Theorem mul2lt0rlt0 9716

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 7950	. . . . 5 |- ( ph -> ( A x. B ) e. RR )
4	3	adantr 274	. . . 4 |- ( ( ph /\ B < 0 ) -> ( A x. B ) e. RR )
5		0red 7921	. . . 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 9645	. . . 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 9711	. . 3 |- ( ( ph /\ B < 0 ) -> ( ( A x. B ) / -u B ) < ( 0 / -u B ) )
12	1	recnd 7948	. . . . . . 7 |- ( ph -> A e. CC )
13	12	adantr 274	. . . . . 6 |- ( ( ph /\ B < 0 ) -> A e. CC )
14	2	recnd 7948	. . . . . . 7 |- ( ph -> B e. CC )
15	14	adantr 274	. . . . . 6 |- ( ( ph /\ B < 0 ) -> B e. CC )
16	13, 15	mulcld 7940	. . . . 5 |- ( ( ph /\ B < 0 ) -> ( A x. B ) e. CC )
17	6, 7	lt0ap0d 8568	. . . . 5 |- ( ( ph /\ B < 0 ) -> B # 0 )
18	16, 15, 17	divneg2apd 8721	. . . 4 |- ( ( ph /\ B < 0 ) -> -u ( ( A x. B ) / B ) = ( ( A x. B ) / -u B ) )
19	13, 15, 17	divcanap4d 8713	. . . . 5 |- ( ( ph /\ B < 0 ) -> ( ( A x. B ) / B ) = A )
20	19	negeqd 8114	. . . 4 |- ( ( ph /\ B < 0 ) -> -u ( ( A x. B ) / B ) = -u A )
21	18, 20	eqtr3d 2205	. . 3 |- ( ( ph /\ B < 0 ) -> ( ( A x. B ) / -u B ) = -u A )
22	15	negcld 8217	. . . 4 |- ( ( ph /\ B < 0 ) -> -u B e. CC )
23	15, 17	negap0d 8550	. . . 4 |- ( ( ph /\ B < 0 ) -> -u B # 0 )
24	22, 23	div0apd 8704	. . 3 |- ( ( ph /\ B < 0 ) -> ( 0 / -u B ) = 0 )
25	11, 21, 24	3brtr3d 4020	. 2 |- ( ( ph /\ B < 0 ) -> -u A < 0 )
26	1	adantr 274	. . 3 |- ( ( ph /\ B < 0 ) -> A e. RR )
27	26	lt0neg2d 8435	. 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 2141   class class class wbr 3989  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   x. cmul 7779   < clt 7954   -ucneg 8091   / cdiv 8589

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-rp 9611

This theorem is referenced by:  mul2lt0llt0  9718