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Theorem prodgt0 9601
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 8838 . . . . 5  |-  0  e.  RR
21a1i 10 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  0  e.  RR )
3 simpl 443 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  e.  RR )
42, 3leloed 8962 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  A  <->  ( 0  <  A  \/  0  =  A )
) )
5 simpll 730 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  A  e.  RR )
6 simplr 731 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  B  e.  RR )
75, 6remulcld 8863 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  ( A  x.  B )  e.  RR )
8 simprl 732 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  A )
98gt0ne0d 9337 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  A  =/=  0 )
105, 9rereccld 9587 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  (
1  /  A )  e.  RR )
11 simprr 733 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  ( A  x.  B
) )
12 recgt0 9600 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( 1  /  A ) )
1312ad2ant2r 727 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  ( 1  /  A
) )
147, 10, 11, 13mulgt0d 8971 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  ( ( A  x.  B )  x.  (
1  /  A ) ) )
157recnd 8861 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  ( A  x.  B )  e.  CC )
165recnd 8861 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  A  e.  CC )
1715, 16, 9divrecd 9539 . . . . . . 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 447 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
1918recnd 8861 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  CC )
2019adantr 451 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  B  e.  CC )
2120, 16, 9divcan3d 9541 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  (
( A  x.  B
)  /  A )  =  B )
2217, 21eqtr3d 2317 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  (
( A  x.  B
)  x.  ( 1  /  A ) )  =  B )
2314, 22breqtrd 4047 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  B )
2423exp32 588 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  A  ->  ( 0  <  ( A  x.  B )  ->  0  <  B ) ) )
251ltnri 8929 . . . . . . 7  |-  -.  0  <  0
2619mul02d 9010 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  x.  B
)  =  0 )
2726breq2d 4035 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  (
0  x.  B )  <->  0  <  0 ) )
2825, 27mtbiri 294 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  0  <  (
0  x.  B ) )
2928pm2.21d 98 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  (
0  x.  B )  ->  0  <  B
) )
30 oveq1 5865 . . . . . . 7  |-  ( 0  =  A  ->  (
0  x.  B )  =  ( A  x.  B ) )
3130breq2d 4035 . . . . . 6  |-  ( 0  =  A  ->  (
0  <  ( 0  x.  B )  <->  0  <  ( A  x.  B ) ) )
3231imbi1d 308 . . . . 5  |-  ( 0  =  A  ->  (
( 0  <  (
0  x.  B )  ->  0  <  B
)  <->  ( 0  < 
( A  x.  B
)  ->  0  <  B ) ) )
3329, 32syl5ibcom 211 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  =  A  ->  ( 0  < 
( A  x.  B
)  ->  0  <  B ) ) )
3424, 33jaod 369 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  < 
A  \/  0  =  A )  ->  (
0  <  ( A  x.  B )  ->  0  <  B ) ) )
354, 34sylbid 206 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <_  A  ->  ( 0  <  ( A  x.  B )  ->  0  <  B ) ) )
3635imp32 422 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 357    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    < clt 8867    <_ cle 8868    / cdiv 9423
This theorem is referenced by:  prodgt02  9602  prodgt0i  9663
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424
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