Theorem iccshftl 10221

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

Hypotheses

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

Assertion

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

Proof of Theorem iccshftl

Step	Hyp	Ref	Expression
1		simpl 109	. . . . 5 |- ( ( X e. RR /\ R e. RR ) -> X e. RR )
2		resubcl 8433	. . . . 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		lesub1 8626	. . . . . 6 |- ( ( A e. RR /\ X e. RR /\ R e. RR ) -> ( A <_ X <-> ( A - R ) <_ ( X - R ) ) )
6	5	3expb 1228	. . . . 5 |- ( ( A e. RR /\ ( X e. RR /\ R e. RR ) ) -> ( A <_ X <-> ( A - R ) <_ ( X - R ) ) )
7	6	adantlr 477	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( A <_ X <-> ( A - R ) <_ ( X - R ) ) )
8		iccshftl.1	. . . . 5 |- ( A - R ) = C
9	8	breq1i 4093	. . . 4 |- ( ( A - R ) <_ ( X - R ) <-> C <_ ( X - R ) )
10	7, 9	bitrdi 196	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( A <_ X <-> C <_ ( X - R ) ) )
11		lesub1 8626	. . . . . . 7 |- ( ( X e. RR /\ B e. RR /\ R e. RR ) -> ( X <_ B <-> ( X - R ) <_ ( B - R ) ) )
12	11	3expb 1228	. . . . . 6 |- ( ( X e. RR /\ ( B e. RR /\ R e. RR ) ) -> ( X <_ B <-> ( X - R ) <_ ( B - R ) ) )
13	12	an12s 565	. . . . 5 |- ( ( B e. RR /\ ( X e. RR /\ R e. RR ) ) -> ( X <_ B <-> ( X - R ) <_ ( B - R ) ) )
14	13	adantll 476	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( X <_ B <-> ( X - R ) <_ ( B - R ) ) )
15		iccshftl.2	. . . . 5 |- ( B - R ) = D
16	15	breq2i 4094	. . . 4 |- ( ( X - R ) <_ ( B - R ) <-> ( X - R ) <_ D )
17	14, 16	bitrdi 196	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( X <_ B <-> ( X - R ) <_ D ) )
18	4, 10, 17	3anbi123d 1346	. 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 ) ) )
19		elicc2 10163	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( X e. ( A [,] B ) <-> ( X e. RR /\ A <_ X /\ X <_ B ) ) )
20	19	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 ) ) )
21		resubcl 8433	. . . . . 6 |- ( ( A e. RR /\ R e. RR ) -> ( A - R ) e. RR )
22	8, 21	eqeltrrid 2317	. . . . 5 |- ( ( A e. RR /\ R e. RR ) -> C e. RR )
23		resubcl 8433	. . . . . 6 |- ( ( B e. RR /\ R e. RR ) -> ( B - R ) e. RR )
24	15, 23	eqeltrrid 2317	. . . . 5 |- ( ( B e. RR /\ R e. RR ) -> D e. RR )
25		elicc2 10163	. . . . 5 |- ( ( C e. RR /\ D e. RR ) -> ( ( X - R ) e. ( C [,] D ) <-> ( ( X - R ) e. RR /\ C <_ ( X - R ) /\ ( X - R ) <_ D ) ) )
26	22, 24, 25	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 ) ) )
27	26	anandirs 595	. . 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 ) ) )
28	27	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 ) ) )
29	18, 20, 28	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 1002   = wceq 1395   e. wcel 2200   class class class wbr 4086  (class class class)co 6013   RRcr 8021   <_ cle 8205   - cmin 8340   [,]cicc 10116

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 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-ltadd 8138

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 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-po 4391  df-iso 4392  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-icc 10120

This theorem is referenced by:  iccshftli  10222  iccf1o  10229