Theorem iccdil 10150

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 9812	. . . . . 6 |- ( R e. RR+ -> R e. RR )
3		remulcl 8083	. . . . . 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 9807	. . . . . . 7 |- ( R e. RR+ <-> ( R e. RR /\ 0 < R ) )
8		lemul1 8696	. . . . . . 7 |- ( ( A e. RR /\ X e. RR /\ ( R e. RR /\ 0 < R ) ) -> ( A <_ X <-> ( A x. R ) <_ ( X x. R ) ) )
9	7, 8	syl3an3b 1288	. . . . . 6 |- ( ( A e. RR /\ X e. RR /\ R e. RR+ ) -> ( A <_ X <-> ( A x. R ) <_ ( X x. R ) ) )
10	9	3expb 1207	. . . . 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 4061	. . . 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 8696	. . . . . . . 8 |- ( ( X e. RR /\ B e. RR /\ ( R e. RR /\ 0 < R ) ) -> ( X <_ B <-> ( X x. R ) <_ ( B x. R ) ) )
16	7, 15	syl3an3b 1288	. . . . . . 7 |- ( ( X e. RR /\ B e. RR /\ R e. RR+ ) -> ( X <_ B <-> ( X x. R ) <_ ( B x. R ) ) )
17	16	3expb 1207	. . . . . 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 4062	. . . 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 1325	. 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 10090	. . 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 8083	. . . . . . 7 |- ( ( A e. RR /\ R e. RR ) -> ( A x. R ) e. RR )
27	12, 26	eqeltrrid 2294	. . . . . 6 |- ( ( A e. RR /\ R e. RR ) -> C e. RR )
28		remulcl 8083	. . . . . . 7 |- ( ( B e. RR /\ R e. RR ) -> ( B x. R ) e. RR )
29	20, 28	eqeltrrid 2294	. . . . . 6 |- ( ( B e. RR /\ R e. RR ) -> D e. RR )
30		elicc2 10090	. . . . . 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 593	. . . 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 981   = wceq 1373   e. wcel 2177   class class class wbr 4054  (class class class)co 5962   RRcr 7954   0cc0 7955   x. cmul 7960   < clt 8137   <_ cle 8138   RR+crp 9805   [,]cicc 10043

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-mulrcl 8054  ax-addcom 8055  ax-mulcom 8056  ax-addass 8057  ax-mulass 8058  ax-distr 8059  ax-i2m1 8060  ax-1rid 8062  ax-0id 8063  ax-rnegex 8064  ax-precex 8065  ax-cnre 8066  ax-pre-ltirr 8067  ax-pre-ltwlin 8068  ax-pre-lttrn 8069  ax-pre-ltadd 8071  ax-pre-mulgt0 8072

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-opab 4117  df-id 4353  df-po 4356  df-iso 4357  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-iota 5246  df-fun 5287  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-pnf 8139  df-mnf 8140  df-xr 8141  df-ltxr 8142  df-le 8143  df-sub 8275  df-neg 8276  df-rp 9806  df-icc 10047

This theorem is referenced by:  iccdili  10151  lincmb01cmp  10155  iccf1o  10156