Theorem lt2mul2div 8788

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 526	. . . . . . 7 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> C e. RR )
2	1	recnd 7941	. . . . . 6 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> C e. CC )
3		simprrl 534	. . . . . . 7 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> D e. RR )
4	3	recnd 7941	. . . . . 6 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> D e. CC )
5	2, 4	mulcomd 7934	. . . . 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 5866	. . . 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 530	. . . . . 6 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> B e. RR )
8	7	recnd 7941	. . . . 5 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> B e. CC )
9		simplrr 531	. . . . . 6 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> 0 < B )
10	7, 9	gt0ap0d 8541	. . . . 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 8736	. . . 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 2203	. . 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 3999	. 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 524	. . 3 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> A e. RR )
15	1, 3	remulcld 7943	. . 3 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> ( C x. D ) e. RR )
16		simplr 525	. . 3 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> ( B e. RR /\ 0 < B ) )
17		ltmuldiv 8783	. . 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 1233	. 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 8745	. . 3 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> ( C / B ) e. RR )
20		simprr 527	. . 3 |- ( ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) /\ ( C e. RR /\ ( D e. RR /\ 0 < D ) ) ) -> ( D e. RR /\ 0 < D ) )
21		ltdivmul 8785	. . 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 1233	. 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 219	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 103   <-> wb 104   e. wcel 2141   class class class wbr 3987  (class class class)co 5851   RRcr 7766   0cc0 7767   x. cmul 7772   < clt 7947   / cdiv 8582

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7858  ax-resscn 7859  ax-1cn 7860  ax-1re 7861  ax-icn 7862  ax-addcl 7863  ax-addrcl 7864  ax-mulcl 7865  ax-mulrcl 7866  ax-addcom 7867  ax-mulcom 7868  ax-addass 7869  ax-mulass 7870  ax-distr 7871  ax-i2m1 7872  ax-0lt1 7873  ax-1rid 7874  ax-0id 7875  ax-rnegex 7876  ax-precex 7877  ax-cnre 7878  ax-pre-ltirr 7879  ax-pre-ltwlin 7880  ax-pre-lttrn 7881  ax-pre-apti 7882  ax-pre-ltadd 7883  ax-pre-mulgt0 7884  ax-pre-mulext 7885

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-po 4279  df-iso 4280  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-pnf 7949  df-mnf 7950  df-xr 7951  df-ltxr 7952  df-le 7953  df-sub 8085  df-neg 8086  df-reap 8487  df-ap 8494  df-div 8583

This theorem is referenced by:  lt2mul2divd  9715