Theorem dvhvaddcbv 36378

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

Step	Hyp	Ref	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 6191	. . . . 5 |- ( f = h -> ( 1st ` f ) = ( 1st ` h ) )
3	2	coeq1d 5283	. . . 4 |- ( f = h -> ( ( 1st ` f ) o. ( 1st ` g ) ) = ( ( 1st ` h ) o. ( 1st ` g ) ) )
4		fveq2 6191	. . . . 5 |- ( f = h -> ( 2nd ` f ) = ( 2nd ` h ) )
5	4	oveq1d 6665	. . . 4 |- ( f = h -> ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) = ( ( 2nd ` h ) .+^ ( 2nd ` g ) ) )
6	3, 5	opeq12d 4410	. . 3 |- ( f = h -> <. ( ( 1st ` f ) o. ( 1st ` g ) ) , ( ( 2nd ` f ) .+^ ( 2nd ` g ) ) >. = <. ( ( 1st ` h ) o. ( 1st ` g ) ) , ( ( 2nd ` h ) .+^ ( 2nd ` g ) ) >. )
7		fveq2 6191	. . . . 5 |- ( g = i -> ( 1st ` g ) = ( 1st ` i ) )
8	7	coeq2d 5284	. . . 4 |- ( g = i -> ( ( 1st ` h ) o. ( 1st ` g ) ) = ( ( 1st ` h ) o. ( 1st ` i ) ) )
9		fveq2 6191	. . . . 5 |- ( g = i -> ( 2nd ` g ) = ( 2nd ` i ) )
10	9	oveq2d 6666	. . . 4 |- ( g = i -> ( ( 2nd ` h ) .+^ ( 2nd ` g ) ) = ( ( 2nd ` h ) .+^ ( 2nd ` i ) ) )
11	8, 10	opeq12d 4410	. . 3 |- ( g = i -> <. ( ( 1st ` h ) o. ( 1st ` g ) ) , ( ( 2nd ` h ) .+^ ( 2nd ` g ) ) >. = <. ( ( 1st ` h ) o. ( 1st ` i ) ) , ( ( 2nd ` h ) .+^ ( 2nd ` i ) ) >. )
12	6, 11	cbvmpt2v 6735	. 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 ) ) >. )
13	1, 12	eqtri 2644	1 |- .+ = ( h e. ( T X. E ) , i e. ( T X. E ) |-> <. ( ( 1st ` h ) o. ( 1st ` i ) ) , ( ( 2nd ` h ) .+^ ( 2nd ` i ) ) >. )

Syntax hints:   = wceq 1483   <.cop 4183   X. cxp 5112   o. ccom 5118   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-co 5123  df-iota 5851  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  dvhvaddval  36379