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Theorem div2neg 9483
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 9052 . . . . 5  |-  ( B  e.  CC  ->  -u B  e.  CC )
213ad2ant2 977 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  -u B  e.  CC )
3 simp1 955 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  A  e.  CC )
4 simp2 956 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  B  e.  CC )
5 simp3 957 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  B  =/=  0 )
6 div12 9446 . . . 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 1186 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( -u B  x.  ( A  /  B ) )  =  ( A  x.  ( -u B  /  B
) ) )
8 divneg 9455 . . . . . . 7  |-  ( ( B  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  -u ( B  /  B )  =  ( -u B  /  B ) )
94, 8syld3an1 1228 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  -u ( B  /  B )  =  ( -u B  /  B ) )
10 divid 9451 . . . . . . . 8  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( B  /  B
)  =  1 )
11103adant1 973 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  /  B )  =  1 )
1211negeqd 9046 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  -u ( B  /  B )  = 
-u 1 )
139, 12eqtr3d 2317 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( -u B  /  B )  =  -u 1 )
1413oveq2d 5874 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  x.  ( -u B  /  B ) )  =  ( A  x.  -u 1
) )
15 ax-1cn 8795 . . . . . . . 8  |-  1  e.  CC
1615negcli 9114 . . . . . . 7  |-  -u 1  e.  CC
17 mulcom 8823 . . . . . . 7  |-  ( ( A  e.  CC  /\  -u 1  e.  CC )  ->  ( A  x.  -u 1 )  =  (
-u 1  x.  A
) )
1816, 17mpan2 652 . . . . . 6  |-  ( A  e.  CC  ->  ( A  x.  -u 1 )  =  ( -u 1  x.  A ) )
19 mulm1 9221 . . . . . 6  |-  ( A  e.  CC  ->  ( -u 1  x.  A )  =  -u A )
2018, 19eqtrd 2315 . . . . 5  |-  ( A  e.  CC  ->  ( A  x.  -u 1 )  =  -u A )
21203ad2ant1 976 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  x.  -u 1 )  =  -u A )
2214, 21eqtrd 2315 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  x.  ( -u B  /  B ) )  = 
-u A )
237, 22eqtrd 2315 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( -u B  x.  ( A  /  B ) )  =  -u A )
24 negcl 9052 . . . 4  |-  ( A  e.  CC  ->  -u A  e.  CC )
25243ad2ant1 976 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  -u A  e.  CC )
26 divcl 9430 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( A  /  B )  e.  CC )
27 negeq0 9101 . . . . . 6  |-  ( B  e.  CC  ->  ( B  =  0  <->  -u B  =  0 ) )
2827necon3bid 2481 . . . . 5  |-  ( B  e.  CC  ->  ( B  =/=  0  <->  -u B  =/=  0 ) )
2928biimpa 470 . . . 4  |-  ( ( B  e.  CC  /\  B  =/=  0 )  ->  -u B  =/=  0 )
30293adant1 973 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  -u B  =/=  0 )
31 divmul 9427 . . 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 1186 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  (
( -u A  /  -u B
)  =  ( A  /  B )  <->  ( -u B  x.  ( A  /  B
) )  =  -u A ) )
3323, 32mpbird 223 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 176    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    x. cmul 8742   -ucneg 9038    / cdiv 9423
This theorem is referenced by:  divneg2  9484  div2negd  9551  div2sub  9585  iblcnlem1  19142  itgcnlem  19144
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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