Theorem icccntr 10333

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 10017	. . . . 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 9988	. . . . . . 7 |- ( R e. RR+ <-> ( R e. RR /\ 0 < R ) )
6		lediv1 9143	. . . . . . 7 |- ( ( A e. RR /\ X e. RR /\ ( R e. RR /\ 0 < R ) ) -> ( A <_ X <-> ( A / R ) <_ ( X / R ) ) )
7	5, 6	syl3an3b 1312	. . . . . 6 |- ( ( A e. RR /\ X e. RR /\ R e. RR+ ) -> ( A <_ X <-> ( A / R ) <_ ( X / R ) ) )
8	7	3expb 1231	. . . . 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 4116	. . . 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 9143	. . . . . . . 8 |- ( ( X e. RR /\ B e. RR /\ ( R e. RR /\ 0 < R ) ) -> ( X <_ B <-> ( X / R ) <_ ( B / R ) ) )
14	5, 13	syl3an3b 1312	. . . . . . 7 |- ( ( X e. RR /\ B e. RR /\ R e. RR+ ) -> ( X <_ B <-> ( X / R ) <_ ( B / R ) ) )
15	14	3expb 1231	. . . . . 6 |- ( ( X e. RR /\ ( B e. RR /\ R e. RR+ ) ) -> ( X <_ B <-> ( X / R ) <_ ( B / R ) ) )
16	15	an12s 567	. . . . 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 4117	. . . 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 1349	. 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 10271	. . 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 10017	. . . . . 6 |- ( ( A e. RR /\ R e. RR+ ) -> ( A / R ) e. RR )
25	10, 24	eqeltrrid 2320	. . . . 5 |- ( ( A e. RR /\ R e. RR+ ) -> C e. RR )
26		rerpdivcl 10017	. . . . . 6 |- ( ( B e. RR /\ R e. RR+ ) -> ( B / R ) e. RR )
27	18, 26	eqeltrrid 2320	. . . . 5 |- ( ( B e. RR /\ R e. RR+ ) -> D e. RR )
28		elicc2 10271	. . . . 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 597	. . 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 1005   = wceq 1398   e. wcel 2203   class class class wbr 4109  (class class class)co 6050   RRcr 8126   0cc0 8127   < clt 8308   <_ cle 8309   / cdiv 8946   RR+crp 9986   [,]cicc 10224

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-rp 9987  df-icc 10228

This theorem is referenced by:  icccntri  10334