Theorem ltdiv23neg 39617

Description: Swap denominator with other side of 'less than', when both are negative. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
ltdiv23neg.1	|- ( ph -> A e. RR )
ltdiv23neg.2	|- ( ph -> B e. RR )
ltdiv23neg.3	|- ( ph -> B < 0 )
ltdiv23neg.4	|- ( ph -> C e. RR )
ltdiv23neg.5	|- ( ph -> C < 0 )

Assertion

Ref	Expression
ltdiv23neg	|- ( ph -> ( ( A / B ) < C <-> ( A / C ) < B ) )

Proof of Theorem ltdiv23neg

Step	Hyp	Ref	Expression
1		ltdiv23neg.1	. . . 4 |- ( ph -> A e. RR )
2		ltdiv23neg.2	. . . 4 |- ( ph -> B e. RR )
3		ltdiv23neg.3	. . . . 5 |- ( ph -> B < 0 )
4	2, 3	ltned 10173	. . . 4 |- ( ph -> B =/= 0 )
5	1, 2, 4	redivcld 10853	. . 3 |- ( ph -> ( A / B ) e. RR )
6		ltdiv23neg.4	. . 3 |- ( ph -> C e. RR )
7	5, 6, 2, 3	ltmulneg 39615	. 2 |- ( ph -> ( ( A / B ) < C <-> ( C x. B ) < ( ( A / B ) x. B ) ) )
8		recn 10026	. . . . 5 |- ( A e. RR -> A e. CC )
9	1, 8	syl 17	. . . 4 |- ( ph -> A e. CC )
10		recn 10026	. . . . 5 |- ( B e. RR -> B e. CC )
11	2, 10	syl 17	. . . 4 |- ( ph -> B e. CC )
12	9, 11, 4	divcan1d 10802	. . 3 |- ( ph -> ( ( A / B ) x. B ) = A )
13	12	breq2d 4665	. 2 |- ( ph -> ( ( C x. B ) < ( ( A / B ) x. B ) <-> ( C x. B ) < A ) )
14		remulcl 10021	. . . . 5 |- ( ( C e. RR /\ B e. RR ) -> ( C x. B ) e. RR )
15	6, 2, 14	syl2anc 693	. . . 4 |- ( ph -> ( C x. B ) e. RR )
16		ltdiv23neg.5	. . . . . 6 |- ( ph -> C < 0 )
17	6, 16	ltned 10173	. . . . 5 |- ( ph -> C =/= 0 )
18	6, 17	rereccld 10852	. . . 4 |- ( ph -> ( 1 / C ) e. RR )
19	6, 16	reclt0d 39607	. . . 4 |- ( ph -> ( 1 / C ) < 0 )
20	15, 1, 18, 19	ltmulneg 39615	. . 3 |- ( ph -> ( ( C x. B ) < A <-> ( A x. ( 1 / C ) ) < ( ( C x. B ) x. ( 1 / C ) ) ) )
21		recn 10026	. . . . . . 7 |- ( C e. RR -> C e. CC )
22	6, 21	syl 17	. . . . . 6 |- ( ph -> C e. CC )
23	9, 22, 17	divrecd 10804	. . . . 5 |- ( ph -> ( A / C ) = ( A x. ( 1 / C ) ) )
24	23	eqcomd 2628	. . . 4 |- ( ph -> ( A x. ( 1 / C ) ) = ( A / C ) )
25	22, 11	mulcld 10060	. . . . . 6 |- ( ph -> ( C x. B ) e. CC )
26	25, 22, 17	divrecd 10804	. . . . 5 |- ( ph -> ( ( C x. B ) / C ) = ( ( C x. B ) x. ( 1 / C ) ) )
27		divcan3 10711	. . . . . . 7 |- ( ( B e. CC /\ C e. CC /\ C =/= 0 ) -> ( ( C x. B ) / C ) = B )
28	27	3expb 1266	. . . . . 6 |- ( ( B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( C x. B ) / C ) = B )
29	11, 22, 17, 28	syl12anc 1324	. . . . 5 |- ( ph -> ( ( C x. B ) / C ) = B )
30	26, 29	eqtr3d 2658	. . . 4 |- ( ph -> ( ( C x. B ) x. ( 1 / C ) ) = B )
31	24, 30	breq12d 4666	. . 3 |- ( ph -> ( ( A x. ( 1 / C ) ) < ( ( C x. B ) x. ( 1 / C ) ) <-> ( A / C ) < B ) )
32	20, 31	bitrd 268	. 2 |- ( ph -> ( ( C x. B ) < A <-> ( A / C ) < B ) )
33	7, 13, 32	3bitrd 294	1 |- ( ph -> ( ( A / B ) < C <-> ( A / C ) < B ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   / cdiv 10684

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:  pimrecltneg  40933