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Theorem iccshift 37568
Description: A closed interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
iccshift.1  |-  ( ph  ->  A  e.  RR )
iccshift.2  |-  ( ph  ->  B  e.  RR )
iccshift.3  |-  ( ph  ->  T  e.  RR )
Assertion
Ref Expression
iccshift  |-  ( ph  ->  ( ( A  +  T ) [,] ( B  +  T )
)  =  { w  e.  CC  |  E. z  e.  ( A [,] B
) w  =  ( z  +  T ) } )
Distinct variable groups:    w, A, z    w, B, z    w, T, z    ph, z
Allowed substitution hint:    ph( w)

Proof of Theorem iccshift
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2426 . . . . . . 7  |-  ( w  =  x  ->  (
w  =  ( z  +  T )  <->  x  =  ( z  +  T
) ) )
21rexbidv 2936 . . . . . 6  |-  ( w  =  x  ->  ( E. z  e.  ( A [,] B ) w  =  ( z  +  T )  <->  E. z  e.  ( A [,] B
) x  =  ( z  +  T ) ) )
32elrab 3228 . . . . 5  |-  ( x  e.  { w  e.  CC  |  E. z  e.  ( A [,] B
) w  =  ( z  +  T ) }  <->  ( x  e.  CC  /\  E. z  e.  ( A [,] B
) x  =  ( z  +  T ) ) )
4 simprr 764 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  E. z  e.  ( A [,] B
) x  =  ( z  +  T ) ) )  ->  E. z  e.  ( A [,] B
) x  =  ( z  +  T ) )
5 nfv 1755 . . . . . . . 8  |-  F/ z
ph
6 nfv 1755 . . . . . . . . 9  |-  F/ z  x  e.  CC
7 nfre1 2883 . . . . . . . . 9  |-  F/ z E. z  e.  ( A [,] B ) x  =  ( z  +  T )
86, 7nfan 1988 . . . . . . . 8  |-  F/ z ( x  e.  CC  /\ 
E. z  e.  ( A [,] B ) x  =  ( z  +  T ) )
95, 8nfan 1988 . . . . . . 7  |-  F/ z ( ph  /\  (
x  e.  CC  /\  E. z  e.  ( A [,] B ) x  =  ( z  +  T ) ) )
10 nfv 1755 . . . . . . 7  |-  F/ z  x  e.  ( ( A  +  T ) [,] ( B  +  T ) )
11 simp3 1007 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( A [,] B )  /\  x  =  ( z  +  T ) )  ->  x  =  ( z  +  T
) )
12 iccshift.1 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  A  e.  RR )
13 iccshift.2 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  B  e.  RR )
1412, 13iccssred 37551 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( A [,] B
)  C_  RR )
1514sselda 3464 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  z  e.  RR )
16 iccshift.3 . . . . . . . . . . . . . . 15  |-  ( ph  ->  T  e.  RR )
1716adantr 466 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  T  e.  RR )
1815, 17readdcld 9677 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( z  +  T )  e.  RR )
1912adantr 466 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  A  e.  RR )
20 simpr 462 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  z  e.  ( A [,] B ) )
2113adantr 466 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  B  e.  RR )
22 elicc2 11706 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( z  e.  ( A [,] B )  <-> 
( z  e.  RR  /\  A  <_  z  /\  z  <_  B ) ) )
2319, 21, 22syl2anc 665 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( z  e.  ( A [,] B
)  <->  ( z  e.  RR  /\  A  <_ 
z  /\  z  <_  B ) ) )
2420, 23mpbid 213 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( z  e.  RR  /\  A  <_ 
z  /\  z  <_  B ) )
2524simp2d 1018 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  A  <_  z )
2619, 15, 17, 25leadd1dd 10234 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( A  +  T )  <_  (
z  +  T ) )
2724simp3d 1019 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  z  <_  B )
2815, 21, 17, 27leadd1dd 10234 . . . . . . . . . . . . 13  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( z  +  T )  <_  ( B  +  T )
)
2918, 26, 283jca 1185 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( (
z  +  T )  e.  RR  /\  ( A  +  T )  <_  ( z  +  T
)  /\  ( z  +  T )  <_  ( B  +  T )
) )
30293adant3 1025 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ( A [,] B )  /\  x  =  ( z  +  T ) )  ->  ( (
z  +  T )  e.  RR  /\  ( A  +  T )  <_  ( z  +  T
)  /\  ( z  +  T )  <_  ( B  +  T )
) )
3112, 16readdcld 9677 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  +  T
)  e.  RR )
32313ad2ant1 1026 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  ( A [,] B )  /\  x  =  ( z  +  T ) )  ->  ( A  +  T )  e.  RR )
3313, 16readdcld 9677 . . . . . . . . . . . . 13  |-  ( ph  ->  ( B  +  T
)  e.  RR )
34333ad2ant1 1026 . . . . . . . . . . . 12  |-  ( (
ph  /\  z  e.  ( A [,] B )  /\  x  =  ( z  +  T ) )  ->  ( B  +  T )  e.  RR )
35 elicc2 11706 . . . . . . . . . . . 12  |-  ( ( ( A  +  T
)  e.  RR  /\  ( B  +  T
)  e.  RR )  ->  ( ( z  +  T )  e.  ( ( A  +  T ) [,] ( B  +  T )
)  <->  ( ( z  +  T )  e.  RR  /\  ( A  +  T )  <_ 
( z  +  T
)  /\  ( z  +  T )  <_  ( B  +  T )
) ) )
3632, 34, 35syl2anc 665 . . . . . . . . . . 11  |-  ( (
ph  /\  z  e.  ( A [,] B )  /\  x  =  ( z  +  T ) )  ->  ( (
z  +  T )  e.  ( ( A  +  T ) [,] ( B  +  T
) )  <->  ( (
z  +  T )  e.  RR  /\  ( A  +  T )  <_  ( z  +  T
)  /\  ( z  +  T )  <_  ( B  +  T )
) ) )
3730, 36mpbird 235 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( A [,] B )  /\  x  =  ( z  +  T ) )  ->  ( z  +  T )  e.  ( ( A  +  T
) [,] ( B  +  T ) ) )
3811, 37eqeltrd 2507 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( A [,] B )  /\  x  =  ( z  +  T ) )  ->  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )
39383exp 1204 . . . . . . . 8  |-  ( ph  ->  ( z  e.  ( A [,] B )  ->  ( x  =  ( z  +  T
)  ->  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) ) ) )
4039adantr 466 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  E. z  e.  ( A [,] B
) x  =  ( z  +  T ) ) )  ->  (
z  e.  ( A [,] B )  -> 
( x  =  ( z  +  T )  ->  x  e.  ( ( A  +  T
) [,] ( B  +  T ) ) ) ) )
419, 10, 40rexlimd 2906 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  E. z  e.  ( A [,] B
) x  =  ( z  +  T ) ) )  ->  ( E. z  e.  ( A [,] B ) x  =  ( z  +  T )  ->  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) ) )
424, 41mpd 15 . . . . 5  |-  ( (
ph  /\  ( x  e.  CC  /\  E. z  e.  ( A [,] B
) x  =  ( z  +  T ) ) )  ->  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )
433, 42sylan2b 477 . . . 4  |-  ( (
ph  /\  x  e.  { w  e.  CC  |  E. z  e.  ( A [,] B ) w  =  ( z  +  T ) } )  ->  x  e.  ( ( A  +  T
) [,] ( B  +  T ) ) )
4431adantr 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  ( A  +  T )  e.  RR )
4533adantr 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  ( B  +  T )  e.  RR )
46 simpr 462 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )
47 eliccre 37552 . . . . . . 7  |-  ( ( ( A  +  T
)  e.  RR  /\  ( B  +  T
)  e.  RR  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T
) ) )  ->  x  e.  RR )
4844, 45, 46, 47syl3anc 1264 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  x  e.  RR )
4948recnd 9676 . . . . 5  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  x  e.  CC )
5012adantr 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  A  e.  RR )
5113adantr 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  B  e.  RR )
5216adantr 466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  T  e.  RR )
5348, 52resubcld 10054 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  (
x  -  T )  e.  RR )
5412recnd 9676 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  CC )
5516recnd 9676 . . . . . . . . . . 11  |-  ( ph  ->  T  e.  CC )
5654, 55pncand 9994 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  +  T )  -  T
)  =  A )
5756eqcomd 2430 . . . . . . . . 9  |-  ( ph  ->  A  =  ( ( A  +  T )  -  T ) )
5857adantr 466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  A  =  ( ( A  +  T )  -  T ) )
59 elicc2 11706 . . . . . . . . . . . 12  |-  ( ( ( A  +  T
)  e.  RR  /\  ( B  +  T
)  e.  RR )  ->  ( x  e.  ( ( A  +  T ) [,] ( B  +  T )
)  <->  ( x  e.  RR  /\  ( A  +  T )  <_  x  /\  x  <_  ( B  +  T )
) ) )
6044, 45, 59syl2anc 665 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  (
x  e.  ( ( A  +  T ) [,] ( B  +  T ) )  <->  ( x  e.  RR  /\  ( A  +  T )  <_  x  /\  x  <_  ( B  +  T )
) ) )
6146, 60mpbid 213 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  (
x  e.  RR  /\  ( A  +  T
)  <_  x  /\  x  <_  ( B  +  T ) ) )
6261simp2d 1018 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  ( A  +  T )  <_  x )
6344, 48, 52, 62lesub1dd 10236 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  (
( A  +  T
)  -  T )  <_  ( x  -  T ) )
6458, 63eqbrtrd 4444 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  A  <_  ( x  -  T
) )
6561simp3d 1019 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  x  <_  ( B  +  T
) )
6648, 45, 52, 65lesub1dd 10236 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  (
x  -  T )  <_  ( ( B  +  T )  -  T ) )
6713recnd 9676 . . . . . . . . . 10  |-  ( ph  ->  B  e.  CC )
6867, 55pncand 9994 . . . . . . . . 9  |-  ( ph  ->  ( ( B  +  T )  -  T
)  =  B )
6968adantr 466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  (
( B  +  T
)  -  T )  =  B )
7066, 69breqtrd 4448 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  (
x  -  T )  <_  B )
7150, 51, 53, 64, 70eliccd 37550 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  (
x  -  T )  e.  ( A [,] B ) )
7255adantr 466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  T  e.  CC )
7349, 72npcand 9997 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  (
( x  -  T
)  +  T )  =  x )
7473eqcomd 2430 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  x  =  ( ( x  -  T )  +  T ) )
75 oveq1 6312 . . . . . . . 8  |-  ( z  =  ( x  -  T )  ->  (
z  +  T )  =  ( ( x  -  T )  +  T ) )
7675eqeq2d 2436 . . . . . . 7  |-  ( z  =  ( x  -  T )  ->  (
x  =  ( z  +  T )  <->  x  =  ( ( x  -  T )  +  T
) ) )
7776rspcev 3182 . . . . . 6  |-  ( ( ( x  -  T
)  e.  ( A [,] B )  /\  x  =  ( (
x  -  T )  +  T ) )  ->  E. z  e.  ( A [,] B ) x  =  ( z  +  T ) )
7871, 74, 77syl2anc 665 . . . . 5  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  E. z  e.  ( A [,] B
) x  =  ( z  +  T ) )
7949, 78, 3sylanbrc 668 . . . 4  |-  ( (
ph  /\  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) )  ->  x  e.  { w  e.  CC  |  E. z  e.  ( A [,] B ) w  =  ( z  +  T ) } )
8043, 79impbida 840 . . 3  |-  ( ph  ->  ( x  e.  {
w  e.  CC  |  E. z  e.  ( A [,] B ) w  =  ( z  +  T ) }  <->  x  e.  ( ( A  +  T ) [,] ( B  +  T )
) ) )
8180eqrdv 2419 . 2  |-  ( ph  ->  { w  e.  CC  |  E. z  e.  ( A [,] B ) w  =  ( z  +  T ) }  =  ( ( A  +  T ) [,] ( B  +  T
) ) )
8281eqcomd 2430 1  |-  ( ph  ->  ( ( A  +  T ) [,] ( B  +  T )
)  =  { w  e.  CC  |  E. z  e.  ( A [,] B
) w  =  ( z  +  T ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   E.wrex 2772   {crab 2775   class class class wbr 4423  (class class class)co 6305   CCcc 9544   RRcr 9545    + caddc 9549    <_ cle 9683    - cmin 9867   [,]cicc 11645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-icc 11649
This theorem is referenced by:  itgiccshift  37797  itgperiod  37798
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