MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  div2neg Structured version   Unicode version

Theorem div2neg 9729
Description: Quotient of two negatives. (Contributed by Paul Chapman, 10-Nov-2012.)
Assertion
Ref Expression
div2neg  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( -u A  /  -u B
)  =  ( A  /  B ) )

Proof of Theorem div2neg
StepHypRef Expression
1 negcl 9298 . . . . 5  |-  ( B  e.  CC  ->  -u B  e.  CC )
213ad2ant2 979 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  -u B  e.  CC )
3 simp1 957 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  A  e.  CC )
4 simp2 958 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  B  e.  CC )
5 simp3 959 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  B  =/=  0 )
6 div12 9692 . . . 4  |-  ( (
-u B  e.  CC  /\  A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( -u B  x.  ( A  /  B
) )  =  ( A  x.  ( -u B  /  B ) ) )
72, 3, 4, 5, 6syl112anc 1188 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( -u B  x.  ( A  /  B ) )  =  ( A  x.  ( -u B  /  B
) ) )
8 divneg 9701 . . . . . . 7  |-  ( ( B  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  -u ( B  /  B )  =  ( -u B  /  B ) )
94, 8syld3an1 1230 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  -u ( B  /  B )  =  ( -u B  /  B ) )
10 divid 9697 . . . . . . . 8  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( B  /  B
)  =  1 )
11103adant1 975 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  /  B )  =  1 )
1211negeqd 9292 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  -u ( B  /  B )  = 
-u 1 )
139, 12eqtr3d 2469 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( -u B  /  B )  =  -u 1 )
1413oveq2d 6089 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  x.  ( -u B  /  B ) )  =  ( A  x.  -u 1
) )
15 ax-1cn 9040 . . . . . . . 8  |-  1  e.  CC
1615negcli 9360 . . . . . . 7  |-  -u 1  e.  CC
17 mulcom 9068 . . . . . . 7  |-  ( ( A  e.  CC  /\  -u 1  e.  CC )  ->  ( A  x.  -u 1 )  =  (
-u 1  x.  A
) )
1816, 17mpan2 653 . . . . . 6  |-  ( A  e.  CC  ->  ( A  x.  -u 1 )  =  ( -u 1  x.  A ) )
19 mulm1 9467 . . . . . 6  |-  ( A  e.  CC  ->  ( -u 1  x.  A )  =  -u A )
2018, 19eqtrd 2467 . . . . 5  |-  ( A  e.  CC  ->  ( A  x.  -u 1 )  =  -u A )
21203ad2ant1 978 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  x.  -u 1 )  =  -u A )
2214, 21eqtrd 2467 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  x.  ( -u B  /  B ) )  = 
-u A )
237, 22eqtrd 2467 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( -u B  x.  ( A  /  B ) )  =  -u A )
24 negcl 9298 . . . 4  |-  ( A  e.  CC  ->  -u A  e.  CC )
25243ad2ant1 978 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  -u A  e.  CC )
26 divcl 9676 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  e.  CC )
27 negeq0 9347 . . . . . 6  |-  ( B  e.  CC  ->  ( B  =  0  <->  -u B  =  0 ) )
2827necon3bid 2633 . . . . 5  |-  ( B  e.  CC  ->  ( B  =/=  0  <->  -u B  =/=  0 ) )
2928biimpa 471 . . . 4  |-  ( ( B  e.  CC  /\  B  =/=  0 )  ->  -u B  =/=  0 )
30293adant1 975 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  -u B  =/=  0 )
31 divmul 9673 . . 3  |-  ( (
-u A  e.  CC  /\  ( A  /  B
)  e.  CC  /\  ( -u B  e.  CC  /\  -u B  =/=  0
) )  ->  (
( -u A  /  -u B
)  =  ( A  /  B )  <->  ( -u B  x.  ( A  /  B
) )  =  -u A ) )
3225, 26, 2, 30, 31syl112anc 1188 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( -u A  /  -u B
)  =  ( A  /  B )  <->  ( -u B  x.  ( A  /  B
) )  =  -u A ) )
3323, 32mpbird 224 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( -u A  /  -u B
)  =  ( A  /  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    x. cmul 8987   -ucneg 9284    / cdiv 9669
This theorem is referenced by:  divneg2  9730  div2negd  9797  div2sub  9831  iblcnlem1  19671  itgcnlem  19673
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
  Copyright terms: Public domain W3C validator