ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  div2negap Unicode version

Theorem div2negap 8176
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 7661 . . . . 5  |-  ( B  e.  CC  ->  -u B  e.  CC )
213ad2ant2 965 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  -u B  e.  CC )
3 simp1 943 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  A  e.  CC )
4 simp2 944 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  B  e.  CC )
5 simp3 945 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  B #  0 )
6 div12ap 8135 . . . 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 1178 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( -u B  x.  ( A  /  B ) )  =  ( A  x.  ( -u B  /  B
) ) )
8 divnegap 8147 . . . . . . 7  |-  ( ( B  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  -u ( B  /  B )  =  ( -u B  /  B ) )
94, 8syld3an1 1220 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  -u ( B  /  B )  =  ( -u B  /  B ) )
10 dividap 8142 . . . . . . . 8  |-  ( ( B  e.  CC  /\  B #  0 )  ->  ( B  /  B )  =  1 )
11103adant1 961 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( B  /  B )  =  1 )
1211negeqd 7656 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  -u ( B  /  B )  = 
-u 1 )
139, 12eqtr3d 2122 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( -u B  /  B )  =  -u 1 )
1413oveq2d 5650 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  x.  ( -u B  /  B ) )  =  ( A  x.  -u 1
) )
15 ax-1cn 7417 . . . . . . . 8  |-  1  e.  CC
1615negcli 7729 . . . . . . 7  |-  -u 1  e.  CC
17 mulcom 7450 . . . . . . 7  |-  ( ( A  e.  CC  /\  -u 1  e.  CC )  ->  ( A  x.  -u 1 )  =  (
-u 1  x.  A
) )
1816, 17mpan2 416 . . . . . 6  |-  ( A  e.  CC  ->  ( A  x.  -u 1 )  =  ( -u 1  x.  A ) )
19 mulm1 7857 . . . . . 6  |-  ( A  e.  CC  ->  ( -u 1  x.  A )  =  -u A )
2018, 19eqtrd 2120 . . . . 5  |-  ( A  e.  CC  ->  ( A  x.  -u 1 )  =  -u A )
21203ad2ant1 964 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  x.  -u 1 )  =  -u A )
2214, 21eqtrd 2120 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  x.  ( -u B  /  B ) )  = 
-u A )
237, 22eqtrd 2120 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( -u B  x.  ( A  /  B ) )  =  -u A )
24 negcl 7661 . . . 4  |-  ( A  e.  CC  ->  -u A  e.  CC )
25243ad2ant1 964 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  -u A  e.  CC )
26 divclap 8119 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  ( A  /  B )  e.  CC )
27 negap0 8082 . . . . 5  |-  ( B  e.  CC  ->  ( B #  0  <->  -u B #  0 ) )
2827biimpa 290 . . . 4  |-  ( ( B  e.  CC  /\  B #  0 )  ->  -u B #  0 )
29283adant1 961 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  -u B #  0 )
30 divmulap 8116 . . 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 1178 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B #  0 )  ->  (
( -u A  /  -u B
)  =  ( A  /  B )  <->  ( -u B  x.  ( A  /  B
) )  =  -u A ) )
3223, 31mpbird 165 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 103    /\ w3a 924    = wceq 1289    e. wcel 1438   class class class wbr 3837  (class class class)co 5634   CCcc 7327   0cc0 7329   1c1 7330    x. cmul 7334   -ucneg 7633   # cap 8034    / cdiv 8113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-po 4114  df-iso 4115  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114
This theorem is referenced by:  divneg2ap  8177  div2negapd  8245  div2subap  8274
  Copyright terms: Public domain W3C validator