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Theorem dvhvaddcbv 31961
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 5731 . . . . 5  |-  ( f  =  h  ->  ( 1st `  f )  =  ( 1st `  h
) )
32coeq1d 5037 . . . 4  |-  ( f  =  h  ->  (
( 1st `  f
)  o.  ( 1st `  g ) )  =  ( ( 1st `  h
)  o.  ( 1st `  g ) ) )
4 fveq2 5731 . . . . 5  |-  ( f  =  h  ->  ( 2nd `  f )  =  ( 2nd `  h
) )
54oveq1d 6099 . . . 4  |-  ( f  =  h  ->  (
( 2nd `  f
)  .+^  ( 2nd `  g
) )  =  ( ( 2nd `  h
)  .+^  ( 2nd `  g
) ) )
63, 5opeq12d 3994 . . 3  |-  ( f  =  h  ->  <. (
( 1st `  f
)  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f
)  .+^  ( 2nd `  g
) ) >.  =  <. ( ( 1st `  h
)  o.  ( 1st `  g ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  g
) ) >. )
7 fveq2 5731 . . . . 5  |-  ( g  =  i  ->  ( 1st `  g )  =  ( 1st `  i
) )
87coeq2d 5038 . . . 4  |-  ( g  =  i  ->  (
( 1st `  h
)  o.  ( 1st `  g ) )  =  ( ( 1st `  h
)  o.  ( 1st `  i ) ) )
9 fveq2 5731 . . . . 5  |-  ( g  =  i  ->  ( 2nd `  g )  =  ( 2nd `  i
) )
109oveq2d 6100 . . . 4  |-  ( g  =  i  ->  (
( 2nd `  h
)  .+^  ( 2nd `  g
) )  =  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) )
118, 10opeq12d 3994 . . 3  |-  ( g  =  i  ->  <. (
( 1st `  h
)  o.  ( 1st `  g ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  g
) ) >.  =  <. ( ( 1st `  h
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) >. )
126, 11cbvmpt2v 6155 . 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 2458 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 1653   <.cop 3819    X. cxp 4879    o. ccom 4885   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   1stc1st 6350   2ndc2nd 6351
This theorem is referenced by:  dvhvaddval  31962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-co 4890  df-iota 5421  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089
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