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Theorem oprpiece1res2 21973
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 9690 . . . . 5  |-  A  e. 
RR*
4 oprpiece1.2 . . . . . 6  |-  B  e.  RR
54rexri 9690 . . . . 5  |-  B  e. 
RR*
6 oprpiece1.3 . . . . 5  |-  A  <_  B
7 ubicc2 11746 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
83, 5, 6, 7mp3an 1363 . . . 4  |-  B  e.  ( A [,] B
)
9 iccss2 11702 . . . 4  |-  ( ( K  e.  ( A [,] B )  /\  B  e.  ( A [,] B ) )  -> 
( K [,] B
)  C_  ( A [,] B ) )
101, 8, 9mp2an 677 . . 3  |-  ( K [,] B )  C_  ( A [,] B )
11 ssid 3450 . . 3  |-  C  C_  C
12 resmpt2 6391 . . 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 677 . 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 5100 . 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 732 . . . . . 6  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  P  =  Q )
19 simpr 463 . . . . . . . 8  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  x  <_  K )
202, 4elicc2i 11697 . . . . . . . . . . . . 13  |-  ( K  e.  ( A [,] B )  <->  ( K  e.  RR  /\  A  <_  K  /\  K  <_  B
) )
2120simp1bi 1022 . . . . . . . . . . . 12  |-  ( K  e.  ( A [,] B )  ->  K  e.  RR )
221, 21ax-mp 5 . . . . . . . . . . 11  |-  K  e.  RR
2322, 4elicc2i 11697 . . . . . . . . . 10  |-  ( x  e.  ( K [,] B )  <->  ( x  e.  RR  /\  K  <_  x  /\  x  <_  B
) )
2423simp2bi 1023 . . . . . . . . 9  |-  ( x  e.  ( K [,] B )  ->  K  <_  x )
2524ad2antrr 731 . . . . . . . 8  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  K  <_  x )
2623simp1bi 1022 . . . . . . . . . 10  |-  ( x  e.  ( K [,] B )  ->  x  e.  RR )
2726ad2antrr 731 . . . . . . . . 9  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  x  e.  RR )
28 letri3 9716 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  K  e.  RR )  ->  ( x  =  K  <-> 
( x  <_  K  /\  K  <_  x ) ) )
2927, 22, 28sylancl 667 . . . . . . . 8  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  ( x  =  K  <->  ( x  <_  K  /\  K  <_  x
) ) )
3019, 25, 29mpbir2and 932 . . . . . . 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 2494 . . . . 5  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  R  =  S )
36 eqidd 2451 . . . . 5  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  -.  x  <_  K )  ->  S  =  S )
3735, 36ifeqda 3913 . . . 4  |-  ( ( x  e.  ( K [,] B )  /\  y  e.  C )  ->  if ( x  <_  K ,  R ,  S )  =  S )
3837mpt2eq3ia 6353 . . 3  |-  ( x  e.  ( K [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S )
)  =  ( x  e.  ( K [,] B ) ,  y  e.  C  |->  S )
3916, 38eqtr4i 2475 . 2  |-  G  =  ( x  e.  ( K [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )
4013, 15, 393eqtr4i 2482 1  |-  ( F  |`  ( ( K [,] B )  X.  C
) )  =  G
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1443    e. wcel 1886   _Vcvv 3044    C_ wss 3403   ifcif 3880   class class class wbr 4401    X. cxp 4831    |` cres 4835  (class class class)co 6288    |-> cmpt2 6290   RRcr 9535   RR*cxr 9671    <_ cle 9673   [,]cicc 11635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-pre-lttri 9610  ax-pre-lttrn 9611
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-po 4754  df-so 4755  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-1st 6790  df-2nd 6791  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-icc 11639
This theorem is referenced by: (None)
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