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Theorem xmulgt0 11478
Description: Extended real version of mulgt0 9651. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xmulgt0  |-  ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e. 
RR*  /\  0  <  B ) )  ->  0  <  ( A xe B ) )

Proof of Theorem xmulgt0
StepHypRef Expression
1 simpr 459 . . . . . 6  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  0  <  A )
2 simpr 459 . . . . . 6  |-  ( ( B  e.  RR*  /\  0  <  B )  ->  0  <  B )
31, 2anim12i 564 . . . . 5  |-  ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e. 
RR*  /\  0  <  B ) )  ->  (
0  <  A  /\  0  <  B ) )
4 mulgt0 9651 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  ( A  x.  B ) )
54an4s 824 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  B ) )  ->  0  <  ( A  x.  B
) )
65ancoms 451 . . . . . 6  |-  ( ( ( 0  <  A  /\  0  <  B )  /\  ( A  e.  RR  /\  B  e.  RR ) )  -> 
0  <  ( A  x.  B ) )
7 rexmul 11466 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A xe B )  =  ( A  x.  B ) )
87adantl 464 . . . . . 6  |-  ( ( ( 0  <  A  /\  0  <  B )  /\  ( A  e.  RR  /\  B  e.  RR ) )  -> 
( A xe B )  =  ( A  x.  B ) )
96, 8breqtrrd 4465 . . . . 5  |-  ( ( ( 0  <  A  /\  0  <  B )  /\  ( A  e.  RR  /\  B  e.  RR ) )  -> 
0  <  ( A xe B ) )
103, 9sylan 469 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  0  <  A )  /\  ( B  e.  RR*  /\  0  <  B ) )  /\  ( A  e.  RR  /\  B  e.  RR ) )  ->  0  <  ( A xe B ) )
1110anassrs 646 . . 3  |-  ( ( ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e.  RR*  /\  0  <  B ) )  /\  A  e.  RR )  /\  B  e.  RR )  ->  0  <  ( A xe B ) )
12 0ltpnf 11335 . . . . 5  |-  0  < +oo
13 oveq2 6278 . . . . . 6  |-  ( B  = +oo  ->  ( A xe B )  =  ( A xe +oo ) )
14 xmulpnf1 11469 . . . . . . 7  |-  ( ( A  e.  RR*  /\  0  <  A )  ->  ( A xe +oo )  = +oo )
1514adantr 463 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e. 
RR*  /\  0  <  B ) )  ->  ( A xe +oo )  = +oo )
1613, 15sylan9eqr 2517 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  0  <  A )  /\  ( B  e.  RR*  /\  0  <  B ) )  /\  B  = +oo )  ->  ( A xe B )  = +oo )
1712, 16syl5breqr 4475 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  0  <  A )  /\  ( B  e.  RR*  /\  0  <  B ) )  /\  B  = +oo )  ->  0  <  ( A xe B ) )
1817adantlr 712 . . 3  |-  ( ( ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e.  RR*  /\  0  <  B ) )  /\  A  e.  RR )  /\  B  = +oo )  ->  0  <  ( A xe B ) )
19 simplrr 760 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  0  <  A )  /\  ( B  e.  RR*  /\  0  <  B ) )  /\  A  e.  RR )  ->  0  <  B )
20 xmulasslem2 11477 . . . 4  |-  ( ( 0  <  B  /\  B  = -oo )  ->  0  <  ( A xe B ) )
2119, 20sylan 469 . . 3  |-  ( ( ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e.  RR*  /\  0  <  B ) )  /\  A  e.  RR )  /\  B  = -oo )  ->  0  <  ( A xe B ) )
22 simprl 754 . . . . 5  |-  ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e. 
RR*  /\  0  <  B ) )  ->  B  e.  RR* )
23 elxr 11328 . . . . 5  |-  ( B  e.  RR*  <->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
2422, 23sylib 196 . . . 4  |-  ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e. 
RR*  /\  0  <  B ) )  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
2524adantr 463 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  0  <  A )  /\  ( B  e.  RR*  /\  0  <  B ) )  /\  A  e.  RR )  ->  ( B  e.  RR  \/  B  = +oo  \/  B  = -oo ) )
2611, 18, 21, 25mpjao3dan 1293 . 2  |-  ( ( ( ( A  e. 
RR*  /\  0  <  A )  /\  ( B  e.  RR*  /\  0  <  B ) )  /\  A  e.  RR )  ->  0  <  ( A xe B ) )
27 oveq1 6277 . . . 4  |-  ( A  = +oo  ->  ( A xe B )  =  ( +oo xe B ) )
28 xmulpnf2 11470 . . . . 5  |-  ( ( B  e.  RR*  /\  0  <  B )  ->  ( +oo xe B )  = +oo )
2928adantl 464 . . . 4  |-  ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e. 
RR*  /\  0  <  B ) )  ->  ( +oo xe B )  = +oo )
3027, 29sylan9eqr 2517 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  0  <  A )  /\  ( B  e.  RR*  /\  0  <  B ) )  /\  A  = +oo )  ->  ( A xe B )  = +oo )
3112, 30syl5breqr 4475 . 2  |-  ( ( ( ( A  e. 
RR*  /\  0  <  A )  /\  ( B  e.  RR*  /\  0  <  B ) )  /\  A  = +oo )  ->  0  <  ( A xe B ) )
32 simplr 753 . . 3  |-  ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e. 
RR*  /\  0  <  B ) )  ->  0  <  A )
33 xmulasslem2 11477 . . 3  |-  ( ( 0  <  A  /\  A  = -oo )  ->  0  <  ( A xe B ) )
3432, 33sylan 469 . 2  |-  ( ( ( ( A  e. 
RR*  /\  0  <  A )  /\  ( B  e.  RR*  /\  0  <  B ) )  /\  A  = -oo )  ->  0  <  ( A xe B ) )
35 simpll 751 . . 3  |-  ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e. 
RR*  /\  0  <  B ) )  ->  A  e.  RR* )
36 elxr 11328 . . 3  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
3735, 36sylib 196 . 2  |-  ( ( ( A  e.  RR*  /\  0  <  A )  /\  ( B  e. 
RR*  /\  0  <  B ) )  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
3826, 31, 34, 37mpjao3dan 1293 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    /\ wa 367    \/ w3o 970    = wceq 1398    e. wcel 1823   class class class wbr 4439  (class class class)co 6270   RRcr 9480   0cc0 9481    x. cmul 9486   +oocpnf 9614   -oocmnf 9615   RR*cxr 9616    < clt 9617   xecxmu 11320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-xmul 11323
This theorem is referenced by:  xmulge0  11479  xmulasslem3  11481
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