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Theorem dvhvaddcbv 31279
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 5525 . . . . 5  |-  ( f  =  h  ->  ( 1st `  f )  =  ( 1st `  h
) )
32coeq1d 4845 . . . 4  |-  ( f  =  h  ->  (
( 1st `  f
)  o.  ( 1st `  g ) )  =  ( ( 1st `  h
)  o.  ( 1st `  g ) ) )
4 fveq2 5525 . . . . 5  |-  ( f  =  h  ->  ( 2nd `  f )  =  ( 2nd `  h
) )
54oveq1d 5873 . . . 4  |-  ( f  =  h  ->  (
( 2nd `  f
)  .+^  ( 2nd `  g
) )  =  ( ( 2nd `  h
)  .+^  ( 2nd `  g
) ) )
63, 5opeq12d 3804 . . 3  |-  ( f  =  h  ->  <. (
( 1st `  f
)  o.  ( 1st `  g ) ) ,  ( ( 2nd `  f
)  .+^  ( 2nd `  g
) ) >.  =  <. ( ( 1st `  h
)  o.  ( 1st `  g ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  g
) ) >. )
7 fveq2 5525 . . . . 5  |-  ( g  =  i  ->  ( 1st `  g )  =  ( 1st `  i
) )
87coeq2d 4846 . . . 4  |-  ( g  =  i  ->  (
( 1st `  h
)  o.  ( 1st `  g ) )  =  ( ( 1st `  h
)  o.  ( 1st `  i ) ) )
9 fveq2 5525 . . . . 5  |-  ( g  =  i  ->  ( 2nd `  g )  =  ( 2nd `  i
) )
109oveq2d 5874 . . . 4  |-  ( g  =  i  ->  (
( 2nd `  h
)  .+^  ( 2nd `  g
) )  =  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) )
118, 10opeq12d 3804 . . 3  |-  ( g  =  i  ->  <. (
( 1st `  h
)  o.  ( 1st `  g ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  g
) ) >.  =  <. ( ( 1st `  h
)  o.  ( 1st `  i ) ) ,  ( ( 2nd `  h
)  .+^  ( 2nd `  i
) ) >. )
126, 11cbvmpt2v 5926 . 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 2303 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 1623   <.cop 3643    X. cxp 4687    o. ccom 4693   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121
This theorem is referenced by:  dvhvaddval  31280
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-co 4698  df-iota 5219  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863
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