Theorem divelunit 11665

Description: A condition for a ratio to be a member of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)

Assertion

Ref	Expression
divelunit	|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) e. ( 0 [,] 1 ) <-> A <_ B ) )

Proof of Theorem divelunit

Step	Hyp	Ref	Expression
1		0re 9585	. . . 4 |- 0 e. RR
2		1re 9584	. . . 4 |- 1 e. RR
3	1, 2	elicc2i 11593	. . 3 |- ( ( A / B ) e. ( 0 [,] 1 ) <-> ( ( A / B ) e. RR /\ 0 <_ ( A / B ) /\ ( A / B ) <_ 1 ) )
4		df-3an 973	. . 3 |- ( ( ( A / B ) e. RR /\ 0 <_ ( A / B ) /\ ( A / B ) <_ 1 ) <-> ( ( ( A / B ) e. RR /\ 0 <_ ( A / B ) ) /\ ( A / B ) <_ 1 ) )
5	3, 4	bitri 249	. 2 |- ( ( A / B ) e. ( 0 [,] 1 ) <-> ( ( ( A / B ) e. RR /\ 0 <_ ( A / B ) ) /\ ( A / B ) <_ 1 ) )
6		ledivmul 10414	. . . . 5 |- ( ( A e. RR /\ 1 e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) <_ 1 <-> A <_ ( B x. 1 ) ) )
7	2, 6	mp3an2 1310	. . . 4 |- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) <_ 1 <-> A <_ ( B x. 1 ) ) )
8	7	adantlr 712	. . 3 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) <_ 1 <-> A <_ ( B x. 1 ) ) )
9		simpll 751	. . . . . 6 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> A e. RR )
10		simprl 754	. . . . . 6 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> B e. RR )
11		gt0ne0 10013	. . . . . . 7 |- ( ( B e. RR /\ 0 < B ) -> B =/= 0 )
12	11	adantl 464	. . . . . 6 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> B =/= 0 )
13	9, 10, 12	redivcld 10368	. . . . 5 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A / B ) e. RR )
14		divge0 10407	. . . . 5 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 <_ ( A / B ) )
15	13, 14	jca 530	. . . 4 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) e. RR /\ 0 <_ ( A / B ) ) )
16	15	biantrurd 506	. . 3 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) <_ 1 <-> ( ( ( A / B ) e. RR /\ 0 <_ ( A / B ) ) /\ ( A / B ) <_ 1 ) ) )
17		recn 9571	. . . . . 6 |- ( B e. RR -> B e. CC )
18	17	ad2antrl 725	. . . . 5 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> B e. CC )
19	18	mulid1d 9602	. . . 4 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> ( B x. 1 ) = B )
20	19	breq2d 4451	. . 3 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A <_ ( B x. 1 ) <-> A <_ B ) )
21	8, 16, 20	3bitr3d 283	. 2 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( ( ( A / B ) e. RR /\ 0 <_ ( A / B ) ) /\ ( A / B ) <_ 1 ) <-> A <_ B ) )
22	5, 21	syl5bb 257	1 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) e. ( 0 [,] 1 ) <-> A <_ B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 971   e. wcel 1823   =/= wne 2649   class class class wbr 4439  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482   x. cmul 9486   < clt 9617   <_ cle 9618   / cdiv 10202   [,]cicc 11535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-icc 11539

This theorem is referenced by:  brbtwn2  24413  axsegconlem7  24431  axcontlem2  24473  axcontlem4  24475  axcontlem7  24478  axcontlem8  24479