Theorem mul2lt0rlt0 9777

Description: If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.)

Hypotheses

Ref	Expression
mul2lt0.1	|- ( ph -> A e. RR )
mul2lt0.2	|- ( ph -> B e. RR )
mul2lt0.3	|- ( ph -> ( A x. B ) < 0 )

Assertion

Ref	Expression
mul2lt0rlt0	|- ( ( ph /\ B < 0 ) -> 0 < A )

Proof of Theorem mul2lt0rlt0

Step	Hyp	Ref	Expression
1		mul2lt0.1	. . . . . 6 |- ( ph -> A e. RR )
2		mul2lt0.2	. . . . . 6 |- ( ph -> B e. RR )
3	1, 2	remulcld 8006	. . . . 5 |- ( ph -> ( A x. B ) e. RR )
4	3	adantr 276	. . . 4 |- ( ( ph /\ B < 0 ) -> ( A x. B ) e. RR )
5		0red 7976	. . . 4 |- ( ( ph /\ B < 0 ) -> 0 e. RR )
6	2	adantr 276	. . . . 5 |- ( ( ph /\ B < 0 ) -> B e. RR )
7		simpr 110	. . . . 5 |- ( ( ph /\ B < 0 ) -> B < 0 )
8	6, 7	negelrpd 9706	. . . 4 |- ( ( ph /\ B < 0 ) -> -u B e. RR+ )
9		mul2lt0.3	. . . . 5 |- ( ph -> ( A x. B ) < 0 )
10	9	adantr 276	. . . 4 |- ( ( ph /\ B < 0 ) -> ( A x. B ) < 0 )
11	4, 5, 8, 10	ltdiv1dd 9772	. . 3 |- ( ( ph /\ B < 0 ) -> ( ( A x. B ) / -u B ) < ( 0 / -u B ) )
12	1	recnd 8004	. . . . . . 7 |- ( ph -> A e. CC )
13	12	adantr 276	. . . . . 6 |- ( ( ph /\ B < 0 ) -> A e. CC )
14	2	recnd 8004	. . . . . . 7 |- ( ph -> B e. CC )
15	14	adantr 276	. . . . . 6 |- ( ( ph /\ B < 0 ) -> B e. CC )
16	13, 15	mulcld 7996	. . . . 5 |- ( ( ph /\ B < 0 ) -> ( A x. B ) e. CC )
17	6, 7	lt0ap0d 8624	. . . . 5 |- ( ( ph /\ B < 0 ) -> B # 0 )
18	16, 15, 17	divneg2apd 8779	. . . 4 |- ( ( ph /\ B < 0 ) -> -u ( ( A x. B ) / B ) = ( ( A x. B ) / -u B ) )
19	13, 15, 17	divcanap4d 8771	. . . . 5 |- ( ( ph /\ B < 0 ) -> ( ( A x. B ) / B ) = A )
20	19	negeqd 8170	. . . 4 |- ( ( ph /\ B < 0 ) -> -u ( ( A x. B ) / B ) = -u A )
21	18, 20	eqtr3d 2224	. . 3 |- ( ( ph /\ B < 0 ) -> ( ( A x. B ) / -u B ) = -u A )
22	15	negcld 8273	. . . 4 |- ( ( ph /\ B < 0 ) -> -u B e. CC )
23	15, 17	negap0d 8606	. . . 4 |- ( ( ph /\ B < 0 ) -> -u B # 0 )
24	22, 23	div0apd 8762	. . 3 |- ( ( ph /\ B < 0 ) -> ( 0 / -u B ) = 0 )
25	11, 21, 24	3brtr3d 4049	. 2 |- ( ( ph /\ B < 0 ) -> -u A < 0 )
26	1	adantr 276	. . 3 |- ( ( ph /\ B < 0 ) -> A e. RR )
27	26	lt0neg2d 8491	. 2 |- ( ( ph /\ B < 0 ) -> ( 0 < A <-> -u A < 0 ) )
28	25, 27	mpbird 167	1 |- ( ( ph /\ B < 0 ) -> 0 < A )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2160   class class class wbr 4018  (class class class)co 5891   CCcc 7827   RRcr 7828   0cc0 7829   x. cmul 7834   < clt 8010   -ucneg 8147   / cdiv 8647

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-mulrcl 7928  ax-addcom 7929  ax-mulcom 7930  ax-addass 7931  ax-mulass 7932  ax-distr 7933  ax-i2m1 7934  ax-0lt1 7935  ax-1rid 7936  ax-0id 7937  ax-rnegex 7938  ax-precex 7939  ax-cnre 7940  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943  ax-pre-apti 7944  ax-pre-ltadd 7945  ax-pre-mulgt0 7946  ax-pre-mulext 7947

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4308  df-po 4311  df-iso 4312  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-iota 5193  df-fun 5233  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-sub 8148  df-neg 8149  df-reap 8550  df-ap 8557  df-div 8648  df-rp 9672

This theorem is referenced by:  mul2lt0llt0  9779