Theorem icccntr 10003

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 9687	. . . . 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 9658	. . . . . . 7 |- ( R e. RR+ <-> ( R e. RR /\ 0 < R ) )
6		lediv1 8829	. . . . . . 7 |- ( ( A e. RR /\ X e. RR /\ ( R e. RR /\ 0 < R ) ) -> ( A <_ X <-> ( A / R ) <_ ( X / R ) ) )
7	5, 6	syl3an3b 1276	. . . . . 6 |- ( ( A e. RR /\ X e. RR /\ R e. RR+ ) -> ( A <_ X <-> ( A / R ) <_ ( X / R ) ) )
8	7	3expb 1204	. . . . 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 4012	. . . 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 8829	. . . . . . . 8 |- ( ( X e. RR /\ B e. RR /\ ( R e. RR /\ 0 < R ) ) -> ( X <_ B <-> ( X / R ) <_ ( B / R ) ) )
14	5, 13	syl3an3b 1276	. . . . . . 7 |- ( ( X e. RR /\ B e. RR /\ R e. RR+ ) -> ( X <_ B <-> ( X / R ) <_ ( B / R ) ) )
15	14	3expb 1204	. . . . . 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 4013	. . . 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 1312	. 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 9941	. . 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 9687	. . . . . 6 |- ( ( A e. RR /\ R e. RR+ ) -> ( A / R ) e. RR )
25	10, 24	eqeltrrid 2265	. . . . 5 |- ( ( A e. RR /\ R e. RR+ ) -> C e. RR )
26		rerpdivcl 9687	. . . . . 6 |- ( ( B e. RR /\ R e. RR+ ) -> ( B / R ) e. RR )
27	18, 26	eqeltrrid 2265	. . . . 5 |- ( ( B e. RR /\ R e. RR+ ) -> D e. RR )
28		elicc2 9941	. . . . 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 978   = wceq 1353   e. wcel 2148   class class class wbr 4005  (class class class)co 5878   RRcr 7813   0cc0 7814   < clt 7995   <_ cle 7996   / cdiv 8632   RR+crp 9656   [,]cicc 9894

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-precex 7924  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930  ax-pre-mulgt0 7931  ax-pre-mulext 7932

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-po 4298  df-iso 4299  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-reap 8535  df-ap 8542  df-div 8633  df-rp 9657  df-icc 9898

This theorem is referenced by:  icccntri  10004