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Theorem oprpiece1res1 18465
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
2 rexr 8893 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  RR* )
31, 2ax-mp 8 . . . . 5  |-  A  e. 
RR*
4 oprpiece1.2 . . . . . 6  |-  B  e.  RR
5 rexr 8893 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  RR* )
64, 5ax-mp 8 . . . . 5  |-  B  e. 
RR*
7 oprpiece1.3 . . . . 5  |-  A  <_  B
8 lbicc2 10768 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
93, 6, 7, 8mp3an 1277 . . . 4  |-  A  e.  ( A [,] B
)
10 oprpiece1.6 . . . 4  |-  K  e.  ( A [,] B
)
11 iccss2 10736 . . . 4  |-  ( ( A  e.  ( A [,] B )  /\  K  e.  ( A [,] B ) )  -> 
( A [,] K
)  C_  ( A [,] B ) )
129, 10, 11mp2an 653 . . 3  |-  ( A [,] K )  C_  ( A [,] B )
13 ssid 3210 . . 3  |-  C  C_  C
14 resmpt2 5958 . . 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 ) ) )
1512, 13, 14mp2an 653 . 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 ) )
16 oprpiece1.7 . . 3  |-  F  =  ( x  e.  ( A [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )
1716reseq1i 4967 . 2  |-  ( F  |`  ( ( A [,] K )  X.  C
) )  =  ( ( x  e.  ( A [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )  |`  ( ( A [,] K )  X.  C
) )
18 oprpiece1.8 . . 3  |-  G  =  ( x  e.  ( A [,] K ) ,  y  e.  C  |->  R )
19 elicc1 10716 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( K  e.  ( A [,] B )  <->  ( K  e.  RR*  /\  A  <_  K  /\  K  <_  B
) ) )
203, 6, 19mp2an 653 . . . . . . . . 9  |-  ( K  e.  ( A [,] B )  <->  ( K  e.  RR*  /\  A  <_  K  /\  K  <_  B
) )
2120simp1bi 970 . . . . . . . 8  |-  ( K  e.  ( A [,] B )  ->  K  e.  RR* )
2210, 21ax-mp 8 . . . . . . 7  |-  K  e. 
RR*
23 iccleub 10723 . . . . . . 7  |-  ( ( A  e.  RR*  /\  K  e.  RR*  /\  x  e.  ( A [,] K
) )  ->  x  <_  K )
243, 22, 23mp3an12 1267 . . . . . 6  |-  ( x  e.  ( A [,] K )  ->  x  <_  K )
25 iftrue 3584 . . . . . 6  |-  ( x  <_  K  ->  if ( x  <_  K ,  R ,  S )  =  R )
2624, 25syl 15 . . . . 5  |-  ( x  e.  ( A [,] K )  ->  if ( x  <_  K ,  R ,  S )  =  R )
2726adantr 451 . . . 4  |-  ( ( x  e.  ( A [,] K )  /\  y  e.  C )  ->  if ( x  <_  K ,  R ,  S )  =  R )
2827mpt2eq3ia 5929 . . 3  |-  ( x  e.  ( A [,] K ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S )
)  =  ( x  e.  ( A [,] K ) ,  y  e.  C  |->  R )
2918, 28eqtr4i 2319 . 2  |-  G  =  ( x  e.  ( A [,] K ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )
3015, 17, 293eqtr4i 2326 1  |-  ( F  |`  ( ( A [,] K )  X.  C
) )  =  G
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   ifcif 3578   class class class wbr 4039    X. cxp 4703    |` cres 4707  (class class class)co 5874    e. cmpt2 5876   RRcr 8752   RR*cxr 8882    <_ cle 8884   [,]cicc 10675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-icc 10679
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