Theorem div2negap 9065

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 8526	. . . . 5 |- ( B e. CC -> -u B e. CC )
2	1	3ad2ant2 1050	. . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u B e. CC )
3		simp1 1028	. . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> A e. CC )
4		simp2 1029	. . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> B e. CC )
5		simp3 1030	. . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> B # 0 )
6		div12ap 9024	. . . 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 1282	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( -u B x. ( A / B ) ) = ( A x. ( -u B / B ) ) )
8		divnegap 9036	. . . . . . 7 |- ( ( B e. CC /\ B e. CC /\ B # 0 ) -> -u ( B / B ) = ( -u B / B ) )
9	4, 8	syld3an1 1324	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u ( B / B ) = ( -u B / B ) )
10		dividap 9031	. . . . . . . 8 |- ( ( B e. CC /\ B # 0 ) -> ( B / B ) = 1 )
11	10	3adant1 1046	. . . . . . 7 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( B / B ) = 1 )
12	11	negeqd 8521	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u ( B / B ) = -u 1 )
13	9, 12	eqtr3d 2273	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( -u B / B ) = -u 1 )
14	13	oveq2d 6101	. . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A x. ( -u B / B ) ) = ( A x. -u 1 ) )
15		ax-1cn 8272	. . . . . . . 8 |- 1 e. CC
16	15	negcli 8594	. . . . . . 7 |- -u 1 e. CC
17		mulcom 8308	. . . . . . 7 |- ( ( A e. CC /\ -u 1 e. CC ) -> ( A x. -u 1 ) = ( -u 1 x. A ) )
18	16, 17	mpan2 429	. . . . . 6 |- ( A e. CC -> ( A x. -u 1 ) = ( -u 1 x. A ) )
19		mulm1 8727	. . . . . 6 |- ( A e. CC -> ( -u 1 x. A ) = -u A )
20	18, 19	eqtrd 2271	. . . . 5 |- ( A e. CC -> ( A x. -u 1 ) = -u A )
21	20	3ad2ant1 1049	. . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A x. -u 1 ) = -u A )
22	14, 21	eqtrd 2271	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A x. ( -u B / B ) ) = -u A )
23	7, 22	eqtrd 2271	. 2 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( -u B x. ( A / B ) ) = -u A )
24		negcl 8526	. . . 4 |- ( A e. CC -> -u A e. CC )
25	24	3ad2ant1 1049	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u A e. CC )
26		divclap 9008	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) e. CC )
27		negap0 8958	. . . . 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 1046	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> -u B # 0 )
30		divmulap 9005	. . 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 1282	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   CCcc 8177   0cc0 8179   1c1 8180   x. cmul 8184   -ucneg 8498   # cap 8909   / cdiv 9002

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003

This theorem is used by:  divneg2ap  9066  div2negapd  9135  div2subap  9167