Theorem ltmul2 8815

Description: Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.)

Assertion

Ref	Expression
ltmul2	|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < B <-> ( C x. A ) < ( C x. B ) ) )

Proof of Theorem ltmul2

Step	Hyp	Ref	Expression
1		ltmul1 8551	. 2 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < B <-> ( A x. C ) < ( B x. C ) ) )
2		recn 7946	. . . 4 |- ( C e. RR -> C e. CC )
3		recn 7946	. . . . . . 7 |- ( A e. RR -> A e. CC )
4		mulcom 7942	. . . . . . 7 |- ( ( A e. CC /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
5	3, 4	sylan 283	. . . . . 6 |- ( ( A e. RR /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
6	5	3adant2 1016	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
7		recn 7946	. . . . . . 7 |- ( B e. RR -> B e. CC )
8		mulcom 7942	. . . . . . 7 |- ( ( B e. CC /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
9	7, 8	sylan 283	. . . . . 6 |- ( ( B e. RR /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
10	9	3adant1 1015	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
11	6, 10	breq12d 4018	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. CC ) -> ( ( A x. C ) < ( B x. C ) <-> ( C x. A ) < ( C x. B ) ) )
12	2, 11	syl3an3 1273	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A x. C ) < ( B x. C ) <-> ( C x. A ) < ( C x. B ) ) )
13	12	3adant3r 1235	. 2 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A x. C ) < ( B x. C ) <-> ( C x. A ) < ( C x. B ) ) )
14	1, 13	bitrd 188	1 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < B <-> ( C x. A ) < ( C x. B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   class class class wbr 4005  (class class class)co 5877   CCcc 7811   RRcr 7812   0cc0 7813   x. cmul 7818   < clt 7994

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltadd 7929  ax-pre-mulgt0 7930

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-sub 8132  df-neg 8133

This theorem is referenced by:  ltmul12a  8819  mulgt1  8822  ltmulgt11  8823  lt2msq1  8844  ltdiv2  8846  ltmul2i  8882  ltmul2d  9741  ef01bndlem  11766  cos01gt0  11772  sin4lt0  11776  pockthg  12357  tangtx  14344  lgsdilem  14513