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Theorem dvhvaddcbv 34728
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 5879 . . . . 5  |-  ( f  =  h  ->  ( 1st `  f )  =  ( 1st `  h
) )
32coeq1d 5001 . . . 4  |-  ( f  =  h  ->  (
( 1st `  f
)  o.  ( 1st `  g ) )  =  ( ( 1st `  h
)  o.  ( 1st `  g ) ) )
4 fveq2 5879 . . . . 5  |-  ( f  =  h  ->  ( 2nd `  f )  =  ( 2nd `  h
) )
54oveq1d 6323 . . . 4  |-  ( f  =  h  ->  (
( 2nd `  f
)  .+^  ( 2nd `  g
) )  =  ( ( 2nd `  h
)  .+^  ( 2nd `  g
) ) )
63, 5opeq12d 4166 . . 3  |-  ( f  =  h  ->  <. (
( 1st `  f
)  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f
)  .+^  ( 2nd `  g
) ) >.  =  <. ( ( 1st `  h
)  o.  ( 1st `  g ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  g
) ) >. )
7 fveq2 5879 . . . . 5  |-  ( g  =  i  ->  ( 1st `  g )  =  ( 1st `  i
) )
87coeq2d 5002 . . . 4  |-  ( g  =  i  ->  (
( 1st `  h
)  o.  ( 1st `  g ) )  =  ( ( 1st `  h
)  o.  ( 1st `  i ) ) )
9 fveq2 5879 . . . . 5  |-  ( g  =  i  ->  ( 2nd `  g )  =  ( 2nd `  i
) )
109oveq2d 6324 . . . 4  |-  ( g  =  i  ->  (
( 2nd `  h
)  .+^  ( 2nd `  g
) )  =  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) )
118, 10opeq12d 4166 . . 3  |-  ( g  =  i  ->  <. (
( 1st `  h
)  o.  ( 1st `  g ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  g
) ) >.  =  <. ( ( 1st `  h
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) >. )
126, 11cbvmpt2v 6390 . 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 2493 1  |-  .+  =  ( h  e.  ( T  X.  E ) ,  i  e.  ( T  X.  E )  |->  <.
( ( 1st `  h
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) >. )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452   <.cop 3965    X. cxp 4837    o. ccom 4843   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   1stc1st 6810   2ndc2nd 6811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-co 4848  df-iota 5553  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313
This theorem is referenced by:  dvhvaddval  34729
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