Theorem iccshftr 10330

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 8255	. . . . 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 8706	. . . . . 6 |- ( ( A e. RR /\ X e. RR /\ R e. RR ) -> ( A <_ X <-> ( A + R ) <_ ( X + R ) ) )
6	5	3expb 1231	. . . . 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 4118	. . . 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 8706	. . . . . . 7 |- ( ( X e. RR /\ B e. RR /\ R e. RR ) -> ( X <_ B <-> ( X + R ) <_ ( B + R ) ) )
12	11	3expb 1231	. . . . . 6 |- ( ( X e. RR /\ ( B e. RR /\ R e. RR ) ) -> ( X <_ B <-> ( X + R ) <_ ( B + R ) ) )
13	12	an12s 567	. . . . 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 4119	. . . 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 1349	. 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 10274	. . 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 8255	. . . . . 6 |- ( ( A e. RR /\ R e. RR ) -> ( A + R ) e. RR )
22	8, 21	eqeltrrid 2322	. . . . 5 |- ( ( A e. RR /\ R e. RR ) -> C e. RR )
23		readdcl 8255	. . . . . 6 |- ( ( B e. RR /\ R e. RR ) -> ( B + R ) e. RR )
24	15, 23	eqeltrrid 2322	. . . . 5 |- ( ( B e. RR /\ R e. RR ) -> D e. RR )
25		elicc2 10274	. . . . 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 597	. . 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 1005   = wceq 1398   e. wcel 2205   class class class wbr 4111  (class class class)co 6052   RRcr 8128   + caddc 8132   <_ cle 8311   [,]cicc 10227

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-i2m1 8234  ax-0id 8237  ax-rnegex 8238  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-po 4419  df-iso 4420  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-icc 10231

This theorem is referenced by:  iccshftri  10331  lincmb01cmp  10339