Theorem div2negap 8694

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

Step	Hyp	Ref	Expression
1		negcl 8159	. . . . 5 |- ( B e. CC -> -u B e. CC )
2	1	3ad2ant2 1019	. . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u B e. CC )
3		simp1 997	. . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> A e. CC )
4		simp2 998	. . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> B e. CC )
5		simp3 999	. . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> B # 0 )
6		div12ap 8653	. . . 4 |- ( ( -u B e. CC /\ A e. CC /\ ( B e. CC /\ B # 0 ) ) -> ( -u B x. ( A / B ) ) = ( A x. ( -u B / B ) ) )
7	2, 3, 4, 5, 6	syl112anc 1242	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( -u B x. ( A / B ) ) = ( A x. ( -u B / B ) ) )
8		divnegap 8665	. . . . . . 7 |- ( ( B e. CC /\ B e. CC /\ B # 0 ) -> -u ( B / B ) = ( -u B / B ) )
9	4, 8	syld3an1 1284	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u ( B / B ) = ( -u B / B ) )
10		dividap 8660	. . . . . . . 8 |- ( ( B e. CC /\ B # 0 ) -> ( B / B ) = 1 )
11	10	3adant1 1015	. . . . . . 7 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( B / B ) = 1 )
12	11	negeqd 8154	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u ( B / B ) = -u 1 )
13	9, 12	eqtr3d 2212	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( -u B / B ) = -u 1 )
14	13	oveq2d 5893	. . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A x. ( -u B / B ) ) = ( A x. -u 1 ) )
15		ax-1cn 7906	. . . . . . . 8 |- 1 e. CC
16	15	negcli 8227	. . . . . . 7 |- -u 1 e. CC
17		mulcom 7942	. . . . . . 7 |- ( ( A e. CC /\ -u 1 e. CC ) -> ( A x. -u 1 ) = ( -u 1 x. A ) )
18	16, 17	mpan2 425	. . . . . 6 |- ( A e. CC -> ( A x. -u 1 ) = ( -u 1 x. A ) )
19		mulm1 8359	. . . . . 6 |- ( A e. CC -> ( -u 1 x. A ) = -u A )
20	18, 19	eqtrd 2210	. . . . 5 |- ( A e. CC -> ( A x. -u 1 ) = -u A )
21	20	3ad2ant1 1018	. . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A x. -u 1 ) = -u A )
22	14, 21	eqtrd 2210	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A x. ( -u B / B ) ) = -u A )
23	7, 22	eqtrd 2210	. 2 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( -u B x. ( A / B ) ) = -u A )
24		negcl 8159	. . . 4 |- ( A e. CC -> -u A e. CC )
25	24	3ad2ant1 1018	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u A e. CC )
26		divclap 8637	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) e. CC )
27		negap0 8589	. . . . 5 |- ( B e. CC -> ( B # 0 <-> -u B # 0 ) )
28	27	biimpa 296	. . . 4 |- ( ( B e. CC /\ B # 0 ) -> -u B # 0 )
29	28	3adant1 1015	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u B # 0 )
30		divmulap 8634	. . 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 ) )
31	25, 26, 2, 29, 30	syl112anc 1242	. 2 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( ( -u A / -u B ) = ( A / B ) <-> ( -u B x. ( A / B ) ) = -u A ) )
32	23, 31	mpbird 167	1 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( -u A / -u B ) = ( A / B ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   class class class wbr 4005  (class class class)co 5877   CCcc 7811   0cc0 7813   1c1 7814   x. cmul 7818   -ucneg 8131   # cap 8540   / cdiv 8631

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-po 4298  df-iso 4299  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632

This theorem is referenced by:  divneg2ap  8695  div2negapd  8764  div2subap  8796