Theorem lt2mul2div 8825

Description: 'Less than' relationship between division and multiplication. (Contributed by NM, 8-Jan-2006.)

Assertion

Ref	Expression
lt2mul2div	|- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> ( ( A x. B ) < ( C x. D ) <-> ( A / D ) < ( C / B ) ) )

Proof of Theorem lt2mul2div

Step	Hyp	Ref	Expression
1		simprl 529	. . . . . . 7 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> C e. RR )
2	1	recnd 7976	. . . . . 6 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> C e. CC )
3		simprrl 539	. . . . . . 7 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> D e. RR )
4	3	recnd 7976	. . . . . 6 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> D e. CC )
5	2, 4	mulcomd 7969	. . . . 5 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> ( C x. D ) = ( D x. C ) )
6	5	oveq1d 5884	. . . 4 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> ( ( C x. D ) / B ) = ( ( D x. C ) / B ) )
7		simplrl 535	. . . . . 6 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> B e. RR )
8	7	recnd 7976	. . . . 5 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> B e. CC )
9		simplrr 536	. . . . . 6 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> 0 < B )
10	7, 9	gt0ap0d 8576	. . . . 5 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> B # 0 )
11	4, 2, 8, 10	divassapd 8772	. . . 4 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> ( ( D x. C ) / B ) = ( D x. ( C / B ) ) )
12	6, 11	eqtrd 2210	. . 3 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> ( ( C x. D ) / B ) = ( D x. ( C / B ) ) )
13	12	breq2d 4012	. 2 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> ( A < ( ( C x. D ) / B ) <-> A < ( D x. ( C / B ) ) ) )
14		simpll 527	. . 3 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> A e. RR )
15	1, 3	remulcld 7978	. . 3 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> ( C x. D ) e. RR )
16		simplr 528	. . 3 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> ( B e. RR /\ 0 < B ) )
17		ltmuldiv 8820	. . 3 |- ( ( A e. RR /\ ( C x. D ) e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( A x. B ) < ( C x. D ) <-> A < ( ( C x. D ) / B ) ) )
18	14, 15, 16, 17	syl3anc 1238	. 2 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> ( ( A x. B ) < ( C x. D ) <-> A < ( ( C x. D ) / B ) ) )
19	1, 7, 10	redivclapd 8781	. . 3 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> ( C / B ) e. RR )
20		simprr 531	. . 3 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> ( D e. RR /\ 0 < D ) )
21		ltdivmul 8822	. . 3 |- ( ( A e. RR /\ ( C / B ) e. RR /\ ( D e. RR /\ 0 < D ) ) -> ( ( A / D ) < ( C / B ) <-> A < ( D x. ( C / B ) ) ) )
22	14, 19, 20, 21	syl3anc 1238	. 2 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> ( ( A / D ) < ( C / B ) <-> A < ( D x. ( C / B ) ) ) )
23	13, 18, 22	3bitr4d 220	1 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> ( ( A x. B ) < ( C x. D ) <-> ( A / D ) < ( C / B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2148   class class class wbr 4000  (class class class)co 5869   RRcr 7801   0cc0 7802   x. cmul 7807   < clt 7982   / cdiv 8618

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 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920

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-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-po 4293  df-iso 4294  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619

This theorem is referenced by:  lt2mul2divd  9752