Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mul2lt0rgt0 Structured version   Unicode version

Theorem mul2lt0rgt0 26044
Description: If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.)
Hypotheses
Ref Expression
mul2lt0.1  |-  ( ph  ->  A  e.  RR )
mul2lt0.2  |-  ( ph  ->  B  e.  RR )
mul2lt0.3  |-  ( ph  ->  ( A  x.  B
)  <  0 )
Assertion
Ref Expression
mul2lt0rgt0  |-  ( (
ph  /\  0  <  B )  ->  A  <  0 )

Proof of Theorem mul2lt0rgt0
StepHypRef Expression
1 mul2lt0.3 . . . 4  |-  ( ph  ->  ( A  x.  B
)  <  0 )
21adantr 465 . . 3  |-  ( (
ph  /\  0  <  B )  ->  ( A  x.  B )  <  0
)
3 mul2lt0.2 . . . . . 6  |-  ( ph  ->  B  e.  RR )
43adantr 465 . . . . 5  |-  ( (
ph  /\  0  <  B )  ->  B  e.  RR )
54recnd 9417 . . . 4  |-  ( (
ph  /\  0  <  B )  ->  B  e.  CC )
65mul02d 9572 . . 3  |-  ( (
ph  /\  0  <  B )  ->  ( 0  x.  B )  =  0 )
72, 6breqtrrd 4323 . 2  |-  ( (
ph  /\  0  <  B )  ->  ( A  x.  B )  <  (
0  x.  B ) )
8 mul2lt0.1 . . . 4  |-  ( ph  ->  A  e.  RR )
98adantr 465 . . 3  |-  ( (
ph  /\  0  <  B )  ->  A  e.  RR )
10 0re 9391 . . . 4  |-  0  e.  RR
1110a1i 11 . . 3  |-  ( (
ph  /\  0  <  B )  ->  0  e.  RR )
12 simpr 461 . . . 4  |-  ( (
ph  /\  0  <  B )  ->  0  <  B )
134, 12elrpd 11030 . . 3  |-  ( (
ph  /\  0  <  B )  ->  B  e.  RR+ )
149, 11, 13ltmul1d 11069 . 2  |-  ( (
ph  /\  0  <  B )  ->  ( A  <  0  <->  ( A  x.  B )  <  (
0  x.  B ) ) )
157, 14mpbird 232 1  |-  ( (
ph  /\  0  <  B )  ->  A  <  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   class class class wbr 4297  (class class class)co 6096   RRcr 9286   0cc0 9287    x. cmul 9292    < clt 9423
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 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
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 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-ltxr 9428  df-sub 9602  df-neg 9603  df-rp 10997
This theorem is referenced by:  mul2lt0lgt0  26046  sgnmul  26930  signsply0  26957
  Copyright terms: Public domain W3C validator