Theorem iccshftri 9778

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

Hypotheses

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

Assertion

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

Proof of Theorem iccshftri

Step	Hyp	Ref	Expression
1		iccshftri.1	. . . 4 |- A e. RR
2		iccshftri.2	. . . 4 |- B e. RR
3		iccssre 9738	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
4	1, 2, 3	mp2an 422	. . 3 |- ( A [,] B ) C_ RR
5	4	sseli 3093	. 2 |- ( X e. ( A [,] B ) -> X e. RR )
6		iccshftri.3	. . . 4 |- R e. RR
7		iccshftri.4	. . . . . 6 |- ( A + R ) = C
8		iccshftri.5	. . . . . 6 |- ( B + R ) = D
9	7, 8	iccshftr 9777	. . . . 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 432	. . . 4 |- ( ( X e. RR /\ R e. RR ) -> ( X e. ( A [,] B ) <-> ( X + R ) e. ( C [,] D ) ) )
11	6, 10	mpan2 421	. . 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 1331   e. wcel 1480   C_ wss 3071  (class class class)co 5774   RRcr 7619   + caddc 7623   [,]cicc 9674

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-i2m1 7725  ax-0id 7728  ax-rnegex 7729  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-ltadd 7736

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-icc 9678

This theorem is referenced by: (None)