_  C  /\  C  <  D ) ) )  ->  ( A  x.  C )  <  ( B  x.  D )
)

Proof of Theorem ltmul12a
StepHypRef Expression
1 simplll 734 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( ( 0  <_  A  /\  A  <  B
)  /\  ( 0  <_  C  /\  C  <  D ) ) )  ->  A  e.  RR )
2 simpllr 735 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( ( 0  <_  A  /\  A  <  B
)  /\  ( 0  <_  C  /\  C  <  D ) ) )  ->  B  e.  RR )
3 simpll 730 . . . . . 6  |-  ( ( ( C  e.  RR  /\  D  e.  RR )  /\  ( 0  <_  C  /\  C  <  D
) )  ->  C  e.  RR )
4 simprl 732 . . . . . 6  |-  ( ( ( C  e.  RR  /\  D  e.  RR )  /\  ( 0  <_  C  /\  C  <  D
) )  ->  0  <_  C )
53, 4jca 518 . . . . 5  |-  ( ( ( C  e.  RR  /\  D  e.  RR )  /\  ( 0  <_  C  /\  C  <  D
) )  ->  ( C  e.  RR  /\  0  <_  C ) )
65ad2ant2l 726 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( ( 0  <_  A  /\  A  <  B
)  /\  ( 0  <_  C  /\  C  <  D ) ) )  ->  ( C  e.  RR  /\  0  <_  C ) )
7 ltle 8926 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  ->  A  <_  B )
)
87imp 418 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <  B
)  ->  A  <_  B )
98adantrl 696 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  A  <  B
) )  ->  A  <_  B )
109ad2ant2r 727 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( ( 0  <_  A  /\  A  <  B
)  /\  ( 0  <_  C  /\  C  <  D ) ) )  ->  A  <_  B
)
11 lemul1a 9626 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <_  C )
)  /\  A  <_  B )  ->  ( A  x.  C )  <_  ( B  x.  C )
)
121, 2, 6, 10, 11syl31anc 1185 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( ( 0  <_  A  /\  A  <  B
)  /\  ( 0  <_  C  /\  C  <  D ) ) )  ->  ( A  x.  C )  <_  ( B  x.  C )
)
13 simplrl 736 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  C  e.  RR )
14 simplrr 737 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  D  e.  RR )
15 simpllr 735 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  B  e.  RR )
16 0re 8854 . . . . . . . . . 10  |-  0  e.  RR
17 lelttr 8928 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  A  e.  RR  /\  B  e.  RR )  ->  (
( 0  <_  A  /\  A  <  B )  ->  0  <  B
) )
1816, 17mp3an1 1264 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  <_  A  /\  A  <  B
)  ->  0  <  B ) )
1918imp 418 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  A  <  B
) )  ->  0  <  B )
2019adantlr 695 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  0  <  B )
21 ltmul2 9623 . . . . . . 7  |-  ( ( C  e.  RR  /\  D  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( C  <  D  <->  ( B  x.  C )  <  ( B  x.  D ) ) )
2213, 14, 15, 20, 21syl112anc 1186 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( 0  <_  A  /\  A  <  B ) )  ->  ( C  <  D  <->  ( B  x.  C )  <  ( B  x.  D )
) )
2322biimpa 470 . . . . 5  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( 0  <_  A  /\  A  <  B ) )  /\  C  < 
D )  ->  ( B  x.  C )  <  ( B  x.  D
) )
2423anasss 628 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( ( 0  <_  A  /\  A  <  B
)  /\  C  <  D ) )  ->  ( B  x.  C )  <  ( B  x.  D
) )
2524adantrrl 704 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( ( 0  <_  A  /\  A  <  B
)  /\  ( 0  <_  C  /\  C  <  D ) ) )  ->  ( B  x.  C )  <  ( B  x.  D )
)
26 remulcl 8838 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  x.  C
)  e.  RR )
2726ad2ant2r 727 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  x.  C
)  e.  RR )
28 remulcl 8838 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( B  x.  C
)  e.  RR )
2928ad2ant2lr 728 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( B  x.  C
)  e.  RR )
30 remulcl 8838 . . . . . 6  |-  ( ( B  e.  RR  /\  D  e.  RR )  ->  ( B  x.  D
)  e.  RR )
3130ad2ant2l 726 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( B  x.  D
)  e.  RR )
32 lelttr 8928 . . . . 5  |-  ( ( ( A  x.  C
)  e.  RR  /\  ( B  x.  C
)  e.  RR  /\  ( B  x.  D
)  e.  RR )  ->  ( ( ( A  x.  C )  <_  ( B  x.  C )  /\  ( B  x.  C )  <  ( B  x.  D
) )  ->  ( A  x.  C )  <  ( B  x.  D
) ) )
3327, 29, 31, 32syl3anc 1182 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( ( A  x.  C )  <_ 
( B  x.  C
)  /\  ( B  x.  C )  <  ( B  x.  D )
)  ->  ( A  x.  C )  <  ( B  x.  D )
) )
3433adantr 451 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( ( 0  <_  A  /\  A  <  B
)  /\  ( 0  <_  C  /\  C  <  D ) ) )  ->  ( ( ( A  x.  C )  <_  ( B  x.  C )  /\  ( B  x.  C )  <  ( B  x.  D
) )  ->  ( A  x.  C )  <  ( B  x.  D
) ) )
3512, 25, 34mp2and 660 . 2  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  /\  ( ( 0  <_  A  /\  A  <  B
)  /\  ( 0  <_  C  /\  C  <  D ) ) )  ->  ( A  x.  C )  <  ( B  x.  D )
)
3635an4s 799 1  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( ( C  e.  RR  /\  D  e.  RR )  /\  (
0  <_  C  /\  C  <  D ) ) )  ->  ( A  x.  C )  <  ( B  x.  D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1696   class class class wbr 4039  (class class class)co 5874   RRcr 8752   0cc0 8753    x. cmul 8758    < clt 8883    <_ cle 8884
This theorem is referenced by:  ltmul12ad  9714  expmordi  27135  stoweidlem3  27855
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056
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