Theorem mul2lt0rlt0 9901

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 8123	. . . . 5 |- ( ph -> ( A x. B ) e. RR )
4	3	adantr 276	. . . 4 |- ( ( ph /\ B < 0 ) -> ( A x. B ) e. RR )
5		0red 8093	. . . 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 9830	. . . 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 9896	. . 3 |- ( ( ph /\ B < 0 ) -> ( ( A x. B ) / -u B ) < ( 0 / -u B ) )
12	1	recnd 8121	. . . . . . 7 |- ( ph -> A e. CC )
13	12	adantr 276	. . . . . 6 |- ( ( ph /\ B < 0 ) -> A e. CC )
14	2	recnd 8121	. . . . . . 7 |- ( ph -> B e. CC )
15	14	adantr 276	. . . . . 6 |- ( ( ph /\ B < 0 ) -> B e. CC )
16	13, 15	mulcld 8113	. . . . 5 |- ( ( ph /\ B < 0 ) -> ( A x. B ) e. CC )
17	6, 7	lt0ap0d 8742	. . . . 5 |- ( ( ph /\ B < 0 ) -> B # 0 )
18	16, 15, 17	divneg2apd 8897	. . . 4 |- ( ( ph /\ B < 0 ) -> -u ( ( A x. B ) / B ) = ( ( A x. B ) / -u B ) )
19	13, 15, 17	divcanap4d 8889	. . . . 5 |- ( ( ph /\ B < 0 ) -> ( ( A x. B ) / B ) = A )
20	19	negeqd 8287	. . . 4 |- ( ( ph /\ B < 0 ) -> -u ( ( A x. B ) / B ) = -u A )
21	18, 20	eqtr3d 2241	. . 3 |- ( ( ph /\ B < 0 ) -> ( ( A x. B ) / -u B ) = -u A )
22	15	negcld 8390	. . . 4 |- ( ( ph /\ B < 0 ) -> -u B e. CC )
23	15, 17	negap0d 8724	. . . 4 |- ( ( ph /\ B < 0 ) -> -u B # 0 )
24	22, 23	div0apd 8880	. . 3 |- ( ( ph /\ B < 0 ) -> ( 0 / -u B ) = 0 )
25	11, 21, 24	3brtr3d 4082	. 2 |- ( ( ph /\ B < 0 ) -> -u A < 0 )
26	1	adantr 276	. . 3 |- ( ( ph /\ B < 0 ) -> A e. RR )
27	26	lt0neg2d 8609	. 2 |- ( ( ph /\ B < 0 ) -> ( 0 < A <-> -u A < 0 ) )
28	25, 27	mpbird 167	1 |- ( ( ph /\ B < 0 ) -> 0 < A )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2177   class class class wbr 4051  (class class class)co 5957   CCcc 7943   RRcr 7944   0cc0 7945   x. cmul 7950   < clt 8127   -ucneg 8264   / cdiv 8765

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-mulrcl 8044  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-precex 8055  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061  ax-pre-mulgt0 8062  ax-pre-mulext 8063

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-id 4348  df-po 4351  df-iso 4352  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-iota 5241  df-fun 5282  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-reap 8668  df-ap 8675  df-div 8766  df-rp 9796

This theorem is referenced by:  mul2lt0llt0  9903