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Theorem oprpiece1res2 18450
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 ressxr 8876 . . . . . 6  |-  RR  C_  RR*
3 oprpiece1.1 . . . . . 6  |-  A  e.  RR
42, 3sselii 3177 . . . . 5  |-  A  e. 
RR*
5 oprpiece1.2 . . . . . 6  |-  B  e.  RR
62, 5sselii 3177 . . . . 5  |-  B  e. 
RR*
7 oprpiece1.3 . . . . 5  |-  A  <_  B
8 ubicc2 10753 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
94, 6, 7, 8mp3an 1277 . . . 4  |-  B  e.  ( A [,] B
)
10 iccss2 10720 . . . 4  |-  ( ( K  e.  ( A [,] B )  /\  B  e.  ( A [,] B ) )  -> 
( K [,] B
)  C_  ( A [,] B ) )
111, 9, 10mp2an 653 . . 3  |-  ( K [,] B )  C_  ( A [,] B )
12 ssid 3197 . . 3  |-  C  C_  C
13 resmpt2 5942 . . 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 ) ) )
1411, 12, 13mp2an 653 . 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 ) )
15 oprpiece1.7 . . 3  |-  F  =  ( x  e.  ( A [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )
1615reseq1i 4951 . 2  |-  ( F  |`  ( ( K [,] B )  X.  C
) )  =  ( ( x  e.  ( A [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )  |`  ( ( K [,] B )  X.  C
) )
17 oprpiece1.12 . . 3  |-  G  =  ( x  e.  ( K [,] B ) ,  y  e.  C  |->  S )
18 eqeq1 2289 . . . . 5  |-  ( R  =  if ( x  <_  K ,  R ,  S )  ->  ( R  =  S  <->  if (
x  <_  K ,  R ,  S )  =  S ) )
19 eqeq1 2289 . . . . 5  |-  ( S  =  if ( x  <_  K ,  R ,  S )  ->  ( S  =  S  <->  if (
x  <_  K ,  R ,  S )  =  S ) )
20 oprpiece1.11 . . . . . . 7  |-  ( y  e.  C  ->  P  =  Q )
2120ad2antlr 707 . . . . . 6  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  P  =  Q )
22 simpr 447 . . . . . . . 8  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  x  <_  K )
233, 5elicc2i 10716 . . . . . . . . . . . . 13  |-  ( K  e.  ( A [,] B )  <->  ( K  e.  RR  /\  A  <_  K  /\  K  <_  B
) )
2423simp1bi 970 . . . . . . . . . . . 12  |-  ( K  e.  ( A [,] B )  ->  K  e.  RR )
251, 24ax-mp 8 . . . . . . . . . . 11  |-  K  e.  RR
2625, 5elicc2i 10716 . . . . . . . . . 10  |-  ( x  e.  ( K [,] B )  <->  ( x  e.  RR  /\  K  <_  x  /\  x  <_  B
) )
2726simp2bi 971 . . . . . . . . 9  |-  ( x  e.  ( K [,] B )  ->  K  <_  x )
2827ad2antrr 706 . . . . . . . 8  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  K  <_  x )
2926simp1bi 970 . . . . . . . . . 10  |-  ( x  e.  ( K [,] B )  ->  x  e.  RR )
3029ad2antrr 706 . . . . . . . . 9  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  x  e.  RR )
31 letri3 8907 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  K  e.  RR )  ->  ( x  =  K  <-> 
( x  <_  K  /\  K  <_  x ) ) )
3230, 25, 31sylancl 643 . . . . . . . 8  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  ( x  =  K  <->  ( x  <_  K  /\  K  <_  x
) ) )
3322, 28, 32mpbir2and 888 . . . . . . 7  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  x  =  K )
34 oprpiece1.9 . . . . . . 7  |-  ( x  =  K  ->  R  =  P )
3533, 34syl 15 . . . . . 6  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  R  =  P )
36 oprpiece1.10 . . . . . . 7  |-  ( x  =  K  ->  S  =  Q )
3733, 36syl 15 . . . . . 6  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  S  =  Q )
3821, 35, 373eqtr4d 2325 . . . . 5  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  x  <_  K )  ->  R  =  S )
39 eqidd 2284 . . . . 5  |-  ( ( ( x  e.  ( K [,] B )  /\  y  e.  C
)  /\  -.  x  <_  K )  ->  S  =  S )
4018, 19, 38, 39ifbothda 3595 . . . 4  |-  ( ( x  e.  ( K [,] B )  /\  y  e.  C )  ->  if ( x  <_  K ,  R ,  S )  =  S )
4140mpt2eq3ia 5913 . . 3  |-  ( x  e.  ( K [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S )
)  =  ( x  e.  ( K [,] B ) ,  y  e.  C  |->  S )
4217, 41eqtr4i 2306 . 2  |-  G  =  ( x  e.  ( K [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )
4314, 16, 423eqtr4i 2313 1  |-  ( F  |`  ( ( K [,] B )  X.  C
) )  =  G
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ifcif 3565   class class class wbr 4023    X. cxp 4687    |` cres 4691  (class class class)co 5858    e. cmpt2 5860   RRcr 8736   RR*cxr 8866    <_ cle 8868   [,]cicc 10659
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-icc 10663
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