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Theorem div2negap 8723
Description: Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.)
Assertion
Ref Expression
div2negap  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( -u A  /  -u B
)  =  ( A  /  B ) )

Proof of Theorem div2negap
StepHypRef Expression
1 negcl 8188 . . . . 5  |-  ( B  e.  CC  ->  -u B  e.  CC )
213ad2ant2 1021 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  -u B  e.  CC )
3 simp1 999 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  A  e.  CC )
4 simp2 1000 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  B  e.  CC )
5 simp3 1001 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  B #  0 )
6 div12ap 8682 . . . 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 1253 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( -u B  x.  ( A  /  B ) )  =  ( A  x.  ( -u B  /  B
) ) )
8 divnegap 8694 . . . . . . 7  |-  ( ( B  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  -u ( B  /  B )  =  ( -u B  /  B ) )
94, 8syld3an1 1295 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  -u ( B  /  B )  =  ( -u B  /  B ) )
10 dividap 8689 . . . . . . . 8  |-  ( ( B  e.  CC  /\  B #  0 )  ->  ( B  /  B )  =  1 )
11103adant1 1017 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( B  /  B )  =  1 )
1211negeqd 8183 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  -u ( B  /  B )  = 
-u 1 )
139, 12eqtr3d 2224 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( -u B  /  B )  =  -u 1 )
1413oveq2d 5913 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  x.  ( -u B  /  B ) )  =  ( A  x.  -u 1
) )
15 ax-1cn 7935 . . . . . . . 8  |-  1  e.  CC
1615negcli 8256 . . . . . . 7  |-  -u 1  e.  CC
17 mulcom 7971 . . . . . . 7  |-  ( ( A  e.  CC  /\  -u 1  e.  CC )  ->  ( A  x.  -u 1 )  =  (
-u 1  x.  A
) )
1816, 17mpan2 425 . . . . . 6  |-  ( A  e.  CC  ->  ( A  x.  -u 1 )  =  ( -u 1  x.  A ) )
19 mulm1 8388 . . . . . 6  |-  ( A  e.  CC  ->  ( -u 1  x.  A )  =  -u A )
2018, 19eqtrd 2222 . . . . 5  |-  ( A  e.  CC  ->  ( A  x.  -u 1 )  =  -u A )
21203ad2ant1 1020 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  x.  -u 1 )  =  -u A )
2214, 21eqtrd 2222 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  x.  ( -u B  /  B ) )  = 
-u A )
237, 22eqtrd 2222 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( -u B  x.  ( A  /  B ) )  =  -u A )
24 negcl 8188 . . . 4  |-  ( A  e.  CC  ->  -u A  e.  CC )
25243ad2ant1 1020 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  -u A  e.  CC )
26 divclap 8666 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  /  B )  e.  CC )
27 negap0 8618 . . . . 5  |-  ( B  e.  CC  ->  ( B #  0  <->  -u B #  0 ) )
2827biimpa 296 . . . 4  |-  ( ( B  e.  CC  /\  B #  0 )  ->  -u B #  0 )
29283adant1 1017 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  -u B #  0 )
30 divmulap 8663 . . 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 ) )
3125, 26, 2, 29, 30syl112anc 1253 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  (
( -u A  /  -u B
)  =  ( A  /  B )  <->  ( -u B  x.  ( A  /  B
) )  =  -u A ) )
3223, 31mpbird 167 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 105    /\ w3a 980    = wceq 1364    e. wcel 2160   class class class wbr 4018  (class class class)co 5897   CCcc 7840   0cc0 7842   1c1 7843    x. cmul 7847   -ucneg 8160   # cap 8569    / cdiv 8660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661
This theorem is referenced by:  divneg2ap  8724  div2negapd  8793  div2subap  8825
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