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Theorem oprpiece1res1 21576
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 9663 . . . . 5  |-  A  e. 
RR*
3 oprpiece1.2 . . . . . 6  |-  B  e.  RR
43rexri 9663 . . . . 5  |-  B  e. 
RR*
5 oprpiece1.3 . . . . 5  |-  A  <_  B
6 lbicc2 11661 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
72, 4, 5, 6mp3an 1324 . . . 4  |-  A  e.  ( A [,] B
)
8 oprpiece1.6 . . . 4  |-  K  e.  ( A [,] B
)
9 iccss2 11620 . . . 4  |-  ( ( A  e.  ( A [,] B )  /\  K  e.  ( A [,] B ) )  -> 
( A [,] K
)  C_  ( A [,] B ) )
107, 8, 9mp2an 672 . . 3  |-  ( A [,] K )  C_  ( A [,] B )
11 ssid 3518 . . 3  |-  C  C_  C
12 resmpt2 6399 . . 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 672 . 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 5279 . 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 11598 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( K  e.  ( A [,] B )  <->  ( K  e.  RR*  /\  A  <_  K  /\  K  <_  B
) ) )
182, 4, 17mp2an 672 . . . . . . . . 9  |-  ( K  e.  ( A [,] B )  <->  ( K  e.  RR*  /\  A  <_  K  /\  K  <_  B
) )
1918simp1bi 1011 . . . . . . . 8  |-  ( K  e.  ( A [,] B )  ->  K  e.  RR* )
208, 19ax-mp 5 . . . . . . 7  |-  K  e. 
RR*
21 iccleub 11605 . . . . . . 7  |-  ( ( A  e.  RR*  /\  K  e.  RR*  /\  x  e.  ( A [,] K
) )  ->  x  <_  K )
222, 20, 21mp3an12 1314 . . . . . 6  |-  ( x  e.  ( A [,] K )  ->  x  <_  K )
2322iftrued 3952 . . . . 5  |-  ( x  e.  ( A [,] K )  ->  if ( x  <_  K ,  R ,  S )  =  R )
2423adantr 465 . . . 4  |-  ( ( x  e.  ( A [,] K )  /\  y  e.  C )  ->  if ( x  <_  K ,  R ,  S )  =  R )
2524mpt2eq3ia 6361 . . 3  |-  ( x  e.  ( A [,] K ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S )
)  =  ( x  e.  ( A [,] K ) ,  y  e.  C  |->  R )
2616, 25eqtr4i 2489 . 2  |-  G  =  ( x  e.  ( A [,] K ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )
2713, 15, 263eqtr4i 2496 1  |-  ( F  |`  ( ( A [,] K )  X.  C
) )  =  G
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ w3a 973    = wceq 1395    e. wcel 1819   _Vcvv 3109    C_ wss 3471   ifcif 3944   class class class wbr 4456    X. cxp 5006    |` cres 5010  (class class class)co 6296    |-> cmpt2 6298   RRcr 9508   RR*cxr 9644    <_ cle 9646   [,]cicc 11557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-pre-lttri 9583  ax-pre-lttrn 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-icc 11561
This theorem is referenced by: (None)
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