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Theorem oprpiece1res2 18979
Description: Restriction to the second part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)
Hypotheses
Ref Expression
oprpiece1.1  |-  A  e.  RR
oprpiece1.2  |-  B  e.  RR
oprpiece1.3  |-  A  <_  B
oprpiece1.4  |-  R  e. 
_V
oprpiece1.5  |-  S  e. 
_V
oprpiece1.6  |-  K  e.  ( A [,] B
)
oprpiece1.7  |-  F  =  ( x  e.  ( A [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )
oprpiece1.9  |-  ( x  =  K  ->  R  =  P )
oprpiece1.10  |-  ( x  =  K  ->  S  =  Q )
oprpiece1.11  |-  ( y  e.  C  ->  P  =  Q )
oprpiece1.12  |-  G  =  ( x  e.  ( K [,] B ) ,  y  e.  C  |->  S )
Assertion
Ref Expression
oprpiece1res2  |-  ( F  |`  ( ( K [,] B )  X.  C
) )  =  G
Distinct variable groups:    x, A, y    x, B, y    x, C, y    x, K, y   
x, P    x, Q
Allowed substitution hints:    P( y)    Q( y)    R( x, y)    S( x, y)    F( x, y)    G( x, y)

Proof of Theorem oprpiece1res2
StepHypRef Expression
1 oprpiece1.6 . . . 4  |-  K  e.  ( A [,] B
)
2 oprpiece1.1 . . . . . 6  |-  A  e.  RR
32rexri 9139 . . . . 5  |-  A  e. 
RR*
4 oprpiece1.2 . . . . . 6  |-  B  e.  RR
54rexri 9139 . . . . 5  |-  B  e. 
RR*
6 oprpiece1.3 . . . . 5  |-  A  <_  B
7 ubicc2 11016 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
83, 5, 6, 7mp3an 1280 . . . 4  |-  B  e.  ( A [,] B
)
9 iccss2 10983 . . . 4  |-  ( ( K  e.  ( A [,] B )  /\  B  e.  ( A [,] B ) )  -> 
( K [,] B
)  C_  ( A [,] B ) )
101, 8, 9mp2an 655 . . 3  |-  ( K [,] B )  C_  ( A [,] B )
11 ssid 3369 . . 3  |-  C  C_  C
12 resmpt2 6170 . . 3  |-  ( ( ( K [,] B
)  C_  ( A [,] B )  /\  C  C_  C )  ->  (
( x  e.  ( A [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )  |`  ( ( K [,] B )  X.  C
) )  =  ( x  e.  ( K [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) ) )
1310, 11, 12mp2an 655 . 2  |-  ( ( x  e.  ( A [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )  |`  ( ( K [,] B )  X.  C ) )  =  ( x  e.  ( K [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )
14 oprpiece1.7 . . 3  |-  F  =  ( x  e.  ( A [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )
1514reseq1i 5144 . 2  |-  ( F  |`  ( ( K [,] B )  X.  C
) )  =  ( ( x  e.  ( A [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )  |`  ( ( K [,] B )  X.  C
) )
16 oprpiece1.12 . . 3  |-  G  =  ( x  e.  ( K [,] B ) ,  y  e.  C  |->  S )
17 eqeq1 2444 . . . . 5  |-  ( R  =  if ( x  <_  K ,  R ,  S )  ->  ( R  =  S  <->  if (
x  <_  K ,  R ,  S )  =  S ) )
18 eqeq1 2444 . . . . 5  |-  ( S  =  if ( x  <_  K ,  R ,  S )  ->  ( S  =  S  <->  if (
x  <_  K ,  R ,  S )  =  S ) )
19 oprpiece1.11 . . . . . . 7  |-  ( y  e.  C  ->  P  =  Q )
2019ad2antlr 709 . . . . . 6  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  P  =  Q )
21 simpr 449 . . . . . . . 8  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  x  <_  K )
222, 4elicc2i 10978 . . . . . . . . . . . . 13  |-  ( K  e.  ( A [,] B )  <->  ( K  e.  RR  /\  A  <_  K  /\  K  <_  B
) )
2322simp1bi 973 . . . . . . . . . . . 12  |-  ( K  e.  ( A [,] B )  ->  K  e.  RR )
241, 23ax-mp 8 . . . . . . . . . . 11  |-  K  e.  RR
2524, 4elicc2i 10978 . . . . . . . . . 10  |-  ( x  e.  ( K [,] B )  <->  ( x  e.  RR  /\  K  <_  x  /\  x  <_  B
) )
2625simp2bi 974 . . . . . . . . 9  |-  ( x  e.  ( K [,] B )  ->  K  <_  x )
2726ad2antrr 708 . . . . . . . 8  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  K  <_  x )
2825simp1bi 973 . . . . . . . . . 10  |-  ( x  e.  ( K [,] B )  ->  x  e.  RR )
2928ad2antrr 708 . . . . . . . . 9  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  x  e.  RR )
30 letri3 9162 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  K  e.  RR )  ->  ( x  =  K  <-> 
( x  <_  K  /\  K  <_  x ) ) )
3129, 24, 30sylancl 645 . . . . . . . 8  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  ( x  =  K  <->  ( x  <_  K  /\  K  <_  x
) ) )
3221, 27, 31mpbir2and 890 . . . . . . 7  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  x  =  K )
33 oprpiece1.9 . . . . . . 7  |-  ( x  =  K  ->  R  =  P )
3432, 33syl 16 . . . . . 6  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  R  =  P )
35 oprpiece1.10 . . . . . . 7  |-  ( x  =  K  ->  S  =  Q )
3632, 35syl 16 . . . . . 6  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  S  =  Q )
3720, 34, 363eqtr4d 2480 . . . . 5  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  R  =  S )
38 eqidd 2439 . . . . 5  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  -.  x  <_  K )  ->  S  =  S )
3917, 18, 37, 38ifbothda 3771 . . . 4  |-  ( ( x  e.  ( K [,] B )  /\  y  e.  C )  ->  if ( x  <_  K ,  R ,  S )  =  S )
4039mpt2eq3ia 6141 . . 3  |-  ( x  e.  ( K [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S )
)  =  ( x  e.  ( K [,] B ) ,  y  e.  C  |->  S )
4116, 40eqtr4i 2461 . 2  |-  G  =  ( x  e.  ( K [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )
4213, 15, 413eqtr4i 2468 1  |-  ( F  |`  ( ( K [,] B )  X.  C
) )  =  G
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   ifcif 3741   class class class wbr 4214    X. cxp 4878    |` cres 4882  (class class class)co 6083    e. cmpt2 6085   RRcr 8991   RR*cxr 9121    <_ cle 9123   [,]cicc 10921
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-pre-lttri 9066  ax-pre-lttrn 9067
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-icc 10925
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