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Theorem oprpiece1res2 21180
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 9635 . . . . 5  |-  A  e. 
RR*
4 oprpiece1.2 . . . . . 6  |-  B  e.  RR
54rexri 9635 . . . . 5  |-  B  e. 
RR*
6 oprpiece1.3 . . . . 5  |-  A  <_  B
7 ubicc2 11626 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
83, 5, 6, 7mp3an 1319 . . . 4  |-  B  e.  ( A [,] B
)
9 iccss2 11584 . . . 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 3516 . . 3  |-  C  C_  C
12 resmpt2 6375 . . 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 5260 . 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 2464 . . . . 5  |-  ( R  =  if ( x  <_  K ,  R ,  S )  ->  ( R  =  S  <->  if (
x  <_  K ,  R ,  S )  =  S ) )
18 eqeq1 2464 . . . . 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 726 . . . . . 6  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  P  =  Q )
21 simpr 461 . . . . . . . 8  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  x  <_  K )
222, 4elicc2i 11579 . . . . . . . . . . . . 13  |-  ( K  e.  ( A [,] B )  <->  ( K  e.  RR  /\  A  <_  K  /\  K  <_  B
) )
2322simp1bi 1006 . . . . . . . . . . . 12  |-  ( K  e.  ( A [,] B )  ->  K  e.  RR )
241, 23ax-mp 5 . . . . . . . . . . 11  |-  K  e.  RR
2524, 4elicc2i 11579 . . . . . . . . . 10  |-  ( x  e.  ( K [,] B )  <->  ( x  e.  RR  /\  K  <_  x  /\  x  <_  B
) )
2625simp2bi 1007 . . . . . . . . 9  |-  ( x  e.  ( K [,] B )  ->  K  <_  x )
2726ad2antrr 725 . . . . . . . 8  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  K  <_  x )
2825simp1bi 1006 . . . . . . . . . 10  |-  ( x  e.  ( K [,] B )  ->  x  e.  RR )
2928ad2antrr 725 . . . . . . . . 9  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  x  e.  RR )
30 letri3 9659 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  K  e.  RR )  ->  ( x  =  K  <-> 
( x  <_  K  /\  K  <_  x ) ) )
3129, 24, 30sylancl 662 . . . . . . . 8  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  ( x  =  K  <->  ( x  <_  K  /\  K  <_  x
) ) )
3221, 27, 31mpbir2and 915 . . . . . . 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 2511 . . . . 5  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  R  =  S )
38 eqidd 2461 . . . . 5  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  -.  x  <_  K )  ->  S  =  S )
3917, 18, 37, 38ifbothda 3967 . . . 4  |-  ( ( x  e.  ( K [,] B )  /\  y  e.  C )  ->  if ( x  <_  K ,  R ,  S )  =  S )
4039mpt2eq3ia 6337 . . 3  |-  ( x  e.  ( K [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S )
)  =  ( x  e.  ( K [,] B ) ,  y  e.  C  |->  S )
4116, 40eqtr4i 2492 . 2  |-  G  =  ( x  e.  ( K [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )
4213, 15, 413eqtr4i 2499 1  |-  ( F  |`  ( ( K [,] B )  X.  C
) )  =  G
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3106    C_ wss 3469   ifcif 3932   class class class wbr 4440    X. cxp 4990    |` cres 4994  (class class class)co 6275    |-> cmpt2 6277   RRcr 9480   RR*cxr 9616    <_ cle 9618   [,]cicc 11521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-pre-lttri 9555  ax-pre-lttrn 9556
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-icc 11525
This theorem is referenced by: (None)
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