Theorem mul2lt0rlt0 10177

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 8362	. . . . 5 |- ( ph -> ( A x. B ) e. RR )
4	3	adantr 276	. . . 4 |- ( ( ph /\ B < 0 ) -> ( A x. B ) e. RR )
5		0red 8333	. . . 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 10106	. . . 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 10172	. . 3 |- ( ( ph /\ B < 0 ) -> ( ( A x. B ) / -u B ) < ( 0 / -u B ) )
12	1	recnd 8360	. . . . . . 7 |- ( ph -> A e. CC )
13	12	adantr 276	. . . . . 6 |- ( ( ph /\ B < 0 ) -> A e. CC )
14	2	recnd 8360	. . . . . . 7 |- ( ph -> B e. CC )
15	14	adantr 276	. . . . . 6 |- ( ( ph /\ B < 0 ) -> B e. CC )
16	13, 15	mulcld 8352	. . . . 5 |- ( ( ph /\ B < 0 ) -> ( A x. B ) e. CC )
17	6, 7	lt0ap0d 8985	. . . . 5 |- ( ( ph /\ B < 0 ) -> B # 0 )
18	16, 15, 17	divneg2apd 9142	. . . 4 |- ( ( ph /\ B < 0 ) -> -u ( ( A x. B ) / B ) = ( ( A x. B ) / -u B ) )
19	13, 15, 17	divcanap4d 9134	. . . . 5 |- ( ( ph /\ B < 0 ) -> ( ( A x. B ) / B ) = A )
20	19	negeqd 8528	. . . 4 |- ( ( ph /\ B < 0 ) -> -u ( ( A x. B ) / B ) = -u A )
21	18, 20	eqtr3d 2273	. . 3 |- ( ( ph /\ B < 0 ) -> ( ( A x. B ) / -u B ) = -u A )
22	15	negcld 8631	. . . 4 |- ( ( ph /\ B < 0 ) -> -u B e. CC )
23	15, 17	negap0d 8967	. . . 4 |- ( ( ph /\ B < 0 ) -> -u B # 0 )
24	22, 23	div0apd 9125	. . 3 |- ( ( ph /\ B < 0 ) -> ( 0 / -u B ) = 0 )
25	11, 21, 24	3brtr3d 4162	. 2 |- ( ( ph /\ B < 0 ) -> -u A < 0 )
26	1	adantr 276	. . 3 |- ( ( ph /\ B < 0 ) -> A e. RR )
27	26	lt0neg2d 8851	. 2 |- ( ( ph /\ B < 0 ) -> ( 0 < A <-> -u A < 0 ) )
28	25, 27	mpbird 167	1 |- ( ( ph /\ B < 0 ) -> 0 < A )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   e. wcel 2209   class class class wbr 4131  (class class class)co 6088   CCcc 8183   RRcr 8184   0cc0 8185   x. cmul 8190   < clt 8366   -ucneg 8505   / cdiv 9010

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-id 4440  df-po 4443  df-iso 4444  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-iota 5340  df-fun 5382  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-rp 10072

This theorem is used by:  mul2lt0llt0  10179