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Theorem oprpiece1res2 21746
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 9678 . . . . 5  |-  A  e. 
RR*
4 oprpiece1.2 . . . . . 6  |-  B  e.  RR
54rexri 9678 . . . . 5  |-  B  e. 
RR*
6 oprpiece1.3 . . . . 5  |-  A  <_  B
7 ubicc2 11693 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
83, 5, 6, 7mp3an 1328 . . . 4  |-  B  e.  ( A [,] B
)
9 iccss2 11651 . . . 4  |-  ( ( K  e.  ( A [,] B )  /\  B  e.  ( A [,] B ) )  -> 
( K [,] B
)  C_  ( A [,] B ) )
101, 8, 9mp2an 672 . . 3  |-  ( K [,] B )  C_  ( A [,] B )
11 ssid 3463 . . 3  |-  C  C_  C
12 resmpt2 6383 . . 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 672 . 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 5092 . 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 oprpiece1.11 . . . . . . 7  |-  ( y  e.  C  ->  P  =  Q )
1817ad2antlr 727 . . . . . 6  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  P  =  Q )
19 simpr 461 . . . . . . . 8  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  x  <_  K )
202, 4elicc2i 11646 . . . . . . . . . . . . 13  |-  ( K  e.  ( A [,] B )  <->  ( K  e.  RR  /\  A  <_  K  /\  K  <_  B
) )
2120simp1bi 1014 . . . . . . . . . . . 12  |-  ( K  e.  ( A [,] B )  ->  K  e.  RR )
221, 21ax-mp 5 . . . . . . . . . . 11  |-  K  e.  RR
2322, 4elicc2i 11646 . . . . . . . . . 10  |-  ( x  e.  ( K [,] B )  <->  ( x  e.  RR  /\  K  <_  x  /\  x  <_  B
) )
2423simp2bi 1015 . . . . . . . . 9  |-  ( x  e.  ( K [,] B )  ->  K  <_  x )
2524ad2antrr 726 . . . . . . . 8  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  K  <_  x )
2623simp1bi 1014 . . . . . . . . . 10  |-  ( x  e.  ( K [,] B )  ->  x  e.  RR )
2726ad2antrr 726 . . . . . . . . 9  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  x  e.  RR )
28 letri3 9703 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  K  e.  RR )  ->  ( x  =  K  <-> 
( x  <_  K  /\  K  <_  x ) ) )
2927, 22, 28sylancl 662 . . . . . . . 8  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  ( x  =  K  <->  ( x  <_  K  /\  K  <_  x
) ) )
3019, 25, 29mpbir2and 925 . . . . . . 7  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  x  =  K )
31 oprpiece1.9 . . . . . . 7  |-  ( x  =  K  ->  R  =  P )
3230, 31syl 17 . . . . . 6  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  R  =  P )
33 oprpiece1.10 . . . . . . 7  |-  ( x  =  K  ->  S  =  Q )
3430, 33syl 17 . . . . . 6  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  S  =  Q )
3518, 32, 343eqtr4d 2455 . . . . 5  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  R  =  S )
36 eqidd 2405 . . . . 5  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  -.  x  <_  K )  ->  S  =  S )
3735, 36ifeqda 3920 . . . 4  |-  ( ( x  e.  ( K [,] B )  /\  y  e.  C )  ->  if ( x  <_  K ,  R ,  S )  =  S )
3837mpt2eq3ia 6345 . . 3  |-  ( x  e.  ( K [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S )
)  =  ( x  e.  ( K [,] B ) ,  y  e.  C  |->  S )
3916, 38eqtr4i 2436 . 2  |-  G  =  ( x  e.  ( K [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )
4013, 15, 393eqtr4i 2443 1  |-  ( F  |`  ( ( K [,] B )  X.  C
) )  =  G
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407    e. wcel 1844   _Vcvv 3061    C_ wss 3416   ifcif 3887   class class class wbr 4397    X. cxp 4823    |` cres 4827  (class class class)co 6280    |-> cmpt2 6282   RRcr 9523   RR*cxr 9659    <_ cle 9661   [,]cicc 11587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-pre-lttri 9598  ax-pre-lttrn 9599
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-po 4746  df-so 4747  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-1st 6786  df-2nd 6787  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-icc 11591
This theorem is referenced by: (None)
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