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Theorem dvhvaddcbv 31814
Description: Change bound variables to isolate them later. (Contributed by NM, 3-Nov-2013.)
Hypothesis
Ref Expression
dvhvaddval.a  |-  .+  =  ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E ) 
|->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f )  .+^  ( 2nd `  g ) ) >.
)
Assertion
Ref Expression
dvhvaddcbv  |-  .+  =  ( h  e.  ( T  X.  E ) ,  i  e.  ( T  X.  E )  |->  <.
( ( 1st `  h
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) >. )
Distinct variable groups:    f, g, h, i, E    .+^ , f, g, h, i    T, f, g, h, i
Allowed substitution hints:    .+ ( f, g, h, i)

Proof of Theorem dvhvaddcbv
StepHypRef Expression
1 dvhvaddval.a . 2  |-  .+  =  ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E ) 
|->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f )  .+^  ( 2nd `  g ) ) >.
)
2 fveq2 5720 . . . . 5  |-  ( f  =  h  ->  ( 1st `  f )  =  ( 1st `  h
) )
32coeq1d 5026 . . . 4  |-  ( f  =  h  ->  (
( 1st `  f
)  o.  ( 1st `  g ) )  =  ( ( 1st `  h
)  o.  ( 1st `  g ) ) )
4 fveq2 5720 . . . . 5  |-  ( f  =  h  ->  ( 2nd `  f )  =  ( 2nd `  h
) )
54oveq1d 6088 . . . 4  |-  ( f  =  h  ->  (
( 2nd `  f
)  .+^  ( 2nd `  g
) )  =  ( ( 2nd `  h
)  .+^  ( 2nd `  g
) ) )
63, 5opeq12d 3984 . . 3  |-  ( f  =  h  ->  <. (
( 1st `  f
)  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f
)  .+^  ( 2nd `  g
) ) >.  =  <. ( ( 1st `  h
)  o.  ( 1st `  g ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  g
) ) >. )
7 fveq2 5720 . . . . 5  |-  ( g  =  i  ->  ( 1st `  g )  =  ( 1st `  i
) )
87coeq2d 5027 . . . 4  |-  ( g  =  i  ->  (
( 1st `  h
)  o.  ( 1st `  g ) )  =  ( ( 1st `  h
)  o.  ( 1st `  i ) ) )
9 fveq2 5720 . . . . 5  |-  ( g  =  i  ->  ( 2nd `  g )  =  ( 2nd `  i
) )
109oveq2d 6089 . . . 4  |-  ( g  =  i  ->  (
( 2nd `  h
)  .+^  ( 2nd `  g
) )  =  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) )
118, 10opeq12d 3984 . . 3  |-  ( g  =  i  ->  <. (
( 1st `  h
)  o.  ( 1st `  g ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  g
) ) >.  =  <. ( ( 1st `  h
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) >. )
126, 11cbvmpt2v 6144 . 2  |-  ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E )  |->  <. (
( 1st `  f
)  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f
)  .+^  ( 2nd `  g
) ) >. )  =  ( h  e.  ( T  X.  E
) ,  i  e.  ( T  X.  E
)  |->  <. ( ( 1st `  h )  o.  ( 1st `  i ) ) ,  ( ( 2nd `  h )  .+^  ( 2nd `  i ) ) >.
)
131, 12eqtri 2455 1  |-  .+  =  ( h  e.  ( T  X.  E ) ,  i  e.  ( T  X.  E )  |->  <.
( ( 1st `  h
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) >. )
Colors of variables: wff set class
Syntax hints:    = wceq 1652   <.cop 3809    X. cxp 4868    o. ccom 4874   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1stc1st 6339   2ndc2nd 6340
This theorem is referenced by:  dvhvaddval  31815
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-co 4879  df-iota 5410  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078
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