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Theorem mulge0 9856
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 0red 9386 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  0  e.  RR )
2 simpl 457 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  RR )
31, 2leloed 9516 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
4 simpr 461 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
51, 4leloed 9516 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  B  <->  ( 0  <  B  \/  0  =  B )
) )
63, 5anbi12d 710 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  <_  A  /\  0  <_  B
)  <->  ( ( 0  <  A  \/  0  =  A )  /\  ( 0  <  B  \/  0  =  B
) ) ) )
7 0red 9386 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  0  e.  RR )
8 simpll 753 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  A  e.  RR )
9 simplr 754 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  B  e.  RR )
108, 9remulcld 9413 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  ( A  x.  B )  e.  RR )
11 mulgt0 9451 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  ( A  x.  B ) )
1211an4s 822 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  0  <  ( A  x.  B
) )
137, 10, 12ltled 9521 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  0  <_  ( A  x.  B
) )
1413ex 434 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  < 
A  /\  0  <  B )  ->  0  <_  ( A  x.  B ) ) )
15 0re 9385 . . . . . . . . 9  |-  0  e.  RR
16 leid 9469 . . . . . . . . 9  |-  ( 0  e.  RR  ->  0  <_  0 )
1715, 16ax-mp 5 . . . . . . . 8  |-  0  <_  0
184recnd 9411 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  CC )
1918mul02d 9566 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  x.  B
)  =  0 )
2017, 19syl5breqr 4327 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  0  <_  ( 0  x.  B ) )
21 oveq1 6097 . . . . . . . 8  |-  ( 0  =  A  ->  (
0  x.  B )  =  ( A  x.  B ) )
2221breq2d 4303 . . . . . . 7  |-  ( 0  =  A  ->  (
0  <_  ( 0  x.  B )  <->  0  <_  ( A  x.  B ) ) )
2320, 22syl5ibcom 220 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  =  A  ->  0  <_  ( A  x.  B )
) )
2423adantrd 468 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  =  A  /\  0  < 
B )  ->  0  <_  ( A  x.  B
) ) )
252recnd 9411 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  CC )
2625mul01d 9567 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  0 )  =  0 )
2717, 26syl5breqr 4327 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  0  <_  ( A  x.  0 ) )
28 oveq2 6098 . . . . . . . 8  |-  ( 0  =  B  ->  ( A  x.  0 )  =  ( A  x.  B ) )
2928breq2d 4303 . . . . . . 7  |-  ( 0  =  B  ->  (
0  <_  ( A  x.  0 )  <->  0  <_  ( A  x.  B ) ) )
3027, 29syl5ibcom 220 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  =  B  ->  0  <_  ( A  x.  B )
) )
3130adantld 467 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  < 
A  /\  0  =  B )  ->  0  <_  ( A  x.  B
) ) )
3230adantld 467 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  =  A  /\  0  =  B )  ->  0  <_  ( A  x.  B
) ) )
3314, 24, 31, 32ccased 938 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( 0  <  A  \/  0  =  A )  /\  ( 0  <  B  \/  0  =  B
) )  ->  0  <_  ( A  x.  B
) ) )
346, 33sylbid 215 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  <_  A  /\  0  <_  B
)  ->  0  <_  ( A  x.  B ) ) )
3534imp 429 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <_  B
) )  ->  0  <_  ( A  x.  B
) )
3635an4s 822 1  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <_  B )
)  ->  0  <_  ( A  x.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   class class class wbr 4291  (class class class)co 6090   RRcr 9280   0cc0 9281    x. cmul 9286    < clt 9417    <_ cle 9418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-po 4640  df-so 4641  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423
This theorem is referenced by:  mulge0OLD  9857  mulge0i  9886  mulge0d  9915  mulge0b  10198  ge0mulcl  11397  expge0  11899  bernneq  11989  sqrmul  12748  sqreulem  12846  amgm2  12856  efcllem  13362  nmoco  20315  iihalf1  20502  iimulcl  20508  mbfi1fseqlem1  21192  mbfi1fseqlem3  21194  mbfi1fseqlem5  21196  dchrisumlem3  22739  dchrvmasumlem2  22746  chpdifbndlem2  22802  cnlnadjlem7  25476  leopmuli  25536  reofld  26307  stoweidlem24  29817
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