Theorem iccshftri 10008

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 9968	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A [,] B ) C_ RR )
4	1, 2, 3	mp2an 426	. . 3 |- ( A [,] B ) C_ RR
5	4	sseli 3163	. 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 10007	. . . . 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 436	. . . 4 |- ( ( X e. RR /\ R e. RR ) -> ( X e. ( A [,] B ) <-> ( X + R ) e. ( C [,] D ) ) )
11	6, 10	mpan2 425	. . 3 |- ( X e. RR -> ( X e. ( A [,] B ) <-> ( X + R ) e. ( C [,] D ) ) )
12	11	biimpd 144	. 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 104   <-> wb 105   = wceq 1363   e. wcel 2158   C_ wss 3141  (class class class)co 5888   RRcr 7823   + caddc 7827   [,]cicc 9904

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-i2m1 7929  ax-0id 7932  ax-rnegex 7933  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-ltadd 7940

This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-icc 9908

This theorem is referenced by: (None)