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Theorem div2negap 8622
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 8089 . . . . 5 |- ( B e. CC -> -u B e. CC )
213ad2ant2 1008 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u B e. CC )
3 simp1 986 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> A e. CC )
4 simp2 987 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> B e. CC )
5 simp3 988 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> B # 0 )
6 div12ap 8581 . . . 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 1231 . . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( -u B x. ( A / B ) ) = ( A x. ( -u B / B ) ) )
8 divnegap 8593 . . . . . . 7 |- ( ( B e. CC /\ B e. CC /\ B # 0 ) -> -u ( B / B ) = ( -u B / B ) )
94, 8syld3an1 1273 . . . . . 6 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u ( B / B ) = ( -u B / B ) )
10 dividap 8588 . . . . . . . 8 |- ( ( B e. CC /\ B # 0 ) -> ( B / B ) = 1 )
11103adant1 1004 . . . . . . 7 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( B / B ) = 1 )
1211negeqd 8084 . . . . . 6 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u ( B / B ) = -u 1 )
139, 12eqtr3d 2199 . . . . 5 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( -u B / B ) = -u 1 )
1413oveq2d 5852 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A x. ( -u B / B ) ) = ( A x. -u 1 ) )
15 ax-1cn 7837 . . . . . . . 8 |- 1 e. CC
1615negcli 8157 . . . . . . 7 |- -u 1 e. CC
17 mulcom 7873 . . . . . . 7 |- ( ( A e. CC /\ -u 1 e. CC ) -> ( A x. -u 1 ) = ( -u 1 x. A ) )
1816, 17mpan2 422 . . . . . 6 |- ( A e. CC -> ( A x. -u 1 ) = ( -u 1 x. A ) )
19 mulm1 8289 . . . . . 6 |- ( A e. CC -> ( -u 1 x. A ) = -u A )
2018, 19eqtrd 2197 . . . . 5 |- ( A e. CC -> ( A x. -u 1 ) = -u A )
21203ad2ant1 1007 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A x. -u 1 ) = -u A )
2214, 21eqtrd 2197 . . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A x. ( -u B / B ) ) = -u A )
237, 22eqtrd 2197 . 2 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( -u B x. ( A / B ) ) = -u A )
24 negcl 8089 . . . 4 |- ( A e. CC -> -u A e. CC )
25243ad2ant1 1007 . . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u A e. CC )
26 divclap 8565 . . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) e. CC )
27 negap0 8519 . . . . 5 |- ( B e. CC -> ( B # 0 <-> -u B # 0 ) )
2827biimpa 294 . . . 4 |- ( ( B e. CC /\ B # 0 ) -> -u B # 0 )
29283adant1 1004 . . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u B # 0 )
30 divmulap 8562 . . 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 1231 . 2 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( ( -u A / -u B ) = ( A / B ) <-> ( -u B x. ( A / B ) ) = -u A ) )
3223, 31mpbird 166 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 104   /\ w3a 967   = wceq 1342   e. wcel 2135   class class class wbr 3976  (class class class)co 5836   CCcc 7742   0cc0 7744   1c1 7745   x. cmul 7749   -ucneg 8061   # cap 8470   / cdiv 8559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-mulrcl 7843  ax-addcom 7844  ax-mulcom 7845  ax-addass 7846  ax-mulass 7847  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-1rid 7851  ax-0id 7852  ax-rnegex 7853  ax-precex 7854  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-apti 7859  ax-pre-ltadd 7860  ax-pre-mulgt0 7861  ax-pre-mulext 7862
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2723  df-sbc 2947  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-id 4265  df-po 4268  df-iso 4269  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-iota 5147  df-fun 5184  df-fv 5190  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-reap 8464  df-ap 8471  df-div 8560
This theorem is referenced by:  divneg2ap  8623  div2negapd  8692  div2subap  8724
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