Theorem iccshftr 9951

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 108	. . . . 5 |- ( ( X e. RR /\ R e. RR ) -> X e. RR )
2		readdcl 7900	. . . . 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		leadd1 8349	. . . . . 6 |- ( ( A e. RR /\ X e. RR /\ R e. RR ) -> ( A <_ X <-> ( A + R ) <_ ( X + R ) ) )
6	5	3expb 1199	. . . . 5 |- ( ( A e. RR /\ ( X e. RR /\ R e. RR ) ) -> ( A <_ X <-> ( A + R ) <_ ( X + R ) ) )
7	6	adantlr 474	. . . 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 3996	. . . 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		leadd1 8349	. . . . . . 7 |- ( ( X e. RR /\ B e. RR /\ R e. RR ) -> ( X <_ B <-> ( X + R ) <_ ( B + R ) ) )
12	11	3expb 1199	. . . . . 6 |- ( ( X e. RR /\ ( B e. RR /\ R e. RR ) ) -> ( X <_ B <-> ( X + R ) <_ ( B + R ) ) )
13	12	an12s 560	. . . . 5 |- ( ( B e. RR /\ ( X e. RR /\ R e. RR ) ) -> ( X <_ B <-> ( X + R ) <_ ( B + R ) ) )
14	13	adantll 473	. . . 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 3997	. . . 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 1307	. 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 9895	. . 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		readdcl 7900	. . . . . 6 |- ( ( A e. RR /\ R e. RR ) -> ( A + R ) e. RR )
22	8, 21	eqeltrrid 2258	. . . . 5 |- ( ( A e. RR /\ R e. RR ) -> C e. RR )
23		readdcl 7900	. . . . . 6 |- ( ( B e. RR /\ R e. RR ) -> ( B + R ) e. RR )
24	15, 23	eqeltrrid 2258	. . . . 5 |- ( ( B e. RR /\ R e. RR ) -> D e. RR )
25		elicc2 9895	. . . . 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 588	. . 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 475	. 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 973   = wceq 1348   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   RRcr 7773   + caddc 7777   <_ cle 7955   [,]cicc 9848

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-icc 9852

This theorem is referenced by:  iccshftri  9952  lincmb01cmp  9960