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Theorem xmulge0 11252
Description: Extended real version of mulge0 9862. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xmulge0  |-  ( ( ( A  e.  RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B )
)  ->  0  <_  ( A xe B ) )

Proof of Theorem xmulge0
StepHypRef Expression
1 xmulgt0 11251 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e. 
RR*  /\  0  <  B ) )  ->  0  <  ( A xe B ) )
21an4s 822 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( 0  <  A  /\  0  <  B ) )  ->  0  <  ( A xe B ) )
3 0xr 9435 . . . . . . . 8  |-  0  e.  RR*
4 xmulcl 11241 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A xe B )  e.  RR* )
54adantr 465 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( 0  <  A  /\  0  <  B ) )  ->  ( A xe B )  e.  RR* )
6 xrltle 11131 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  ( A xe B )  e.  RR* )  ->  (
0  <  ( A xe B )  ->  0  <_  ( A xe B ) ) )
73, 5, 6sylancr 663 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( 0  <  A  /\  0  <  B ) )  ->  ( 0  <  ( A xe B )  -> 
0  <_  ( A xe B ) ) )
82, 7mpd 15 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( 0  <  A  /\  0  <  B ) )  ->  0  <_  ( A xe B ) )
98ex 434 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  A  /\  0  <  B )  ->  0  <_  ( A xe B ) ) )
109ad2ant2r 746 . . . 4  |-  ( ( ( A  e.  RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B )
)  ->  ( (
0  <  A  /\  0  <  B )  -> 
0  <_  ( A xe B ) ) )
1110impl 620 . . 3  |-  ( ( ( ( ( A  e.  RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B ) )  /\  0  <  A )  /\  0  <  B )  -> 
0  <_  ( A xe B ) )
12 0le0 10416 . . . . 5  |-  0  <_  0
13 oveq2 6104 . . . . . . 7  |-  ( 0  =  B  ->  ( A xe 0 )  =  ( A xe B ) )
1413eqcomd 2448 . . . . . 6  |-  ( 0  =  B  ->  ( A xe B )  =  ( A xe 0 ) )
15 xmul01 11235 . . . . . . 7  |-  ( A  e.  RR*  ->  ( A xe 0 )  =  0 )
1615ad2antrr 725 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B )
)  ->  ( A xe 0 )  =  0 )
1714, 16sylan9eqr 2497 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B ) )  /\  0  =  B )  ->  ( A xe B )  =  0 )
1812, 17syl5breqr 4333 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B ) )  /\  0  =  B )  ->  0  <_  ( A xe B ) )
1918adantlr 714 . . 3  |-  ( ( ( ( ( A  e.  RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B ) )  /\  0  <  A )  /\  0  =  B )  ->  0  <_  ( A xe B ) )
20 xrleloe 11126 . . . . . 6  |-  ( ( 0  e.  RR*  /\  B  e.  RR* )  ->  (
0  <_  B  <->  ( 0  <  B  \/  0  =  B ) ) )
213, 20mpan 670 . . . . 5  |-  ( B  e.  RR*  ->  ( 0  <_  B  <->  ( 0  <  B  \/  0  =  B ) ) )
2221biimpa 484 . . . 4  |-  ( ( B  e.  RR*  /\  0  <_  B )  ->  (
0  <  B  \/  0  =  B )
)
2322ad2antlr 726 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B ) )  /\  0  <  A )  -> 
( 0  <  B  \/  0  =  B
) )
2411, 19, 23mpjaodan 784 . 2  |-  ( ( ( ( A  e. 
RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B ) )  /\  0  <  A )  -> 
0  <_  ( A xe B ) )
25 oveq1 6103 . . . . 5  |-  ( 0  =  A  ->  (
0 xe B )  =  ( A xe B ) )
2625eqcomd 2448 . . . 4  |-  ( 0  =  A  ->  ( A xe B )  =  ( 0 xe B ) )
27 xmul02 11236 . . . . 5  |-  ( B  e.  RR*  ->  ( 0 xe B )  =  0 )
2827ad2antrl 727 . . . 4  |-  ( ( ( A  e.  RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B )
)  ->  ( 0 xe B )  =  0 )
2926, 28sylan9eqr 2497 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B ) )  /\  0  =  A )  ->  ( A xe B )  =  0 )
3012, 29syl5breqr 4333 . 2  |-  ( ( ( ( A  e. 
RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B ) )  /\  0  =  A )  ->  0  <_  ( A xe B ) )
31 xrleloe 11126 . . . . 5  |-  ( ( 0  e.  RR*  /\  A  e.  RR* )  ->  (
0  <_  A  <->  ( 0  <  A  \/  0  =  A ) ) )
323, 31mpan 670 . . . 4  |-  ( A  e.  RR*  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A ) ) )
3332biimpa 484 . . 3  |-  ( ( A  e.  RR*  /\  0  <_  A )  ->  (
0  <  A  \/  0  =  A )
)
3433adantr 465 . 2  |-  ( ( ( A  e.  RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B )
)  ->  ( 0  <  A  \/  0  =  A ) )
3524, 30, 34mpjaodan 784 1  |-  ( ( ( A  e.  RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B )
)  ->  0  <_  ( A xe B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   class class class wbr 4297  (class class class)co 6096   0cc0 9287   RR*cxr 9422    < clt 9423    <_ cle 9424   xecxmu 11093
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 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-xmul 11096
This theorem is referenced by:  xadddi2  11265  ge0xmulcl  11405
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