Theorem iccdil 10206

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 109	. . . . 5 |- ( ( X e. RR /\ R e. RR+ ) -> X e. RR )
2		rpre 9868	. . . . . 6 |- ( R e. RR+ -> R e. RR )
3		remulcl 8138	. . . . . 6 |- ( ( X e. RR /\ R e. RR ) -> ( X x. R ) e. RR )
4	2, 3	sylan2 286	. . . . 5 |- ( ( X e. RR /\ R e. RR+ ) -> ( X x. R ) e. RR )
5	1, 4	2thd 175	. . . 4 |- ( ( X e. RR /\ R e. RR+ ) -> ( X e. RR <-> ( X x. R ) e. RR ) )
6	5	adantl 277	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X e. RR <-> ( X x. R ) e. RR ) )
7		elrp 9863	. . . . . . 7 |- ( R e. RR+ <-> ( R e. RR /\ 0 < R ) )
8		lemul1 8751	. . . . . . 7 |- ( ( A e. RR /\ X e. RR /\ ( R e. RR /\ 0 < R ) ) -> ( A <_ X <-> ( A x. R ) <_ ( X x. R ) ) )
9	7, 8	syl3an3b 1309	. . . . . 6 |- ( ( A e. RR /\ X e. RR /\ R e. RR+ ) -> ( A <_ X <-> ( A x. R ) <_ ( X x. R ) ) )
10	9	3expb 1228	. . . . 5 |- ( ( A e. RR /\ ( X e. RR /\ R e. RR+ ) ) -> ( A <_ X <-> ( A x. R ) <_ ( X x. R ) ) )
11	10	adantlr 477	. . . 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 4090	. . . 4 |- ( ( A x. R ) <_ ( X x. R ) <-> C <_ ( X x. R ) )
14	11, 13	bitrdi 196	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( A <_ X <-> C <_ ( X x. R ) ) )
15		lemul1 8751	. . . . . . . 8 |- ( ( X e. RR /\ B e. RR /\ ( R e. RR /\ 0 < R ) ) -> ( X <_ B <-> ( X x. R ) <_ ( B x. R ) ) )
16	7, 15	syl3an3b 1309	. . . . . . 7 |- ( ( X e. RR /\ B e. RR /\ R e. RR+ ) -> ( X <_ B <-> ( X x. R ) <_ ( B x. R ) ) )
17	16	3expb 1228	. . . . . 6 |- ( ( X e. RR /\ ( B e. RR /\ R e. RR+ ) ) -> ( X <_ B <-> ( X x. R ) <_ ( B x. R ) ) )
18	17	an12s 565	. . . . 5 |- ( ( B e. RR /\ ( X e. RR /\ R e. RR+ ) ) -> ( X <_ B <-> ( X x. R ) <_ ( B x. R ) ) )
19	18	adantll 476	. . . 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 4091	. . . 4 |- ( ( X x. R ) <_ ( B x. R ) <-> ( X x. R ) <_ D )
22	19, 21	bitrdi 196	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR+ ) ) -> ( X <_ B <-> ( X x. R ) <_ D ) )
23	6, 14, 22	3anbi123d 1346	. 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 10146	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( X e. ( A [,] B ) <-> ( X e. RR /\ A <_ X /\ X <_ B ) ) )
25	24	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 ) ) )
26		remulcl 8138	. . . . . . 7 |- ( ( A e. RR /\ R e. RR ) -> ( A x. R ) e. RR )
27	12, 26	eqeltrrid 2317	. . . . . 6 |- ( ( A e. RR /\ R e. RR ) -> C e. RR )
28		remulcl 8138	. . . . . . 7 |- ( ( B e. RR /\ R e. RR ) -> ( B x. R ) e. RR )
29	20, 28	eqeltrrid 2317	. . . . . 6 |- ( ( B e. RR /\ R e. RR ) -> D e. RR )
30		elicc2 10146	. . . . . 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 289	. . . . 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 595	. . . 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 286	. . 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 478	. 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 220	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 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4083  (class class class)co 6007   RRcr 8009   0cc0 8010   x. cmul 8015   < clt 8192   <_ cle 8193   RR+crp 9861   [,]cicc 10099

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-ltadd 8126  ax-pre-mulgt0 8127

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-rp 9862  df-icc 10103

This theorem is referenced by:  iccdili  10207  lincmb01cmp  10211  iccf1o  10212