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Theorem div2negap 8905
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 8369 . . . . 5 |- ( B e. CC -> -u B e. CC )
213ad2ant2 1043 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u B e. CC )
3 simp1 1021 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> A e. CC )
4 simp2 1022 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> B e. CC )
5 simp3 1023 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> B # 0 )
6 div12ap 8864 . . . 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 1275 . . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( -u B x. ( A / B ) ) = ( A x. ( -u B / B ) ) )
8 divnegap 8876 . . . . . . 7 |- ( ( B e. CC /\ B e. CC /\ B # 0 ) -> -u ( B / B ) = ( -u B / B ) )
94, 8syld3an1 1317 . . . . . 6 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u ( B / B ) = ( -u B / B ) )
10 dividap 8871 . . . . . . . 8 |- ( ( B e. CC /\ B # 0 ) -> ( B / B ) = 1 )
11103adant1 1039 . . . . . . 7 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( B / B ) = 1 )
1211negeqd 8364 . . . . . 6 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u ( B / B ) = -u 1 )
139, 12eqtr3d 2264 . . . . 5 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( -u B / B ) = -u 1 )
1413oveq2d 6029 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A x. ( -u B / B ) ) = ( A x. -u 1 ) )
15 ax-1cn 8115 . . . . . . . 8 |- 1 e. CC
1615negcli 8437 . . . . . . 7 |- -u 1 e. CC
17 mulcom 8151 . . . . . . 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 8569 . . . . . 6 |- ( A e. CC -> ( -u 1 x. A ) = -u A )
2018, 19eqtrd 2262 . . . . 5 |- ( A e. CC -> ( A x. -u 1 ) = -u A )
21203ad2ant1 1042 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A x. -u 1 ) = -u A )
2214, 21eqtrd 2262 . . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A x. ( -u B / B ) ) = -u A )
237, 22eqtrd 2262 . 2 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( -u B x. ( A / B ) ) = -u A )
24 negcl 8369 . . . 4 |- ( A e. CC -> -u A e. CC )
25243ad2ant1 1042 . . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u A e. CC )
26 divclap 8848 . . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) e. CC )
27 negap0 8800 . . . . 5 |- ( B e. CC -> ( B # 0 <-> -u B # 0 ) )
2827biimpa 296 . . . 4 |- ( ( B e. CC /\ B # 0 ) -> -u B # 0 )
29283adant1 1039 . . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u B # 0 )
30 divmulap 8845 . . 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 1275 . 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 1002   = wceq 1395   e. wcel 2200   class class class wbr 4086  (class class class)co 6013   CCcc 8020   0cc0 8022   1c1 8023   x. cmul 8027   -ucneg 8341   # cap 8751   / cdiv 8842
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140
This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-po 4391  df-iso 4392  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843
This theorem is referenced by:  divneg2ap  8906  div2negapd  8975  div2subap  9007
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