Theorem iccshftr 10060

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

Hypotheses

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

Assertion

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

Proof of Theorem iccshftr

Step	Hyp	Ref	Expression
1		simpl 109	. . . . 5 |- ( ( X e. RR /\ R e. RR ) -> X e. RR )
2		readdcl 7998	. . . . 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		leadd1 8449	. . . . . 6 |- ( ( A e. RR /\ X e. RR /\ R e. RR ) -> ( A <_ X <-> ( A + R ) <_ ( X + R ) ) )
6	5	3expb 1206	. . . . 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		iccshftr.1	. . . . 5 |- ( A + R ) = C
9	8	breq1i 4036	. . . 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		leadd1 8449	. . . . . . 7 |- ( ( X e. RR /\ B e. RR /\ R e. RR ) -> ( X <_ B <-> ( X + R ) <_ ( B + R ) ) )
12	11	3expb 1206	. . . . . 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		iccshftr.2	. . . . 5 |- ( B + R ) = D
16	15	breq2i 4037	. . . 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 1323	. 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 10004	. . 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		readdcl 7998	. . . . . 6 |- ( ( A e. RR /\ R e. RR ) -> ( A + R ) e. RR )
22	8, 21	eqeltrrid 2281	. . . . 5 |- ( ( A e. RR /\ R e. RR ) -> C e. RR )
23		readdcl 7998	. . . . . 6 |- ( ( B e. RR /\ R e. RR ) -> ( B + R ) e. RR )
24	15, 23	eqeltrrid 2281	. . . . 5 |- ( ( B e. RR /\ R e. RR ) -> D e. RR )
25		elicc2 10004	. . . . 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 593	. . 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 980   = wceq 1364   e. wcel 2164   class class class wbr 4029  (class class class)co 5918   RRcr 7871   + caddc 7875   <_ cle 8055   [,]cicc 9957

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-po 4327  df-iso 4328  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-icc 9961

This theorem is referenced by:  iccshftri  10061  lincmb01cmp  10069