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Theorem prodgt0 10449
Description: Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 24-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
prodgt0  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  <  B )

Proof of Theorem prodgt0
StepHypRef Expression
1 0red 9643 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  0  e.  RR )
2 simpl 458 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  RR )
31, 2leloed 9777 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
4 simpll 758 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  A  e.  RR )
5 simplr 760 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  B  e.  RR )
64, 5remulcld 9670 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  ( A  x.  B )  e.  RR )
7 simprl 762 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  A )
87gt0ne0d 10177 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  A  =/=  0 )
94, 8rereccld 10433 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  (
1  /  A )  e.  RR )
10 simprr 764 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  ( A  x.  B
) )
11 recgt0 10448 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( 1  /  A ) )
1211ad2ant2r 751 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  ( 1  /  A
) )
136, 9, 10, 12mulgt0d 9789 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  ( ( A  x.  B )  x.  (
1  /  A ) ) )
146recnd 9668 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  ( A  x.  B )  e.  CC )
154recnd 9668 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  A  e.  CC )
1614, 15, 8divrecd 10385 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  (
( A  x.  B
)  /  A )  =  ( ( A  x.  B )  x.  ( 1  /  A
) ) )
17 simpr 462 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
1817recnd 9668 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  CC )
1918adantr 466 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  B  e.  CC )
2019, 15, 8divcan3d 10387 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  (
( A  x.  B
)  /  A )  =  B )
2116, 20eqtr3d 2472 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  (
( A  x.  B
)  x.  ( 1  /  A ) )  =  B )
2213, 21breqtrd 4450 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  B )
2322exp32 608 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  A  ->  ( 0  <  ( A  x.  B )  ->  0  <  B ) ) )
24 0re 9642 . . . . . . . 8  |-  0  e.  RR
2524ltnri 9742 . . . . . . 7  |-  -.  0  <  0
2618mul02d 9830 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  x.  B
)  =  0 )
2726breq2d 4438 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  (
0  x.  B )  <->  0  <  0 ) )
2825, 27mtbiri 304 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  0  <  (
0  x.  B ) )
2928pm2.21d 109 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  (
0  x.  B )  ->  0  <  B
) )
30 oveq1 6312 . . . . . . 7  |-  ( 0  =  A  ->  (
0  x.  B )  =  ( A  x.  B ) )
3130breq2d 4438 . . . . . 6  |-  ( 0  =  A  ->  (
0  <  ( 0  x.  B )  <->  0  <  ( A  x.  B ) ) )
3231imbi1d 318 . . . . 5  |-  ( 0  =  A  ->  (
( 0  <  (
0  x.  B )  ->  0  <  B
)  <->  ( 0  < 
( A  x.  B
)  ->  0  <  B ) ) )
3329, 32syl5ibcom 223 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  =  A  ->  ( 0  < 
( A  x.  B
)  ->  0  <  B ) ) )
3423, 33jaod 381 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  < 
A  \/  0  =  A )  ->  (
0  <  ( A  x.  B )  ->  0  <  B ) ) )
353, 34sylbid 218 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  A  ->  ( 0  <  ( A  x.  B )  ->  0  <  B ) ) )
3635imp32 434 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  <  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870   class class class wbr 4426  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539    x. cmul 9543    < clt 9674    <_ cle 9675    / cdiv 10268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
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 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  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-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269
This theorem is referenced by:  prodgt02  10450  prodgt0i  10513  sgnmul  29201
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