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Theorem prodgt0 10186
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 9399 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  0  e.  RR )
2 simpl 457 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  RR )
31, 2leloed 9529 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
4 simpll 753 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  A  e.  RR )
5 simplr 754 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  B  e.  RR )
64, 5remulcld 9426 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  ( A  x.  B )  e.  RR )
7 simprl 755 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  A )
87gt0ne0d 9916 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  A  =/=  0 )
94, 8rereccld 10170 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  (
1  /  A )  e.  RR )
10 simprr 756 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  ( A  x.  B
) )
11 recgt0 10185 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( 1  /  A ) )
1211ad2ant2r 746 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  ( 1  /  A
) )
136, 9, 10, 12mulgt0d 9538 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  ( ( A  x.  B )  x.  (
1  /  A ) ) )
146recnd 9424 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  ( A  x.  B )  e.  CC )
154recnd 9424 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  A  e.  CC )
1614, 15, 8divrecd 10122 . . . . . . 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 461 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
1817recnd 9424 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  CC )
1918adantr 465 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  B  e.  CC )
2019, 15, 8divcan3d 10124 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  (
( A  x.  B
)  /  A )  =  B )
2116, 20eqtr3d 2477 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  (
( A  x.  B
)  x.  ( 1  /  A ) )  =  B )
2213, 21breqtrd 4328 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  B )
2322exp32 605 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  A  ->  ( 0  <  ( A  x.  B )  ->  0  <  B ) ) )
24 0re 9398 . . . . . . . 8  |-  0  e.  RR
2524ltnri 9495 . . . . . . 7  |-  -.  0  <  0
2618mul02d 9579 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  x.  B
)  =  0 )
2726breq2d 4316 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  (
0  x.  B )  <->  0  <  0 ) )
2825, 27mtbiri 303 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  0  <  (
0  x.  B ) )
2928pm2.21d 106 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  (
0  x.  B )  ->  0  <  B
) )
30 oveq1 6110 . . . . . . 7  |-  ( 0  =  A  ->  (
0  x.  B )  =  ( A  x.  B ) )
3130breq2d 4316 . . . . . 6  |-  ( 0  =  A  ->  (
0  <  ( 0  x.  B )  <->  0  <  ( A  x.  B ) ) )
3231imbi1d 317 . . . . 5  |-  ( 0  =  A  ->  (
( 0  <  (
0  x.  B )  ->  0  <  B
)  <->  ( 0  < 
( A  x.  B
)  ->  0  <  B ) ) )
3329, 32syl5ibcom 220 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  =  A  ->  ( 0  < 
( A  x.  B
)  ->  0  <  B ) ) )
3423, 33jaod 380 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  < 
A  \/  0  =  A )  ->  (
0  <  ( A  x.  B )  ->  0  <  B ) ) )
353, 34sylbid 215 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  A  ->  ( 0  <  ( A  x.  B )  ->  0  <  B ) ) )
3635imp32 433 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 368    /\ wa 369    = wceq 1369    e. wcel 1756   class class class wbr 4304  (class class class)co 6103   CCcc 9292   RRcr 9293   0cc0 9294   1c1 9295    x. cmul 9299    < clt 9430    <_ cle 9431    / cdiv 10005
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 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
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 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-po 4653  df-so 4654  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006
This theorem is referenced by:  prodgt02  10187  prodgt0i  10251  sgnmul  26937
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