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Theorem mulge0 10091
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 9614 . . . . . 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 9745 . . . . 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 9745 . . . . 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 9614 . . . . . . 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 755 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  B  e.  RR )
108, 9remulcld 9641 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  ( A  x.  B )  e.  RR )
11 mulgt0 9679 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  ( A  x.  B ) )
1211an4s 826 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  0  <  ( A  x.  B
) )
137, 10, 12ltled 9750 . . . . . 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 9613 . . . . . . . . 9  |-  0  e.  RR
16 leid 9697 . . . . . . . . 9  |-  ( 0  e.  RR  ->  0  <_  0 )
1715, 16ax-mp 5 . . . . . . . 8  |-  0  <_  0
184recnd 9639 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  CC )
1918mul02d 9795 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  x.  B
)  =  0 )
2017, 19syl5breqr 4492 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  0  <_  ( 0  x.  B ) )
21 oveq1 6303 . . . . . . . 8  |-  ( 0  =  A  ->  (
0  x.  B )  =  ( A  x.  B ) )
2221breq2d 4468 . . . . . . 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 9639 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  CC )
2625mul01d 9796 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  0 )  =  0 )
2717, 26syl5breqr 4492 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  0  <_  ( A  x.  0 ) )
28 oveq2 6304 . . . . . . . 8  |-  ( 0  =  B  ->  ( A  x.  0 )  =  ( A  x.  B ) )
2928breq2d 4468 . . . . . . 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 947 . . . 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 826 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 1395    e. wcel 1819   class class class wbr 4456  (class class class)co 6296   RRcr 9508   0cc0 9509    x. cmul 9514    < clt 9645    <_ cle 9646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651
This theorem is referenced by:  mulge0OLD  10092  mulge0i  10121  mulge0d  10150  mulge0b  10433  ge0mulcl  11658  expge0  12204  bernneq  12294  sqrtmul  13104  sqreulem  13203  amgm2  13213  efcllem  13824  nmoco  21369  iihalf1  21556  iimulcl  21562  mbfi1fseqlem1  22247  mbfi1fseqlem3  22249  mbfi1fseqlem5  22251  dchrisumlem3  23801  dchrvmasumlem2  23808  chpdifbndlem2  23864  cnlnadjlem7  27118  leopmuli  27178  reofld  27983  stoweidlem24  31967
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