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Theorem mulge0 9545
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 9091 . . . . . . 7  |-  0  e.  RR
21a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  0  e.  RR )
3 simpl 444 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  RR )
42, 3leloed 9216 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
5 simpr 448 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
62, 5leloed 9216 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  B  <->  ( 0  <  B  \/  0  =  B )
) )
74, 6anbi12d 692 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  <_  A  /\  0  <_  B
)  <->  ( ( 0  <  A  \/  0  =  A )  /\  ( 0  <  B  \/  0  =  B
) ) ) )
81a1i 11 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  0  e.  RR )
9 simpll 731 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  A  e.  RR )
10 simplr 732 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  B  e.  RR )
119, 10remulcld 9116 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  ( A  x.  B )  e.  RR )
12 mulgt0 9153 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  ( A  x.  B ) )
1312an4s 800 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  0  <  ( A  x.  B
) )
148, 11, 13ltled 9221 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  0  <_  ( A  x.  B
) )
1514ex 424 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  < 
A  /\  0  <  B )  ->  0  <_  ( A  x.  B ) ) )
16 leid 9169 . . . . . . . . 9  |-  ( 0  e.  RR  ->  0  <_  0 )
171, 16ax-mp 8 . . . . . . . 8  |-  0  <_  0
185recnd 9114 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  CC )
1918mul02d 9264 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  x.  B
)  =  0 )
2017, 19syl5breqr 4248 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  0  <_  ( 0  x.  B ) )
21 oveq1 6088 . . . . . . . 8  |-  ( 0  =  A  ->  (
0  x.  B )  =  ( A  x.  B ) )
2221breq2d 4224 . . . . . . 7  |-  ( 0  =  A  ->  (
0  <_  ( 0  x.  B )  <->  0  <_  ( A  x.  B ) ) )
2320, 22syl5ibcom 212 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  =  A  ->  0  <_  ( A  x.  B )
) )
2423adantrd 455 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  =  A  /\  0  < 
B )  ->  0  <_  ( A  x.  B
) ) )
253recnd 9114 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  CC )
2625mul01d 9265 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  0 )  =  0 )
2717, 26syl5breqr 4248 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  0  <_  ( A  x.  0 ) )
28 oveq2 6089 . . . . . . . 8  |-  ( 0  =  B  ->  ( A  x.  0 )  =  ( A  x.  B ) )
2928breq2d 4224 . . . . . . 7  |-  ( 0  =  B  ->  (
0  <_  ( A  x.  0 )  <->  0  <_  ( A  x.  B ) ) )
3027, 29syl5ibcom 212 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  =  B  ->  0  <_  ( A  x.  B )
) )
3130adantld 454 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  < 
A  /\  0  =  B )  ->  0  <_  ( A  x.  B
) ) )
3230adantld 454 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  =  A  /\  0  =  B )  ->  0  <_  ( A  x.  B
) ) )
3315, 24, 31, 32ccased 914 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( 0  <  A  \/  0  =  A )  /\  ( 0  <  B  \/  0  =  B
) )  ->  0  <_  ( A  x.  B
) ) )
347, 33sylbid 207 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  <_  A  /\  0  <_  B
)  ->  0  <_  ( A  x.  B ) ) )
3534imp 419 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <_  B
) )  ->  0  <_  ( A  x.  B
) )
3635an4s 800 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 358    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4212  (class class class)co 6081   RRcr 8989   0cc0 8990    x. cmul 8995    < clt 9120    <_ cle 9121
This theorem is referenced by:  mulge0OLD  9546  mulge0i  9574  mulge0d  9603  ge0mulcl  11010  expge0  11416  bernneq  11505  sqrmul  12065  sqreulem  12163  amgm2  12173  efcllem  12680  nmoco  18771  iihalf1  18956  iimulcl  18962  mbfi1fseqlem1  19607  mbfi1fseqlem3  19609  mbfi1fseqlem5  19611  dchrisumlem3  21185  dchrvmasumlem2  21192  chpdifbndlem2  21248  cnlnadjlem7  23576  leopmuli  23636  reofld  24280  mulge0b  25191  stoweidlem24  27749
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126
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