Theorem ltmul2 8751

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 8490	. 2 |- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < B <-> ( A x. C ) < ( B x. C ) ) )
2		recn 7886	. . . 4 |- ( C e. RR -> C e. CC )
3		recn 7886	. . . . . . 7 |- ( A e. RR -> A e. CC )
4		mulcom 7882	. . . . . . 7 |- ( ( A e. CC /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
5	3, 4	sylan 281	. . . . . 6 |- ( ( A e. RR /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
6	5	3adant2 1006	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
7		recn 7886	. . . . . . 7 |- ( B e. RR -> B e. CC )
8		mulcom 7882	. . . . . . 7 |- ( ( B e. CC /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
9	7, 8	sylan 281	. . . . . 6 |- ( ( B e. RR /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
10	9	3adant1 1005	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
11	6, 10	breq12d 3995	. . . 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 1263	. . 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 1225	. 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 187	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 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   CCcc 7751   RRcr 7752   0cc0 7753   x. cmul 7758   < clt 7933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltadd 7869  ax-pre-mulgt0 7870

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-ltxr 7938  df-sub 8071  df-neg 8072

This theorem is referenced by:  ltmul12a  8755  mulgt1  8758  ltmulgt11  8759  lt2msq1  8780  ltdiv2  8782  ltmul2i  8818  ltmul2d  9675  ef01bndlem  11697  cos01gt0  11703  sin4lt0  11707  pockthg  12287  tangtx  13399  lgsdilem  13568