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Theorem div2neg 10748
Description: Quotient of two negatives. (Contributed by Paul Chapman, 10-Nov-2012.)
Assertion
Ref Expression
div2neg |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( -u A / -u B ) = ( A / B ) )
Proof of Theorem div2neg
StepHypRef Expression
1 negcl 10281 . . . . 5 |- ( B e. CC -> -u B e. CC )
213ad2ant2 1083 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> -u B e. CC )
3 simp1 1061 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> A e. CC )
4 simp2 1062 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> B e. CC )
5 simp3 1063 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> B =/= 0 )
6 div12 10707 . . . 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 1330 . . 3 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( -u B x. ( A / B ) ) = ( A x. ( -u B / B ) ) )
8 divneg 10719 . . . . . . 7 |- ( ( B e. CC /\ B e. CC /\ B =/= 0 ) -> -u ( B / B ) = ( -u B / B ) )
94, 8syld3an1 1372 . . . . . 6 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> -u ( B / B ) = ( -u B / B ) )
10 divid 10714 . . . . . . . 8 |- ( ( B e. CC /\ B =/= 0 ) -> ( B / B ) = 1 )
11103adant1 1079 . . . . . . 7 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( B / B ) = 1 )
1211negeqd 10275 . . . . . 6 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> -u ( B / B ) = -u 1 )
139, 12eqtr3d 2658 . . . . 5 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( -u B / B ) = -u 1 )
1413oveq2d 6666 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( A x. ( -u B / B ) ) = ( A x. -u 1 ) )
15 ax-1cn 9994 . . . . . . . 8 |- 1 e. CC
1615negcli 10349 . . . . . . 7 |- -u 1 e. CC
17 mulcom 10022 . . . . . . 7 |- ( ( A e. CC /\ -u 1 e. CC ) -> ( A x. -u 1 ) = ( -u 1 x. A ) )
1816, 17mpan2 707 . . . . . 6 |- ( A e. CC -> ( A x. -u 1 ) = ( -u 1 x. A ) )
19 mulm1 10471 . . . . . 6 |- ( A e. CC -> ( -u 1 x. A ) = -u A )
2018, 19eqtrd 2656 . . . . 5 |- ( A e. CC -> ( A x. -u 1 ) = -u A )
21203ad2ant1 1082 . . . 4 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( A x. -u 1 ) = -u A )
2214, 21eqtrd 2656 . . 3 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( A x. ( -u B / B ) ) = -u A )
237, 22eqtrd 2656 . 2 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( -u B x. ( A / B ) ) = -u A )
24 negcl 10281 . . . 4 |- ( A e. CC -> -u A e. CC )
25243ad2ant1 1082 . . 3 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> -u A e. CC )
26 divcl 10691 . . 3 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( A / B ) e. CC )
27 negeq0 10335 . . . . . 6 |- ( B e. CC -> ( B = 0 <-> -u B = 0 ) )
2827necon3bid 2838 . . . . 5 |- ( B e. CC -> ( B =/= 0 <-> -u B =/= 0 ) )
2928biimpa 501 . . . 4 |- ( ( B e. CC /\ B =/= 0 ) -> -u B =/= 0 )
30293adant1 1079 . . 3 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> -u B =/= 0 )
31 divmul 10688 . . 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 ) )
3225, 26, 2, 30, 31syl112anc 1330 . 2 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( ( -u A / -u B ) = ( A / B ) <-> ( -u B x. ( A / B ) ) = -u A ) )
3323, 32mpbird 247 1 |- ( ( A e. CC /\ B e. CC /\ B =/= 0 ) -> ( -u A / -u B ) = ( A / B ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   x. cmul 9941   -ucneg 10267   / cdiv 10684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685
This theorem is referenced by:  divneg2  10749  div2negd  10816  div2sub  10850  iblcnlem1  23554  itgcnlem  23556
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