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Theorem dvhvaddcbv 31205
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 5669 . . . . 5  |-  ( f  =  h  ->  ( 1st `  f )  =  ( 1st `  h
) )
32coeq1d 4975 . . . 4  |-  ( f  =  h  ->  (
( 1st `  f
)  o.  ( 1st `  g ) )  =  ( ( 1st `  h
)  o.  ( 1st `  g ) ) )
4 fveq2 5669 . . . . 5  |-  ( f  =  h  ->  ( 2nd `  f )  =  ( 2nd `  h
) )
54oveq1d 6036 . . . 4  |-  ( f  =  h  ->  (
( 2nd `  f
)  .+^  ( 2nd `  g
) )  =  ( ( 2nd `  h
)  .+^  ( 2nd `  g
) ) )
63, 5opeq12d 3935 . . 3  |-  ( f  =  h  ->  <. (
( 1st `  f
)  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f
)  .+^  ( 2nd `  g
) ) >.  =  <. ( ( 1st `  h
)  o.  ( 1st `  g ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  g
) ) >. )
7 fveq2 5669 . . . . 5  |-  ( g  =  i  ->  ( 1st `  g )  =  ( 1st `  i
) )
87coeq2d 4976 . . . 4  |-  ( g  =  i  ->  (
( 1st `  h
)  o.  ( 1st `  g ) )  =  ( ( 1st `  h
)  o.  ( 1st `  i ) ) )
9 fveq2 5669 . . . . 5  |-  ( g  =  i  ->  ( 2nd `  g )  =  ( 2nd `  i
) )
109oveq2d 6037 . . . 4  |-  ( g  =  i  ->  (
( 2nd `  h
)  .+^  ( 2nd `  g
) )  =  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) )
118, 10opeq12d 3935 . . 3  |-  ( g  =  i  ->  <. (
( 1st `  h
)  o.  ( 1st `  g ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  g
) ) >.  =  <. ( ( 1st `  h
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) >. )
126, 11cbvmpt2v 6092 . 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 2408 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 1649   <.cop 3761    X. cxp 4817    o. ccom 4823   ` cfv 5395  (class class class)co 6021    e. cmpt2 6023   1stc1st 6287   2ndc2nd 6288
This theorem is referenced by:  dvhvaddval  31206
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-co 4828  df-iota 5359  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026
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