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Theorem mulge0 9381
Description: The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
mulge0  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  0  <_  ( A  x.  B ) )

Proof of Theorem mulge0
StepHypRef Expression
1 0re 8928 . . . . . . 7  |-  0  e.  RR
21a1i 10 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  0  e.  RR )
3 simpl 443 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  RR )
42, 3leloed 9052 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
5 simpr 447 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
62, 5leloed 9052 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  B  <->  ( 0  <  B  \/  0  =  B )
) )
74, 6anbi12d 691 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  <_  A  /\  0  <_  B
)  <->  ( ( 0  <  A  \/  0  =  A )  /\  ( 0  <  B  \/  0  =  B
) ) ) )
81a1i 10 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  0  e.  RR )
9 simpll 730 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  A  e.  RR )
10 simplr 731 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  B  e.  RR )
119, 10remulcld 8953 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  ( A  x.  B )  e.  RR )
12 mulgt0 8990 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  ( A  x.  B ) )
1312an4s 799 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  0  <  ( A  x.  B
) )
148, 11, 13ltled 9057 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  0  <_  ( A  x.  B
) )
1514ex 423 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  < 
A  /\  0  <  B )  ->  0  <_  ( A  x.  B ) ) )
16 leid 9006 . . . . . . . . 9  |-  ( 0  e.  RR  ->  0  <_  0 )
171, 16ax-mp 8 . . . . . . . 8  |-  0  <_  0
185recnd 8951 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  CC )
1918mul02d 9100 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  x.  B
)  =  0 )
2017, 19syl5breqr 4140 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  0  <_  ( 0  x.  B ) )
21 oveq1 5952 . . . . . . . 8  |-  ( 0  =  A  ->  (
0  x.  B )  =  ( A  x.  B ) )
2221breq2d 4116 . . . . . . 7  |-  ( 0  =  A  ->  (
0  <_  ( 0  x.  B )  <->  0  <_  ( A  x.  B ) ) )
2320, 22syl5ibcom 211 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  =  A  ->  0  <_  ( A  x.  B )
) )
2423adantrd 454 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  =  A  /\  0  < 
B )  ->  0  <_  ( A  x.  B
) ) )
253recnd 8951 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  CC )
2625mul01d 9101 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  0 )  =  0 )
2717, 26syl5breqr 4140 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  0  <_  ( A  x.  0 ) )
28 oveq2 5953 . . . . . . . 8  |-  ( 0  =  B  ->  ( A  x.  0 )  =  ( A  x.  B ) )
2928breq2d 4116 . . . . . . 7  |-  ( 0  =  B  ->  (
0  <_  ( A  x.  0 )  <->  0  <_  ( A  x.  B ) ) )
3027, 29syl5ibcom 211 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  =  B  ->  0  <_  ( A  x.  B )
) )
3130adantld 453 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  < 
A  /\  0  =  B )  ->  0  <_  ( A  x.  B
) ) )
3230adantld 453 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  =  A  /\  0  =  B )  ->  0  <_  ( A  x.  B
) ) )
3315, 24, 31, 32ccased 913 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( 0  <  A  \/  0  =  A )  /\  ( 0  <  B  \/  0  =  B
) )  ->  0  <_  ( A  x.  B
) ) )
347, 33sylbid 206 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  <_  A  /\  0  <_  B
)  ->  0  <_  ( A  x.  B ) ) )
3534imp 418 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <_  B
) )  ->  0  <_  ( A  x.  B
) )
3635an4s 799 1  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  0  <_  ( A  x.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1642    e. wcel 1710   class class class wbr 4104  (class class class)co 5945   RRcr 8826   0cc0 8827    x. cmul 8832    < clt 8957    <_ cle 8958
This theorem is referenced by:  mulge0OLD  9382  mulge0i  9410  mulge0d  9439  ge0mulcl  10841  expge0  11231  bernneq  11320  sqrmul  11841  sqreulem  11939  amgm2  11949  efcllem  12456  nmoco  18348  iihalf1  18533  iimulcl  18539  mbfi1fseqlem1  19174  mbfi1fseqlem3  19176  mbfi1fseqlem5  19178  dchrisumlem3  20752  dchrvmasumlem2  20759  chpdifbndlem2  20815  cnlnadjlem7  22767  leopmuli  22827  mulge0b  24492  fmul01  27033  stoweidlem1  27073  stoweidlem16  27088  stoweidlem24  27096  stoweidlem26  27098  stoweidlem38  27110  stoweidlem51  27123
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-po 4396  df-so 4397  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963
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