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

Theorem offval 6528
Description: Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
Hypotheses
Ref Expression
offval.1  |-  ( ph  ->  F  Fn  A )
offval.2  |-  ( ph  ->  G  Fn  B )
offval.3  |-  ( ph  ->  A  e.  V )
offval.4  |-  ( ph  ->  B  e.  W )
offval.5  |-  ( A  i^i  B )  =  S
offval.6  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  C )
offval.7  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  D )
Assertion
Ref Expression
offval  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  S  |->  ( C R D ) ) )
Distinct variable groups:    x, A    x, F    x, G    ph, x    x, S    x, R
Allowed substitution hints:    B( x)    C( x)    D( x)    V( x)    W( x)

Proof of Theorem offval
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 offval.1 . . . 4  |-  ( ph  ->  F  Fn  A )
2 offval.3 . . . 4  |-  ( ph  ->  A  e.  V )
3 fnex 6120 . . . 4  |-  ( ( F  Fn  A  /\  A  e.  V )  ->  F  e.  _V )
41, 2, 3syl2anc 661 . . 3  |-  ( ph  ->  F  e.  _V )
5 offval.2 . . . 4  |-  ( ph  ->  G  Fn  B )
6 offval.4 . . . 4  |-  ( ph  ->  B  e.  W )
7 fnex 6120 . . . 4  |-  ( ( G  Fn  B  /\  B  e.  W )  ->  G  e.  _V )
85, 6, 7syl2anc 661 . . 3  |-  ( ph  ->  G  e.  _V )
9 fndm 5666 . . . . . . . 8  |-  ( F  Fn  A  ->  dom  F  =  A )
101, 9syl 16 . . . . . . 7  |-  ( ph  ->  dom  F  =  A )
11 fndm 5666 . . . . . . . 8  |-  ( G  Fn  B  ->  dom  G  =  B )
125, 11syl 16 . . . . . . 7  |-  ( ph  ->  dom  G  =  B )
1310, 12ineq12d 3683 . . . . . 6  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  ( A  i^i  B ) )
14 offval.5 . . . . . 6  |-  ( A  i^i  B )  =  S
1513, 14syl6eq 2498 . . . . 5  |-  ( ph  ->  ( dom  F  i^i  dom 
G )  =  S )
1615mpteq1d 4514 . . . 4  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) )  =  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) ) )
17 inex1g 4576 . . . . . 6  |-  ( A  e.  V  ->  ( A  i^i  B )  e. 
_V )
1814, 17syl5eqelr 2534 . . . . 5  |-  ( A  e.  V  ->  S  e.  _V )
19 mptexg 6123 . . . . 5  |-  ( S  e.  _V  ->  (
x  e.  S  |->  ( ( F `  x
) R ( G `
 x ) ) )  e.  _V )
202, 18, 193syl 20 . . . 4  |-  ( ph  ->  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) )  e.  _V )
2116, 20eqeltrd 2529 . . 3  |-  ( ph  ->  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) )  e. 
_V )
22 dmeq 5189 . . . . . 6  |-  ( f  =  F  ->  dom  f  =  dom  F )
23 dmeq 5189 . . . . . 6  |-  ( g  =  G  ->  dom  g  =  dom  G )
2422, 23ineqan12d 3684 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( dom  f  i^i 
dom  g )  =  ( dom  F  i^i  dom 
G ) )
25 fveq1 5851 . . . . . 6  |-  ( f  =  F  ->  (
f `  x )  =  ( F `  x ) )
26 fveq1 5851 . . . . . 6  |-  ( g  =  G  ->  (
g `  x )  =  ( G `  x ) )
2725, 26oveqan12d 6296 . . . . 5  |-  ( ( f  =  F  /\  g  =  G )  ->  ( ( f `  x ) R ( g `  x ) )  =  ( ( F `  x ) R ( G `  x ) ) )
2824, 27mpteq12dv 4511 . . . 4  |-  ( ( f  =  F  /\  g  =  G )  ->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x ) R ( g `  x ) ) )  =  ( x  e.  ( dom  F  i^i  dom 
G )  |->  ( ( F `  x ) R ( G `  x ) ) ) )
29 df-of 6521 . . . 4  |-  oF R  =  ( f  e.  _V ,  g  e.  _V  |->  ( x  e.  ( dom  f  i^i  dom  g )  |->  ( ( f `  x
) R ( g `
 x ) ) ) )
3028, 29ovmpt2ga 6413 . . 3  |-  ( ( F  e.  _V  /\  G  e.  _V  /\  (
x  e.  ( dom 
F  i^i  dom  G ) 
|->  ( ( F `  x ) R ( G `  x ) ) )  e.  _V )  ->  ( F  oF R G )  =  ( x  e.  ( dom  F  i^i  dom 
G )  |->  ( ( F `  x ) R ( G `  x ) ) ) )
314, 8, 21, 30syl3anc 1227 . 2  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  ( dom  F  i^i  dom  G )  |->  ( ( F `
 x ) R ( G `  x
) ) ) )
3214eleq2i 2519 . . . . 5  |-  ( x  e.  ( A  i^i  B )  <->  x  e.  S
)
33 elin 3669 . . . . 5  |-  ( x  e.  ( A  i^i  B )  <->  ( x  e.  A  /\  x  e.  B ) )
3432, 33bitr3i 251 . . . 4  |-  ( x  e.  S  <->  ( x  e.  A  /\  x  e.  B ) )
35 offval.6 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  C )
3635adantrr 716 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  x  e.  B ) )  -> 
( F `  x
)  =  C )
37 offval.7 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  ( G `  x )  =  D )
3837adantrl 715 . . . . 5  |-  ( (
ph  /\  ( x  e.  A  /\  x  e.  B ) )  -> 
( G `  x
)  =  D )
3936, 38oveq12d 6295 . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  x  e.  B ) )  -> 
( ( F `  x ) R ( G `  x ) )  =  ( C R D ) )
4034, 39sylan2b 475 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  (
( F `  x
) R ( G `
 x ) )  =  ( C R D ) )
4140mpteq2dva 4519 . 2  |-  ( ph  ->  ( x  e.  S  |->  ( ( F `  x ) R ( G `  x ) ) )  =  ( x  e.  S  |->  ( C R D ) ) )
4231, 16, 413eqtrd 2486 1  |-  ( ph  ->  ( F  oF R G )  =  ( x  e.  S  |->  ( C R D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802   _Vcvv 3093    i^i cin 3457    |-> cmpt 4491   dom cdm 4985    Fn wfn 5569   ` cfv 5574  (class class class)co 6277    oFcof 6519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6521
This theorem is referenced by:  ofval  6530  offn  6532  off  6535  ofres  6536  offval2  6537  suppssof1OLD  6540  ofco  6541  offveqb  6543  suppssof1  6931  o1rlimmul  13415  gsumbagdiaglem  17895  evlslem1  18052  psrplusgpropd  18145  frlmipval  18677  frlmphllem  18678  frlmphl  18679  mat1dimscm  18844  rrxcph  21690  rrxds  21691  mbfadd  21934  mbfsub  21935  mbfmullem2  21997  mbfmul  21999  bddmulibl  22111  dvcmulf  22214  ofrn2  27345  off2  27346  ofresid  27347  offval2f  27371  ofcof  27972  plymul02  28369  signsplypnf  28373  signsply0  28374  itg2addnc  30037  ftc1anclem8  30065  dflinc2  32721
  Copyright terms: Public domain W3C validator