Theorem iccshftli 9901

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

Hypotheses

Ref	Expression
iccshftli.1	|- A e. RR
iccshftli.2	|- B e. RR
iccshftli.3	|- R e. RR
iccshftli.4	|- ( A - R ) = C
iccshftli.5	|- ( B - R ) = D

Assertion

Ref	Expression
iccshftli	|- ( X e. ( A [,] B ) -> ( X - R ) e. ( C [,] D ) )

Proof of Theorem iccshftli

Step	Hyp	Ref	Expression
1		iccshftli.1	. . . 4 |- A e. RR
2		iccshftli.2	. . . 4 |- B e. RR
3		iccssre 9859	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
4	1, 2, 3	mp2an 423	. . 3 |- ( A [,] B ) C_ RR
5	4	sseli 3124	. 2 |- ( X e. ( A [,] B ) -> X e. RR )
6		iccshftli.3	. . . 4 |- R e. RR
7		iccshftli.4	. . . . . 6 |- ( A - R ) = C
8		iccshftli.5	. . . . . 6 |- ( B - R ) = D
9	7, 8	iccshftl 9900	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( X e. RR /\ R e. RR ) ) -> ( X e. ( A [,] B ) <-> ( X - R ) e. ( C [,] D ) ) )
10	1, 2, 9	mpanl12 433	. . . 4 |- ( ( X e. RR /\ R e. RR ) -> ( X e. ( A [,] B ) <-> ( X - R ) e. ( C [,] D ) ) )
11	6, 10	mpan2 422	. . 3 |- ( X e. RR -> ( X e. ( A [,] B ) <-> ( X - R ) e. ( C [,] D ) ) )
12	11	biimpd 143	. 2 |- ( X e. RR -> ( X e. ( A [,] B ) -> ( X - R ) e. ( C [,] D ) ) )
13	5, 12	mpcom 36	1 |- ( X e. ( A [,] B ) -> ( X - R ) e. ( C [,] D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   C_ wss 3102  (class class class)co 5824   RRcr 7731   - cmin 8046   [,]cicc 9795

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-addcom 7832  ax-addass 7834  ax-distr 7836  ax-i2m1 7837  ax-0id 7840  ax-rnegex 7841  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-ltadd 7848

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-po 4256  df-iso 4257  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-icc 9799

This theorem is referenced by: (None)