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Theorem oprpiece1res1 18976
Description: Restriction to the first 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 ) )
oprpiece1.8  |-  G  =  ( x  e.  ( A [,] K ) ,  y  e.  C  |->  R )
Assertion
Ref Expression
oprpiece1res1  |-  ( F  |`  ( ( A [,] K )  X.  C
) )  =  G
Distinct variable groups:    x, A, y    x, B, y    x, C, y    x, K, y
Allowed substitution hints:    R( x, y)    S( x, y)    F( x, y)    G( x, y)

Proof of Theorem oprpiece1res1
StepHypRef Expression
1 oprpiece1.1 . . . . . 6  |-  A  e.  RR
21rexri 9137 . . . . 5  |-  A  e. 
RR*
3 oprpiece1.2 . . . . . 6  |-  B  e.  RR
43rexri 9137 . . . . 5  |-  B  e. 
RR*
5 oprpiece1.3 . . . . 5  |-  A  <_  B
6 lbicc2 11013 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
72, 4, 5, 6mp3an 1279 . . . 4  |-  A  e.  ( A [,] B
)
8 oprpiece1.6 . . . 4  |-  K  e.  ( A [,] B
)
9 iccss2 10981 . . . 4  |-  ( ( A  e.  ( A [,] B )  /\  K  e.  ( A [,] B ) )  -> 
( A [,] K
)  C_  ( A [,] B ) )
107, 8, 9mp2an 654 . . 3  |-  ( A [,] K )  C_  ( A [,] B )
11 ssid 3367 . . 3  |-  C  C_  C
12 resmpt2 6168 . . 3  |-  ( ( ( A [,] K
)  C_  ( A [,] B )  /\  C  C_  C )  ->  (
( x  e.  ( A [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )  |`  ( ( A [,] K )  X.  C
) )  =  ( x  e.  ( A [,] K ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) ) )
1310, 11, 12mp2an 654 . 2  |-  ( ( x  e.  ( A [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )  |`  ( ( A [,] K )  X.  C ) )  =  ( x  e.  ( A [,] K ) ,  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 5142 . 2  |-  ( F  |`  ( ( A [,] K )  X.  C
) )  =  ( ( x  e.  ( A [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )  |`  ( ( A [,] K )  X.  C
) )
16 oprpiece1.8 . . 3  |-  G  =  ( x  e.  ( A [,] K ) ,  y  e.  C  |->  R )
17 elicc1 10960 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( K  e.  ( A [,] B )  <->  ( K  e.  RR*  /\  A  <_  K  /\  K  <_  B
) ) )
182, 4, 17mp2an 654 . . . . . . . . 9  |-  ( K  e.  ( A [,] B )  <->  ( K  e.  RR*  /\  A  <_  K  /\  K  <_  B
) )
1918simp1bi 972 . . . . . . . 8  |-  ( K  e.  ( A [,] B )  ->  K  e.  RR* )
208, 19ax-mp 8 . . . . . . 7  |-  K  e. 
RR*
21 iccleub 10967 . . . . . . 7  |-  ( ( A  e.  RR*  /\  K  e.  RR*  /\  x  e.  ( A [,] K
) )  ->  x  <_  K )
222, 20, 21mp3an12 1269 . . . . . 6  |-  ( x  e.  ( A [,] K )  ->  x  <_  K )
23 iftrue 3745 . . . . . 6  |-  ( x  <_  K  ->  if ( x  <_  K ,  R ,  S )  =  R )
2422, 23syl 16 . . . . 5  |-  ( x  e.  ( A [,] K )  ->  if ( x  <_  K ,  R ,  S )  =  R )
2524adantr 452 . . . 4  |-  ( ( x  e.  ( A [,] K )  /\  y  e.  C )  ->  if ( x  <_  K ,  R ,  S )  =  R )
2625mpt2eq3ia 6139 . . 3  |-  ( x  e.  ( A [,] K ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S )
)  =  ( x  e.  ( A [,] K ) ,  y  e.  C  |->  R )
2716, 26eqtr4i 2459 . 2  |-  G  =  ( x  e.  ( A [,] K ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )
2813, 15, 273eqtr4i 2466 1  |-  ( F  |`  ( ( A [,] K )  X.  C
) )  =  G
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2956    C_ wss 3320   ifcif 3739   class class class wbr 4212    X. cxp 4876    |` cres 4880  (class class class)co 6081    e. cmpt2 6083   RRcr 8989   RR*cxr 9119    <_ cle 9121   [,]cicc 10919
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-pre-lttri 9064  ax-pre-lttrn 9065
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-icc 10923
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