MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oprpiece1res1 Structured version   Unicode version

Theorem oprpiece1res1 21186
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 9642 . . . . 5  |-  A  e. 
RR*
3 oprpiece1.2 . . . . . 6  |-  B  e.  RR
43rexri 9642 . . . . 5  |-  B  e. 
RR*
5 oprpiece1.3 . . . . 5  |-  A  <_  B
6 lbicc2 11632 . . . . 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 11591 . . . 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 3523 . . 3  |-  C  C_  C
12 resmpt2 6382 . . 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 5267 . 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 11569 . . . . . . . . . 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 11576 . . . . . . 7  |-  ( ( A  e.  RR*  /\  K  e.  RR*  /\  x  e.  ( A [,] K
) )  ->  x  <_  K )
222, 20, 21mp3an12 1314 . . . . . 6  |-  ( x  e.  ( A [,] K )  ->  x  <_  K )
23 iftrue 3945 . . . . . 6  |-  ( x  <_  K  ->  if ( x  <_  K ,  R ,  S )  =  R )
2422, 23syl 16 . . . . 5  |-  ( x  e.  ( A [,] K )  ->  if ( x  <_  K ,  R ,  S )  =  R )
2524adantr 465 . . . 4  |-  ( ( x  e.  ( A [,] K )  /\  y  e.  C )  ->  if ( x  <_  K ,  R ,  S )  =  R )
2625mpt2eq3ia 6344 . . 3  |-  ( x  e.  ( A [,] K ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S )
)  =  ( x  e.  ( A [,] K ) ,  y  e.  C  |->  R )
2716, 26eqtr4i 2499 . 2  |-  G  =  ( x  e.  ( A [,] K ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )
2813, 15, 273eqtr4i 2506 1  |-  ( F  |`  ( ( A [,] K )  X.  C
) )  =  G
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   ifcif 3939   class class class wbr 4447    X. cxp 4997    |` cres 5001  (class class class)co 6282    |-> cmpt2 6284   RRcr 9487   RR*cxr 9623    <_ cle 9625   [,]cicc 11528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-pre-lttri 9562  ax-pre-lttrn 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-icc 11532
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator