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Theorem truni3 25610
Description: Closure of translation in a half-infinite interval. (Contributed by FL, 26-Jan-2009.)
Assertion
Ref Expression
truni3  |-  ( ( D  e.  RR  /\  0  <  D )  -> 
( C  e.  ( 
-oo (,) A )  -> 
( C  -  D
)  e.  (  -oo (,) A ) ) )

Proof of Theorem truni3
StepHypRef Expression
1 eliooxr 10725 . . 3  |-  ( C  e.  (  -oo (,) A )  ->  (  -oo  e.  RR*  /\  A  e. 
RR* ) )
2 elioore 10702 . . . . . . 7  |-  ( C  e.  (  -oo (,) A )  ->  C  e.  RR )
3 resubcl 9127 . . . . . . . 8  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  -  D
)  e.  RR )
43adantrr 697 . . . . . . 7  |-  ( ( C  e.  RR  /\  ( D  e.  RR  /\  0  <  D ) )  ->  ( C  -  D )  e.  RR )
52, 4sylan 457 . . . . . 6  |-  ( ( C  e.  (  -oo (,) A )  /\  ( D  e.  RR  /\  0  <  D ) )  -> 
( C  -  D
)  e.  RR )
653adant1 973 . . . . 5  |-  ( ( (  -oo  e.  RR*  /\  A  e.  RR* )  /\  C  e.  (  -oo (,) A )  /\  ( D  e.  RR  /\  0  <  D ) )  ->  ( C  -  D )  e.  RR )
7 mnflt 10480 . . . . . 6  |-  ( ( C  -  D )  e.  RR  ->  -oo  <  ( C  -  D ) )
86, 7syl 15 . . . . 5  |-  ( ( (  -oo  e.  RR*  /\  A  e.  RR* )  /\  C  e.  (  -oo (,) A )  /\  ( D  e.  RR  /\  0  <  D ) )  ->  -oo  <  ( C  -  D )
)
923ad2ant2 977 . . . . . . 7  |-  ( ( (  -oo  e.  RR*  /\  A  e.  RR* )  /\  C  e.  (  -oo (,) A )  /\  ( D  e.  RR  /\  0  <  D ) )  ->  C  e.  RR )
10 simp3l 983 . . . . . . 7  |-  ( ( (  -oo  e.  RR*  /\  A  e.  RR* )  /\  C  e.  (  -oo (,) A )  /\  ( D  e.  RR  /\  0  <  D ) )  ->  D  e.  RR )
113rexrd 8897 . . . . . . 7  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  -  D
)  e.  RR* )
129, 10, 11syl2anc 642 . . . . . 6  |-  ( ( (  -oo  e.  RR*  /\  A  e.  RR* )  /\  C  e.  (  -oo (,) A )  /\  ( D  e.  RR  /\  0  <  D ) )  ->  ( C  -  D )  e.  RR* )
132rexrd 8897 . . . . . . 7  |-  ( C  e.  (  -oo (,) A )  ->  C  e.  RR* )
14133ad2ant2 977 . . . . . 6  |-  ( ( (  -oo  e.  RR*  /\  A  e.  RR* )  /\  C  e.  (  -oo (,) A )  /\  ( D  e.  RR  /\  0  <  D ) )  ->  C  e.  RR* )
15 simp1r 980 . . . . . 6  |-  ( ( (  -oo  e.  RR*  /\  A  e.  RR* )  /\  C  e.  (  -oo (,) A )  /\  ( D  e.  RR  /\  0  <  D ) )  ->  A  e.  RR* )
16 ltsubpos 9282 . . . . . . . . . . 11  |-  ( ( D  e.  RR  /\  C  e.  RR )  ->  ( 0  <  D  <->  ( C  -  D )  <  C ) )
1716biimpd 198 . . . . . . . . . 10  |-  ( ( D  e.  RR  /\  C  e.  RR )  ->  ( 0  <  D  ->  ( C  -  D
)  <  C )
)
1817impancom 427 . . . . . . . . 9  |-  ( ( D  e.  RR  /\  0  <  D )  -> 
( C  e.  RR  ->  ( C  -  D
)  <  C )
)
192, 18syl5com 26 . . . . . . . 8  |-  ( C  e.  (  -oo (,) A )  ->  (
( D  e.  RR  /\  0  <  D )  ->  ( C  -  D )  <  C
) )
2019a1i 10 . . . . . . 7  |-  ( ( 
-oo  e.  RR*  /\  A  e.  RR* )  ->  ( C  e.  (  -oo (,) A )  ->  (
( D  e.  RR  /\  0  <  D )  ->  ( C  -  D )  <  C
) ) )
21203imp 1145 . . . . . 6  |-  ( ( (  -oo  e.  RR*  /\  A  e.  RR* )  /\  C  e.  (  -oo (,) A )  /\  ( D  e.  RR  /\  0  <  D ) )  ->  ( C  -  D )  <  C
)
22 elioo2 10713 . . . . . . . . 9  |-  ( ( 
-oo  e.  RR*  /\  A  e.  RR* )  ->  ( C  e.  (  -oo (,) A )  <->  ( C  e.  RR  /\  -oo  <  C  /\  C  <  A
) ) )
23 simp3 957 . . . . . . . . 9  |-  ( ( C  e.  RR  /\  -oo 
<  C  /\  C  < 
A )  ->  C  <  A )
2422, 23syl6bi 219 . . . . . . . 8  |-  ( ( 
-oo  e.  RR*  /\  A  e.  RR* )  ->  ( C  e.  (  -oo (,) A )  ->  C  <  A ) )
2524a1dd 42 . . . . . . 7  |-  ( ( 
-oo  e.  RR*  /\  A  e.  RR* )  ->  ( C  e.  (  -oo (,) A )  ->  (
( D  e.  RR  /\  0  <  D )  ->  C  <  A
) ) )
26253imp 1145 . . . . . 6  |-  ( ( (  -oo  e.  RR*  /\  A  e.  RR* )  /\  C  e.  (  -oo (,) A )  /\  ( D  e.  RR  /\  0  <  D ) )  ->  C  <  A )
2712, 14, 15, 21, 26xrlttrd 10506 . . . . 5  |-  ( ( (  -oo  e.  RR*  /\  A  e.  RR* )  /\  C  e.  (  -oo (,) A )  /\  ( D  e.  RR  /\  0  <  D ) )  ->  ( C  -  D )  <  A
)
28 elioo2 10713 . . . . . 6  |-  ( ( 
-oo  e.  RR*  /\  A  e.  RR* )  ->  (
( C  -  D
)  e.  (  -oo (,) A )  <->  ( ( C  -  D )  e.  RR  /\  -oo  <  ( C  -  D )  /\  ( C  -  D )  <  A
) ) )
29283ad2ant1 976 . . . . 5  |-  ( ( (  -oo  e.  RR*  /\  A  e.  RR* )  /\  C  e.  (  -oo (,) A )  /\  ( D  e.  RR  /\  0  <  D ) )  ->  ( ( C  -  D )  e.  (  -oo (,) A
)  <->  ( ( C  -  D )  e.  RR  /\  -oo  <  ( C  -  D )  /\  ( C  -  D )  <  A
) ) )
306, 8, 27, 29mpbir3and 1135 . . . 4  |-  ( ( (  -oo  e.  RR*  /\  A  e.  RR* )  /\  C  e.  (  -oo (,) A )  /\  ( D  e.  RR  /\  0  <  D ) )  ->  ( C  -  D )  e.  ( 
-oo (,) A ) )
31303exp 1150 . . 3  |-  ( ( 
-oo  e.  RR*  /\  A  e.  RR* )  ->  ( C  e.  (  -oo (,) A )  ->  (
( D  e.  RR  /\  0  <  D )  ->  ( C  -  D )  e.  ( 
-oo (,) A ) ) ) )
321, 31mpcom 32 . 2  |-  ( C  e.  (  -oo (,) A )  ->  (
( D  e.  RR  /\  0  <  D )  ->  ( C  -  D )  e.  ( 
-oo (,) A ) ) )
3332com12 27 1  |-  ( ( D  e.  RR  /\  0  <  D )  -> 
( C  e.  ( 
-oo (,) A )  -> 
( C  -  D
)  e.  (  -oo (,) A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1696   class class class wbr 4039  (class class class)co 5874   RRcr 8752   0cc0 8753    -oocmnf 8881   RR*cxr 8882    < clt 8883    - cmin 9053   (,)cioo 10672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-ioo 10676
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