Theorem mul2lt0rlt0 11932

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 10070	. . . . 5 |- ( ph -> ( A x. B ) e. RR )
4	3	adantr 481	. . . 4 |- ( ( ph /\ B < 0 ) -> ( A x. B ) e. RR )
5		0red 10041	. . . 4 |- ( ( ph /\ B < 0 ) -> 0 e. RR )
6		negelrp 11864	. . . . . 6 |- ( B e. RR -> ( -u B e. RR+ <-> B < 0 ) )
7	2, 6	syl 17	. . . . 5 |- ( ph -> ( -u B e. RR+ <-> B < 0 ) )
8	7	biimpar 502	. . . 4 |- ( ( ph /\ B < 0 ) -> -u B e. RR+ )
9		mul2lt0.3	. . . . 5 |- ( ph -> ( A x. B ) < 0 )
10	9	adantr 481	. . . 4 |- ( ( ph /\ B < 0 ) -> ( A x. B ) < 0 )
11	4, 5, 8, 10	ltdiv1dd 11929	. . 3 |- ( ( ph /\ B < 0 ) -> ( ( A x. B ) / -u B ) < ( 0 / -u B ) )
12	1	recnd 10068	. . . . . . 7 |- ( ph -> A e. CC )
13	12	adantr 481	. . . . . 6 |- ( ( ph /\ B < 0 ) -> A e. CC )
14	2	recnd 10068	. . . . . . 7 |- ( ph -> B e. CC )
15	14	adantr 481	. . . . . 6 |- ( ( ph /\ B < 0 ) -> B e. CC )
16	13, 15	mulcld 10060	. . . . 5 |- ( ( ph /\ B < 0 ) -> ( A x. B ) e. CC )
17		simpr 477	. . . . . 6 |- ( ( ph /\ B < 0 ) -> B < 0 )
18	17	lt0ne0d 10593	. . . . 5 |- ( ( ph /\ B < 0 ) -> B =/= 0 )
19	16, 15, 18	divneg2d 10815	. . . 4 |- ( ( ph /\ B < 0 ) -> -u ( ( A x. B ) / B ) = ( ( A x. B ) / -u B ) )
20	13, 15, 18	divcan4d 10807	. . . . 5 |- ( ( ph /\ B < 0 ) -> ( ( A x. B ) / B ) = A )
21	20	negeqd 10275	. . . 4 |- ( ( ph /\ B < 0 ) -> -u ( ( A x. B ) / B ) = -u A )
22	19, 21	eqtr3d 2658	. . 3 |- ( ( ph /\ B < 0 ) -> ( ( A x. B ) / -u B ) = -u A )
23	15	negcld 10379	. . . 4 |- ( ( ph /\ B < 0 ) -> -u B e. CC )
24	15, 18	negne0d 10390	. . . 4 |- ( ( ph /\ B < 0 ) -> -u B =/= 0 )
25	23, 24	div0d 10800	. . 3 |- ( ( ph /\ B < 0 ) -> ( 0 / -u B ) = 0 )
26	11, 22, 25	3brtr3d 4684	. 2 |- ( ( ph /\ B < 0 ) -> -u A < 0 )
27	1	adantr 481	. . 3 |- ( ( ph /\ B < 0 ) -> A e. RR )
28	27	lt0neg2d 10598	. 2 |- ( ( ph /\ B < 0 ) -> ( 0 < A <-> -u A < 0 ) )
29	26, 28	mpbird 247	1 |- ( ( ph /\ B < 0 ) -> 0 < A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   x. cmul 9941   < clt 10074   -ucneg 10267   / cdiv 10684   RR+crp 11832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-rp 11833

This theorem is referenced by:  mul2lt0llt0  11934  mul2lt0bi  11936  sgnmul  30604  signsply0  30628