Theorem icccntr 10352

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 10035	. . . . 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 10006	. . . . . . 7 |- ( R e. RR+ <-> ( R e. RR /\ 0 < R ) )
6		lediv1 9160	. . . . . . 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 4121	. . . 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 9160	. . . . . . . 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 4122	. . . 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 10290	. . 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 10035	. . . . . 6 |- ( ( A e. RR /\ R e. RR+ ) -> ( A / R ) e. RR )
25	10, 24	eqeltrrid 2322	. . . . 5 |- ( ( A e. RR /\ R e. RR+ ) -> C e. RR )
26		rerpdivcl 10035	. . . . . 6 |- ( ( B e. RR /\ R e. RR+ ) -> ( B / R ) e. RR )
27	18, 26	eqeltrrid 2322	. . . . 5 |- ( ( B e. RR /\ R e. RR+ ) -> D e. RR )
28		elicc2 10290	. . . . 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 2205   class class class wbr 4114  (class class class)co 6058   RRcr 8142   0cc0 8143   < clt 8324   <_ cle 8325   / cdiv 8963   RR+crp 10004   [,]cicc 10243

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-po 4422  df-iso 4423  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-rp 10005  df-icc 10247

This theorem is referenced by:  icccntri  10353