Theorem iccshftl 10118

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 8336	. . . . 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 8529	. . . . . 6 |- ( ( A e. RR /\ X e. RR /\ R e. RR ) -> ( A <_ X <-> ( A - R ) <_ ( X - R ) ) )
6	5	3expb 1207	. . . . 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 4051	. . . 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 8529	. . . . . . 7 |- ( ( X e. RR /\ B e. RR /\ R e. RR ) -> ( X <_ B <-> ( X - R ) <_ ( B - R ) ) )
12	11	3expb 1207	. . . . . 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 4052	. . . 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 1325	. 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 10060	. . 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 8336	. . . . . 6 |- ( ( A e. RR /\ R e. RR ) -> ( A - R ) e. RR )
22	8, 21	eqeltrrid 2293	. . . . 5 |- ( ( A e. RR /\ R e. RR ) -> C e. RR )
23		resubcl 8336	. . . . . 6 |- ( ( B e. RR /\ R e. RR ) -> ( B - R ) e. RR )
24	15, 23	eqeltrrid 2293	. . . . 5 |- ( ( B e. RR /\ R e. RR ) -> D e. RR )
25		elicc2 10060	. . . . 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 981   = wceq 1373   e. wcel 2176   class class class wbr 4044  (class class class)co 5944   RRcr 7924   <_ cle 8108   - cmin 8243   [,]cicc 10013

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-po 4343  df-iso 4344  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-icc 10017

This theorem is referenced by:  iccshftli  10119  iccf1o  10126