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Theorem truni3 25507
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 10709 . . 3  |-  ( C  e.  (  -oo (,) A )  ->  (  -oo  e.  RR*  /\  A  e. 
RR* ) )
2 elioore 10686 . . . . . . 7  |-  ( C  e.  (  -oo (,) A )  ->  C  e.  RR )
3 resubcl 9111 . . . . . . . 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 10464 . . . . . 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 8881 . . . . . . 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 8881 . . . . . . 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 9266 . . . . . . . . . . 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 10697 . . . . . . . . 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 10490 . . . . 5  |-  ( ( (  -oo  e.  RR*  /\  A  e.  RR* )  /\  C  e.  (  -oo (,) A )  /\  ( D  e.  RR  /\  0  <  D ) )  ->  ( C  -  D )  <  A
)
28 elioo2 10697 . . . . . 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 1684   class class class wbr 4023  (class class class)co 5858   RRcr 8736   0cc0 8737    -oocmnf 8865   RR*cxr 8866    < clt 8867    - cmin 9037   (,)cioo 10656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-ioo 10660
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