Theorem iccshftl 9932

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 108	. . . . 5 |- ( ( X e. RR /\ R e. RR ) -> X e. RR )
2		resubcl 8162	. . . . 5 |- ( ( X e. RR /\ R e. RR ) -> ( X - R ) e. RR )
3	1, 2	2thd 174	. . . 4 |- ( ( X e. RR /\ R e. RR ) -> ( X e. RR <-> ( X - R ) e. RR ) )
4	3	adantl 275	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( X e. RR <-> ( X - R ) e. RR ) )
5		lesub1 8354	. . . . . 6 |- ( ( A e. RR /\ X e. RR /\ R e. RR ) -> ( A <_ X <-> ( A - R ) <_ ( X - R ) ) )
6	5	3expb 1194	. . . . 5 |- ( ( A e. RR /\ ( X e. RR /\ R e. RR ) ) -> ( A <_ X <-> ( A - R ) <_ ( X - R ) ) )
7	6	adantlr 469	. . . 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 3989	. . . 4 |- ( ( A - R ) <_ ( X - R ) <-> C <_ ( X - R ) )
10	7, 9	bitrdi 195	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( A <_ X <-> C <_ ( X - R ) ) )
11		lesub1 8354	. . . . . . 7 |- ( ( X e. RR /\ B e. RR /\ R e. RR ) -> ( X <_ B <-> ( X - R ) <_ ( B - R ) ) )
12	11	3expb 1194	. . . . . 6 |- ( ( X e. RR /\ ( B e. RR /\ R e. RR ) ) -> ( X <_ B <-> ( X - R ) <_ ( B - R ) ) )
13	12	an12s 555	. . . . 5 |- ( ( B e. RR /\ ( X e. RR /\ R e. RR ) ) -> ( X <_ B <-> ( X - R ) <_ ( B - R ) ) )
14	13	adantll 468	. . . 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 3990	. . . 4 |- ( ( X - R ) <_ ( B - R ) <-> ( X - R ) <_ D )
17	14, 16	bitrdi 195	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( X <_ B <-> ( X - R ) <_ D ) )
18	4, 10, 17	3anbi123d 1302	. 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 9874	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( X e. ( A [,] B ) <-> ( X e. RR /\ A <_ X /\ X <_ B ) ) )
20	19	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 ) ) )
21		resubcl 8162	. . . . . 6 |- ( ( A e. RR /\ R e. RR ) -> ( A - R ) e. RR )
22	8, 21	eqeltrrid 2254	. . . . 5 |- ( ( A e. RR /\ R e. RR ) -> C e. RR )
23		resubcl 8162	. . . . . 6 |- ( ( B e. RR /\ R e. RR ) -> ( B - R ) e. RR )
24	15, 23	eqeltrrid 2254	. . . . 5 |- ( ( B e. RR /\ R e. RR ) -> D e. RR )
25		elicc2 9874	. . . . 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 287	. . . 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 583	. . 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 470	. 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 219	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 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   RRcr 7752   <_ cle 7934   - cmin 8069   [,]cicc 9827

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-icc 9831

This theorem is referenced by:  iccshftli  9933  iccf1o  9940