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Theorem prodgt0 9847
Description: Infer that a multiplicand is positive from a nonnegative muliplier 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 0re 9083 . . . . 5  |-  0  e.  RR
21a1i 11 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  0  e.  RR )
3 simpl 444 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  RR )
42, 3leloed 9208 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
5 simpll 731 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  A  e.  RR )
6 simplr 732 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  B  e.  RR )
75, 6remulcld 9108 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  ( A  x.  B )  e.  RR )
8 simprl 733 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  A )
98gt0ne0d 9583 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  A  =/=  0 )
105, 9rereccld 9833 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  (
1  /  A )  e.  RR )
11 simprr 734 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  ( A  x.  B
) )
12 recgt0 9846 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( 1  /  A ) )
1312ad2ant2r 728 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  ( 1  /  A
) )
147, 10, 11, 13mulgt0d 9217 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  ( ( A  x.  B )  x.  (
1  /  A ) ) )
157recnd 9106 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  ( A  x.  B )  e.  CC )
165recnd 9106 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  A  e.  CC )
1715, 16, 9divrecd 9785 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  (
( A  x.  B
)  /  A )  =  ( ( A  x.  B )  x.  ( 1  /  A
) ) )
18 simpr 448 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
1918recnd 9106 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  CC )
2019adantr 452 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  B  e.  CC )
2120, 16, 9divcan3d 9787 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  (
( A  x.  B
)  /  A )  =  B )
2217, 21eqtr3d 2469 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  (
( A  x.  B
)  x.  ( 1  /  A ) )  =  B )
2314, 22breqtrd 4228 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  B )
2423exp32 589 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  A  ->  ( 0  <  ( A  x.  B )  ->  0  <  B ) ) )
251ltnri 9174 . . . . . . 7  |-  -.  0  <  0
2619mul02d 9256 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  x.  B
)  =  0 )
2726breq2d 4216 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  (
0  x.  B )  <->  0  <  0 ) )
2825, 27mtbiri 295 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  0  <  (
0  x.  B ) )
2928pm2.21d 100 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  (
0  x.  B )  ->  0  <  B
) )
30 oveq1 6080 . . . . . . 7  |-  ( 0  =  A  ->  (
0  x.  B )  =  ( A  x.  B ) )
3130breq2d 4216 . . . . . 6  |-  ( 0  =  A  ->  (
0  <  ( 0  x.  B )  <->  0  <  ( A  x.  B ) ) )
3231imbi1d 309 . . . . 5  |-  ( 0  =  A  ->  (
( 0  <  (
0  x.  B )  ->  0  <  B
)  <->  ( 0  < 
( A  x.  B
)  ->  0  <  B ) ) )
3329, 32syl5ibcom 212 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  =  A  ->  ( 0  < 
( A  x.  B
)  ->  0  <  B ) ) )
3424, 33jaod 370 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  < 
A  \/  0  =  A )  ->  (
0  <  ( A  x.  B )  ->  0  <  B ) ) )
354, 34sylbid 207 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  A  ->  ( 0  <  ( A  x.  B )  ->  0  <  B ) ) )
3635imp32 423 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  <  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4204  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    x. cmul 8987    < clt 9112    <_ cle 9113    / cdiv 9669
This theorem is referenced by:  prodgt02  9848  prodgt0i  9909
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670
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