Theorem iccdil 9880

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

Hypotheses

Ref	Expression
iccdil.1	|- ( A x. R ) = C
iccdil.2	|- ( B x. R ) = D

Assertion

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

Proof of Theorem iccdil

Step	Hyp	Ref	Expression
1		simpl 108	. . . . 5 |- ( ( X e. RR /\ R e. RR+ ) -> X e. RR )
2		rpre 9545	. . . . . 6 |- ( R e. RR+ -> R e. RR )
3		remulcl 7839	. . . . . 6 |- ( ( X e. RR /\ R e. RR ) -> ( X x. R ) e. RR )
4	2, 3	sylan2 284	. . . . 5 |- ( ( X e. RR /\ R e. RR+ ) -> ( X x. R ) e. RR )
5	1, 4	2thd 174	. . . 4 |- ( ( X e. RR /\ R e. RR+ ) -> ( X e. RR <-> ( X x. R ) e. RR ) )
6	5	adantl 275	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X e. RR <-> ( X x. R ) e. RR ) )
7		elrp 9540	. . . . . . 7 |- ( R e. RR+ <-> ( R e. RR /\ 0 < R ) )
8		lemul1 8447	. . . . . . 7 |- ( ( A e. RR /\ X e. RR /\ ( R e. RR /\ 0 < R ) ) -> ( A <_ X <-> ( A x. R ) <_ ( X x. R ) ) )
9	7, 8	syl3an3b 1255	. . . . . 6 |- ( ( A e. RR /\ X e. RR /\ R e. RR+ ) -> ( A <_ X <-> ( A x. R ) <_ ( X x. R ) ) )
10	9	3expb 1183	. . . . 5 |- ( ( A e. RR /\ ( X e. RR /\ R e. RR+ ) ) -> ( A <_ X <-> ( A x. R ) <_ ( X x. R ) ) )
11	10	adantlr 469	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( A <_ X <-> ( A x. R ) <_ ( X x. R ) ) )
12		iccdil.1	. . . . 5 |- ( A x. R ) = C
13	12	breq1i 3968	. . . 4 |- ( ( A x. R ) <_ ( X x. R ) <-> C <_ ( X x. R ) )
14	11, 13	bitrdi 195	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( A <_ X <-> C <_ ( X x. R ) ) )
15		lemul1 8447	. . . . . . . 8 |- ( ( X e. RR /\ B e. RR /\ ( R e. RR /\ 0 < R ) ) -> ( X <_ B <-> ( X x. R ) <_ ( B x. R ) ) )
16	7, 15	syl3an3b 1255	. . . . . . 7 |- ( ( X e. RR /\ B e. RR /\ R e. RR+ ) -> ( X <_ B <-> ( X x. R ) <_ ( B x. R ) ) )
17	16	3expb 1183	. . . . . 6 |- ( ( X e. RR /\ ( B e. RR /\ R e. RR+ ) ) -> ( X <_ B <-> ( X x. R ) <_ ( B x. R ) ) )
18	17	an12s 555	. . . . 5 |- ( ( B e. RR /\ ( X e. RR /\ R e. RR+ ) ) -> ( X <_ B <-> ( X x. R ) <_ ( B x. R ) ) )
19	18	adantll 468	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X <_ B <-> ( X x. R ) <_ ( B x. R ) ) )
20		iccdil.2	. . . . 5 |- ( B x. R ) = D
21	20	breq2i 3969	. . . 4 |- ( ( X x. R ) <_ ( B x. R ) <-> ( X x. R ) <_ D )
22	19, 21	bitrdi 195	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X <_ B <-> ( X x. R ) <_ D ) )
23	6, 14, 22	3anbi123d 1291	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( ( X e. RR /\ A <_ X /\ X <_ B ) <-> ( ( X x. R ) e. RR /\ C <_ ( X x. R ) /\ ( X x. R ) <_ D ) ) )
24		elicc2 9820	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( X e. ( A [,] B ) <-> ( X e. RR /\ A <_ X /\ X <_ B ) ) )
25	24	adantr 274	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X e. ( A [,] B ) <-> ( X e. RR /\ A <_ X /\ X <_ B ) ) )
26		remulcl 7839	. . . . . . 7 |- ( ( A e. RR /\ R e. RR ) -> ( A x. R ) e. RR )
27	12, 26	eqeltrrid 2242	. . . . . 6 |- ( ( A e. RR /\ R e. RR ) -> C e. RR )
28		remulcl 7839	. . . . . . 7 |- ( ( B e. RR /\ R e. RR ) -> ( B x. R ) e. RR )
29	20, 28	eqeltrrid 2242	. . . . . 6 |- ( ( B e. RR /\ R e. RR ) -> D e. RR )
30		elicc2 9820	. . . . . 6 |- ( ( C e. RR /\ D e. RR ) -> ( ( X x. R ) e. ( C [,] D ) <-> ( ( X x. R ) e. RR /\ C <_ ( X x. R ) /\ ( X x. R ) <_ D ) ) )
31	27, 29, 30	syl2an 287	. . . . 5 |- ( ( ( A e. RR /\ R e. RR ) /\ ( B e. RR /\ R e. RR ) ) -> ( ( X x. R ) e. ( C [,] D ) <-> ( ( X x. R ) e. RR /\ C <_ ( X x. R ) /\ ( X x. R ) <_ D ) ) )
32	31	anandirs 583	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ R e. RR ) -> ( ( X x. R ) e. ( C [,] D ) <-> ( ( X x. R ) e. RR /\ C <_ ( X x. R ) /\ ( X x. R ) <_ D ) ) )
33	2, 32	sylan2 284	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ R e. RR+ ) -> ( ( X x. R ) e. ( C [,] D ) <-> ( ( X x. R ) e. RR /\ C <_ ( X x. R ) /\ ( X x. R ) <_ D ) ) )
34	33	adantrl 470	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( ( X x. R ) e. ( C [,] D ) <-> ( ( X x. R ) e. RR /\ C <_ ( X x. R ) /\ ( X x. R ) <_ D ) ) )
35	23, 25, 34	3bitr4d 219	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X e. ( A [,] B ) <-> ( X x. R ) e. ( C [,] D ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 963   = wceq 1332   e. wcel 2125   class class class wbr 3961  (class class class)co 5814   RRcr 7710   0cc0 7711   x. cmul 7716   < clt 7891   <_ cle 7892   RR+crp 9538   [,]cicc 9773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-cnex 7802  ax-resscn 7803  ax-1cn 7804  ax-1re 7805  ax-icn 7806  ax-addcl 7807  ax-addrcl 7808  ax-mulcl 7809  ax-mulrcl 7810  ax-addcom 7811  ax-mulcom 7812  ax-addass 7813  ax-mulass 7814  ax-distr 7815  ax-i2m1 7816  ax-1rid 7818  ax-0id 7819  ax-rnegex 7820  ax-precex 7821  ax-cnre 7822  ax-pre-ltirr 7823  ax-pre-ltwlin 7824  ax-pre-lttrn 7825  ax-pre-ltadd 7827  ax-pre-mulgt0 7828

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-nel 2420  df-ral 2437  df-rex 2438  df-reu 2439  df-rab 2441  df-v 2711  df-sbc 2934  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-opab 4022  df-id 4248  df-po 4251  df-iso 4252  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-iota 5128  df-fun 5165  df-fv 5171  df-riota 5770  df-ov 5817  df-oprab 5818  df-mpo 5819  df-pnf 7893  df-mnf 7894  df-xr 7895  df-ltxr 7896  df-le 7897  df-sub 8027  df-neg 8028  df-rp 9539  df-icc 9777

This theorem is referenced by:  iccdili  9881  lincmb01cmp  9885  iccf1o  9886