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Theorem div2negap 9009
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 8473 . . . . 5 |- ( B e. CC -> -u B e. CC )
213ad2ant2 1046 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u B e. CC )
3 simp1 1024 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> A e. CC )
4 simp2 1025 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> B e. CC )
5 simp3 1026 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> B # 0 )
6 div12ap 8968 . . . 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 1278 . . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( -u B x. ( A / B ) ) = ( A x. ( -u B / B ) ) )
8 divnegap 8980 . . . . . . 7 |- ( ( B e. CC /\ B e. CC /\ B # 0 ) -> -u ( B / B ) = ( -u B / B ) )
94, 8syld3an1 1320 . . . . . 6 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u ( B / B ) = ( -u B / B ) )
10 dividap 8975 . . . . . . . 8 |- ( ( B e. CC /\ B # 0 ) -> ( B / B ) = 1 )
11103adant1 1042 . . . . . . 7 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( B / B ) = 1 )
1211negeqd 8468 . . . . . 6 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u ( B / B ) = -u 1 )
139, 12eqtr3d 2267 . . . . 5 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( -u B / B ) = -u 1 )
1413oveq2d 6066 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A x. ( -u B / B ) ) = ( A x. -u 1 ) )
15 ax-1cn 8220 . . . . . . . 8 |- 1 e. CC
1615negcli 8541 . . . . . . 7 |- -u 1 e. CC
17 mulcom 8256 . . . . . . 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 8673 . . . . . 6 |- ( A e. CC -> ( -u 1 x. A ) = -u A )
2018, 19eqtrd 2265 . . . . 5 |- ( A e. CC -> ( A x. -u 1 ) = -u A )
21203ad2ant1 1045 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A x. -u 1 ) = -u A )
2214, 21eqtrd 2265 . . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A x. ( -u B / B ) ) = -u A )
237, 22eqtrd 2265 . 2 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( -u B x. ( A / B ) ) = -u A )
24 negcl 8473 . . . 4 |- ( A e. CC -> -u A e. CC )
25243ad2ant1 1045 . . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u A e. CC )
26 divclap 8952 . . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) e. CC )
27 negap0 8904 . . . . 5 |- ( B e. CC -> ( B # 0 <-> -u B # 0 ) )
2827biimpa 296 . . . 4 |- ( ( B e. CC /\ B # 0 ) -> -u B # 0 )
29283adant1 1042 . . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u B # 0 )
30 divmulap 8949 . . 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 1278 . 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 1005   = wceq 1398   e. wcel 2203   class class class wbr 4109  (class class class)co 6050   CCcc 8125   0cc0 8127   1c1 8128   x. cmul 8132   -ucneg 8445   # cap 8855   / cdiv 8946
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245
This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947
This theorem is referenced by:  divneg2ap  9010  div2negapd  9079  div2subap  9111
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