Theorem lt2mul2div 7054

Description: 'Less than' relationship between division and multiplication.

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		mulcom 6459	. . . . . . . . 9 |- ((C e. CC /\ D e. CC) -> (C x. D) = (D x. C))
2		recn 6466	. . . . . . . . 9 |- (C e. RR -> C e. CC)
3		recn 6466	. . . . . . . . 9 |- (D e. RR -> D e. CC)
4	1, 2, 3	syl2an 503	. . . . . . . 8 |- ((C e. RR /\ D e. RR) -> (C x. D) = (D x. C))
5	4	opreq1d 4897	. . . . . . 7 |- ((C e. RR /\ D e. RR) -> ((C x. D) / B) = ((D x. C) / B))
6	5	adantl 424	. . . . . 6 |- (((B e. RR /\ 0 < B) /\ (C e. RR /\ D e. RR)) -> ((C x. D) / B) = ((D x. C) / B))
7	3	ad2antll 443	. . . . . . 7 |- (((B e. RR /\ 0 < B) /\ (C e. RR /\ D e. RR)) -> D e. CC)
8	2	ad2antrl 442	. . . . . . 7 |- (((B e. RR /\ 0 < B) /\ (C e. RR /\ D e. RR)) -> C e. CC)
9		recn 6466	. . . . . . . . . 10 |- (B e. RR -> B e. CC)
10	9	adantr 425	. . . . . . . . 9 |- ((B e. RR /\ 0 < B) -> B e. CC)
11		gt0ne0 6800	. . . . . . . . 9 |- ((B e. RR /\ 0 < B) -> B =/= 0)
12	10, 11	jca 310	. . . . . . . 8 |- ((B e. RR /\ 0 < B) -> (B e. CC /\ B =/= 0))
13	12	adantr 425	. . . . . . 7 |- (((B e. RR /\ 0 < B) /\ (C e. RR /\ D e. RR)) -> (B e. CC /\ B =/= 0))
14		divass 6924	. . . . . . 7 |- ((D e. CC /\ C e. CC /\ (B e. CC /\ B =/= 0)) -> ((D x. C) / B) = (D x. (C / B)))
15	7, 8, 13, 14	syl111anc 1100	. . . . . 6 |- (((B e. RR /\ 0 < B) /\ (C e. RR /\ D e. RR)) -> ((D x. C) / B) = (D x. (C / B)))
16	6, 15	eqtrd 1925	. . . . 5 |- (((B e. RR /\ 0 < B) /\ (C e. RR /\ D e. RR)) -> ((C x. D) / B) = (D x. (C / B)))
17	16	adantrrr 439	. . . 4 |- (((B e. RR /\ 0 < B) /\ (C e. RR /\ (D e. RR /\ 0 < D))) -> ((C x. D) / B) = (D x. (C / B)))
18	17	adantll 428	. . 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)))
19	18	breq2d 3350	. 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))))
20		simpll 448	. . 3 |- (((A e. RR /\ (B e. RR /\ 0 < B)) /\ (C e. RR /\ (D e. RR /\ 0 < D))) -> A e. RR)
21		axmulrcl 6427	. . . . 5 |- ((C e. RR /\ D e. RR) -> (C x. D) e. RR)
22	21	adantrr 431	. . . 4 |- ((C e. RR /\ (D e. RR /\ 0 < D)) -> (C x. D) e. RR)
23	22	adantl 424	. . 3 |- (((A e. RR /\ (B e. RR /\ 0 < B)) /\ (C e. RR /\ (D e. RR /\ 0 < D))) -> (C x. D) e. RR)
24		simplr 449	. . 3 |- (((A e. RR /\ (B e. RR /\ 0 < B)) /\ (C e. RR /\ (D e. RR /\ 0 < D))) -> (B e. RR /\ 0 < B))
25		ltmuldiv 7045	. . 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)))
26	20, 23, 24, 25	syl111anc 1100	. 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)))
27		redivcl 6978	. . . . . . 7 |- ((C e. RR /\ B e. RR /\ B =/= 0) -> (C / B) e. RR)
28	27	3expb 1068	. . . . . 6 |- ((C e. RR /\ (B e. RR /\ B =/= 0)) -> (C / B) e. RR)
29		simpl 346	. . . . . . 7 |- ((B e. RR /\ 0 < B) -> B e. RR)
30	29, 11	jca 310	. . . . . 6 |- ((B e. RR /\ 0 < B) -> (B e. RR /\ B =/= 0))
31	28, 30	sylan2 500	. . . . 5 |- ((C e. RR /\ (B e. RR /\ 0 < B)) -> (C / B) e. RR)
32	31	ancoms 484	. . . 4 |- (((B e. RR /\ 0 < B) /\ C e. RR) -> (C / B) e. RR)
33	32	ad2ant2lr 446	. . 3 |- (((A e. RR /\ (B e. RR /\ 0 < B)) /\ (C e. RR /\ (D e. RR /\ 0 < D))) -> (C / B) e. RR)
34		simprr 451	. . 3 |- (((A e. RR /\ (B e. RR /\ 0 < B)) /\ (C e. RR /\ (D e. RR /\ 0 < D))) -> (D e. RR /\ 0 < D))
35		ltdivmul 7049	. . 3 |- ((A e. RR /\ (C / B) e. RR /\ (D e. RR /\ 0 < D)) -> ((A / D) < (C / B) <-> A < (D x. (C / B))))
36	20, 33, 34, 35	syl111anc 1100	. 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))))
37	19, 26, 36	3bitr4d 609	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 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017   class class class wbr 3338  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386   x. cmul 6391   / cdiv 6447   < clt 6653

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892

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