Theorem mul2lt0rlt0 9695

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 7929	. . . . 5 |- ( ph -> ( A x. B ) e. RR )
4	3	adantr 274	. . . 4 |- ( ( ph /\ B < 0 ) -> ( A x. B ) e. RR )
5		0red 7900	. . . 4 |- ( ( ph /\ B < 0 ) -> 0 e. RR )
6	2	adantr 274	. . . . 5 |- ( ( ph /\ B < 0 ) -> B e. RR )
7		simpr 109	. . . . 5 |- ( ( ph /\ B < 0 ) -> B < 0 )
8	6, 7	negelrpd 9624	. . . 4 |- ( ( ph /\ B < 0 ) -> -u B e. RR+ )
9		mul2lt0.3	. . . . 5 |- ( ph -> ( A x. B ) < 0 )
10	9	adantr 274	. . . 4 |- ( ( ph /\ B < 0 ) -> ( A x. B ) < 0 )
11	4, 5, 8, 10	ltdiv1dd 9690	. . 3 |- ( ( ph /\ B < 0 ) -> ( ( A x. B ) / -u B ) < ( 0 / -u B ) )
12	1	recnd 7927	. . . . . . 7 |- ( ph -> A e. CC )
13	12	adantr 274	. . . . . 6 |- ( ( ph /\ B < 0 ) -> A e. CC )
14	2	recnd 7927	. . . . . . 7 |- ( ph -> B e. CC )
15	14	adantr 274	. . . . . 6 |- ( ( ph /\ B < 0 ) -> B e. CC )
16	13, 15	mulcld 7919	. . . . 5 |- ( ( ph /\ B < 0 ) -> ( A x. B ) e. CC )
17	6, 7	lt0ap0d 8547	. . . . 5 |- ( ( ph /\ B < 0 ) -> B # 0 )
18	16, 15, 17	divneg2apd 8700	. . . 4 |- ( ( ph /\ B < 0 ) -> -u ( ( A x. B ) / B ) = ( ( A x. B ) / -u B ) )
19	13, 15, 17	divcanap4d 8692	. . . . 5 |- ( ( ph /\ B < 0 ) -> ( ( A x. B ) / B ) = A )
20	19	negeqd 8093	. . . 4 |- ( ( ph /\ B < 0 ) -> -u ( ( A x. B ) / B ) = -u A )
21	18, 20	eqtr3d 2200	. . 3 |- ( ( ph /\ B < 0 ) -> ( ( A x. B ) / -u B ) = -u A )
22	15	negcld 8196	. . . 4 |- ( ( ph /\ B < 0 ) -> -u B e. CC )
23	15, 17	negap0d 8529	. . . 4 |- ( ( ph /\ B < 0 ) -> -u B # 0 )
24	22, 23	div0apd 8683	. . 3 |- ( ( ph /\ B < 0 ) -> ( 0 / -u B ) = 0 )
25	11, 21, 24	3brtr3d 4013	. 2 |- ( ( ph /\ B < 0 ) -> -u A < 0 )
26	1	adantr 274	. . 3 |- ( ( ph /\ B < 0 ) -> A e. RR )
27	26	lt0neg2d 8414	. 2 |- ( ( ph /\ B < 0 ) -> ( 0 < A <-> -u A < 0 ) )
28	25, 27	mpbird 166	1 |- ( ( ph /\ B < 0 ) -> 0 < A )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   CCcc 7751   RRcr 7752   0cc0 7753   x. cmul 7758   < clt 7933   -ucneg 8070   / cdiv 8568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-rp 9590

This theorem is referenced by:  mul2lt0llt0  9697