Theorem icccntr 10069

Description: Membership in a contracted interval. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

Ref	Expression
icccntr.1	|- ( A / R ) = C
icccntr.2	|- ( B / R ) = D

Assertion

Ref	Expression
icccntr	|- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X e. ( A [,] B ) <-> ( X / R ) e. ( C [,] D ) ) )

Proof of Theorem icccntr

Step	Hyp	Ref	Expression
1		simpl 109	. . . . 5 |- ( ( X e. RR /\ R e. RR+ ) -> X e. RR )
2		rerpdivcl 9753	. . . . 5 |- ( ( X e. RR /\ R e. RR+ ) -> ( X / R ) e. RR )
3	1, 2	2thd 175	. . . 4 |- ( ( X e. RR /\ R e. RR+ ) -> ( X e. RR <-> ( X / R ) e. RR ) )
4	3	adantl 277	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X e. RR <-> ( X / R ) e. RR ) )
5		elrp 9724	. . . . . . 7 |- ( R e. RR+ <-> ( R e. RR /\ 0 < R ) )
6		lediv1 8890	. . . . . . 7 |- ( ( A e. RR /\ X e. RR /\ ( R e. RR /\ 0 < R ) ) -> ( A <_ X <-> ( A / R ) <_ ( X / R ) ) )
7	5, 6	syl3an3b 1287	. . . . . 6 |- ( ( A e. RR /\ X e. RR /\ R e. RR+ ) -> ( A <_ X <-> ( A / R ) <_ ( X / R ) ) )
8	7	3expb 1206	. . . . 5 |- ( ( A e. RR /\ ( X e. RR /\ R e. RR+ ) ) -> ( A <_ X <-> ( A / R ) <_ ( X / R ) ) )
9	8	adantlr 477	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( A <_ X <-> ( A / R ) <_ ( X / R ) ) )
10		icccntr.1	. . . . 5 |- ( A / R ) = C
11	10	breq1i 4037	. . . 4 |- ( ( A / R ) <_ ( X / R ) <-> C <_ ( X / R ) )
12	9, 11	bitrdi 196	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( A <_ X <-> C <_ ( X / R ) ) )
13		lediv1 8890	. . . . . . . 8 |- ( ( X e. RR /\ B e. RR /\ ( R e. RR /\ 0 < R ) ) -> ( X <_ B <-> ( X / R ) <_ ( B / R ) ) )
14	5, 13	syl3an3b 1287	. . . . . . 7 |- ( ( X e. RR /\ B e. RR /\ R e. RR+ ) -> ( X <_ B <-> ( X / R ) <_ ( B / R ) ) )
15	14	3expb 1206	. . . . . 6 |- ( ( X e. RR /\ ( B e. RR /\ R e. RR+ ) ) -> ( X <_ B <-> ( X / R ) <_ ( B / R ) ) )
16	15	an12s 565	. . . . 5 |- ( ( B e. RR /\ ( X e. RR /\ R e. RR+ ) ) -> ( X <_ B <-> ( X / R ) <_ ( B / R ) ) )
17	16	adantll 476	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X <_ B <-> ( X / R ) <_ ( B / R ) ) )
18		icccntr.2	. . . . 5 |- ( B / R ) = D
19	18	breq2i 4038	. . . 4 |- ( ( X / R ) <_ ( B / R ) <-> ( X / R ) <_ D )
20	17, 19	bitrdi 196	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X <_ B <-> ( X / R ) <_ D ) )
21	4, 12, 20	3anbi123d 1323	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( ( X e. RR /\ A <_ X /\ X <_ B ) <-> ( ( X / R ) e. RR /\ C <_ ( X / R ) /\ ( X / R ) <_ D ) ) )
22		elicc2 10007	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( X e. ( A [,] B ) <-> ( X e. RR /\ A <_ X /\ X <_ B ) ) )
23	22	adantr 276	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X e. ( A [,] B ) <-> ( X e. RR /\ A <_ X /\ X <_ B ) ) )
24		rerpdivcl 9753	. . . . . 6 |- ( ( A e. RR /\ R e. RR+ ) -> ( A / R ) e. RR )
25	10, 24	eqeltrrid 2281	. . . . 5 |- ( ( A e. RR /\ R e. RR+ ) -> C e. RR )
26		rerpdivcl 9753	. . . . . 6 |- ( ( B e. RR /\ R e. RR+ ) -> ( B / R ) e. RR )
27	18, 26	eqeltrrid 2281	. . . . 5 |- ( ( B e. RR /\ R e. RR+ ) -> D e. RR )
28		elicc2 10007	. . . . 5 |- ( ( C e. RR /\ D e. RR ) -> ( ( X / R ) e. ( C [,] D ) <-> ( ( X / R ) e. RR /\ C <_ ( X / R ) /\ ( X / R ) <_ D ) ) )
29	25, 27, 28	syl2an 289	. . . 4 |- ( ( ( A e. RR /\ R e. RR+ ) /\ ( B e. RR /\ R e. RR+ ) ) -> ( ( X / R ) e. ( C [,] D ) <-> ( ( X / R ) e. RR /\ C <_ ( X / R ) /\ ( X / R ) <_ D ) ) )
30	29	anandirs 593	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ R e. RR+ ) -> ( ( X / R ) e. ( C [,] D ) <-> ( ( X / R ) e. RR /\ C <_ ( X / R ) /\ ( X / R ) <_ D ) ) )
31	30	adantrl 478	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( ( X / R ) e. ( C [,] D ) <-> ( ( X / R ) e. RR /\ C <_ ( X / R ) /\ ( X / R ) <_ D ) ) )
32	21, 23, 31	3bitr4d 220	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X e. ( A [,] B ) <-> ( X / R ) e. ( C [,] D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   class class class wbr 4030  (class class class)co 5919   RRcr 7873   0cc0 7874   < clt 8056   <_ cle 8057   / cdiv 8693   RR+crp 9722   [,]cicc 9960

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-po 4328  df-iso 4329  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-rp 9723  df-icc 9964

This theorem is referenced by:  icccntri  10070