Theorem mul2lt0rgt0 9731

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
mul2lt0rgt0	|- ( ( ph /\ 0 < B ) -> A < 0 )

Proof of Theorem mul2lt0rgt0

Step	Hyp	Ref	Expression
1		mul2lt0.3	. . . 4 |- ( ph -> ( A x. B ) < 0 )
2	1	adantr 276	. . 3 |- ( ( ph /\ 0 < B ) -> ( A x. B ) < 0 )
3		mul2lt0.2	. . . . . 6 |- ( ph -> B e. RR )
4	3	adantr 276	. . . . 5 |- ( ( ph /\ 0 < B ) -> B e. RR )
5	4	recnd 7960	. . . 4 |- ( ( ph /\ 0 < B ) -> B e. CC )
6	5	mul02d 8323	. . 3 |- ( ( ph /\ 0 < B ) -> ( 0 x. B ) = 0 )
7	2, 6	breqtrrd 4026	. 2 |- ( ( ph /\ 0 < B ) -> ( A x. B ) < ( 0 x. B ) )
8		mul2lt0.1	. . . 4 |- ( ph -> A e. RR )
9	8	adantr 276	. . 3 |- ( ( ph /\ 0 < B ) -> A e. RR )
10		0red 7933	. . 3 |- ( ( ph /\ 0 < B ) -> 0 e. RR )
11		simpr 110	. . . 4 |- ( ( ph /\ 0 < B ) -> 0 < B )
12	4, 11	elrpd 9664	. . 3 |- ( ( ph /\ 0 < B ) -> B e. RR+ )
13	9, 10, 12	ltmul1d 9709	. 2 |- ( ( ph /\ 0 < B ) -> ( A < 0 <-> ( A x. B ) < ( 0 x. B ) ) )
14	7, 13	mpbird 167	1 |- ( ( ph /\ 0 < B ) -> A < 0 )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2146   class class class wbr 3998  (class class class)co 5865   RRcr 7785   0cc0 7786   x. cmul 7791   < clt 7966

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltadd 7902  ax-pre-mulgt0 7903

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-ltxr 7971  df-sub 8104  df-neg 8105  df-rp 9625

This theorem is referenced by:  mul2lt0lgt0  9733