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Theorem xmulge0 11578
Description: Extended real version of mulge0 10140. (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 11577 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e. 
RR*  /\  0  <  B ) )  ->  0  <  ( A xe B ) )
21an4s 833 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( 0  <  A  /\  0  <  B ) )  ->  0  <  ( A xe B ) )
3 0xr 9695 . . . . . . . 8  |-  0  e.  RR*
4 xmulcl 11567 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A xe B )  e.  RR* )
54adantr 466 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( 0  <  A  /\  0  <  B ) )  ->  ( A xe B )  e.  RR* )
6 xrltle 11456 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  ( A xe B )  e.  RR* )  ->  (
0  <  ( A xe B )  ->  0  <_  ( A xe B ) ) )
73, 5, 6sylancr 667 . . . . . . 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 435 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  A  /\  0  <  B )  ->  0  <_  ( A xe B ) ) )
109ad2ant2r 751 . . . 4  |-  ( ( ( A  e.  RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B )
)  ->  ( (
0  <  A  /\  0  <  B )  -> 
0  <_  ( A xe B ) ) )
1110impl 624 . . 3  |-  ( ( ( ( ( A  e.  RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B ) )  /\  0  <  A )  /\  0  <  B )  -> 
0  <_  ( A xe B ) )
12 0le0 10707 . . . . 5  |-  0  <_  0
13 oveq2 6314 . . . . . . 7  |-  ( 0  =  B  ->  ( A xe 0 )  =  ( A xe B ) )
1413eqcomd 2430 . . . . . 6  |-  ( 0  =  B  ->  ( A xe B )  =  ( A xe 0 ) )
15 xmul01 11561 . . . . . . 7  |-  ( A  e.  RR*  ->  ( A xe 0 )  =  0 )
1615ad2antrr 730 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B )
)  ->  ( A xe 0 )  =  0 )
1714, 16sylan9eqr 2485 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B ) )  /\  0  =  B )  ->  ( A xe B )  =  0 )
1812, 17syl5breqr 4460 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B ) )  /\  0  =  B )  ->  0  <_  ( A xe B ) )
1918adantlr 719 . . 3  |-  ( ( ( ( ( A  e.  RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B ) )  /\  0  <  A )  /\  0  =  B )  ->  0  <_  ( A xe B ) )
20 xrleloe 11451 . . . . . 6  |-  ( ( 0  e.  RR*  /\  B  e.  RR* )  ->  (
0  <_  B  <->  ( 0  <  B  \/  0  =  B ) ) )
213, 20mpan 674 . . . . 5  |-  ( B  e.  RR*  ->  ( 0  <_  B  <->  ( 0  <  B  \/  0  =  B ) ) )
2221biimpa 486 . . . 4  |-  ( ( B  e.  RR*  /\  0  <_  B )  ->  (
0  <  B  \/  0  =  B )
)
2322ad2antlr 731 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B ) )  /\  0  <  A )  -> 
( 0  <  B  \/  0  =  B
) )
2411, 19, 23mpjaodan 793 . 2  |-  ( ( ( ( A  e. 
RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B ) )  /\  0  <  A )  -> 
0  <_  ( A xe B ) )
25 oveq1 6313 . . . . 5  |-  ( 0  =  A  ->  (
0 xe B )  =  ( A xe B ) )
2625eqcomd 2430 . . . 4  |-  ( 0  =  A  ->  ( A xe B )  =  ( 0 xe B ) )
27 xmul02 11562 . . . . 5  |-  ( B  e.  RR*  ->  ( 0 xe B )  =  0 )
2827ad2antrl 732 . . . 4  |-  ( ( ( A  e.  RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B )
)  ->  ( 0 xe B )  =  0 )
2926, 28sylan9eqr 2485 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B ) )  /\  0  =  A )  ->  ( A xe B )  =  0 )
3012, 29syl5breqr 4460 . 2  |-  ( ( ( ( A  e. 
RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B ) )  /\  0  =  A )  ->  0  <_  ( A xe B ) )
31 xrleloe 11451 . . . . 5  |-  ( ( 0  e.  RR*  /\  A  e.  RR* )  ->  (
0  <_  A  <->  ( 0  <  A  \/  0  =  A ) ) )
323, 31mpan 674 . . . 4  |-  ( A  e.  RR*  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A ) ) )
3332biimpa 486 . . 3  |-  ( ( A  e.  RR*  /\  0  <_  A )  ->  (
0  <  A  \/  0  =  A )
)
3433adantr 466 . 2  |-  ( ( ( A  e.  RR*  /\  0  <_  A )  /\  ( B  e.  RR*  /\  0  <_  B )
)  ->  ( 0  <  A  \/  0  =  A ) )
3524, 30, 34mpjaodan 793 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 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872   class class class wbr 4423  (class class class)co 6306   0cc0 9547   RR*cxr 9682    < clt 9683    <_ cle 9684   xecxmu 11416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-1st 6808  df-2nd 6809  df-er 7375  df-en 7582  df-dom 7583  df-sdom 7584  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-xmul 11419
This theorem is referenced by:  xadddi2  11591  ge0xmulcl  11755
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