Theorem dvhvscaval 36388

Description: The scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Nov-2013.)

Hypothesis

Ref	Expression
dvhvscaval.s	|- .x. = ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. )

Assertion

Ref	Expression
dvhvscaval	|- ( ( U e. E /\ F e. ( T X. E ) ) -> ( U .x. F ) = <. ( U ` ( 1st ` F ) ) , ( U o. ( 2nd ` F ) ) >. )

Distinct variable groups:   f, s, E   T, s, f

Allowed substitution hints:   .x. ( f, s)   U( f, s)   F( f, s)

Proof of Theorem dvhvscaval

Dummy variables t g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6190	. . 3 |- ( t = U -> ( t ` ( 1st ` g ) ) = ( U ` ( 1st ` g ) ) )
2		coeq1 5279	. . 3 |- ( t = U -> ( t o. ( 2nd ` g ) ) = ( U o. ( 2nd ` g ) ) )
3	1, 2	opeq12d 4410	. 2 |- ( t = U -> <. ( t ` ( 1st ` g ) ) , ( t o. ( 2nd ` g ) ) >. = <. ( U ` ( 1st ` g ) ) , ( U o. ( 2nd ` g ) ) >. )
4		fveq2 6191	. . . 4 |- ( g = F -> ( 1st ` g ) = ( 1st ` F ) )
5	4	fveq2d 6195	. . 3 |- ( g = F -> ( U ` ( 1st ` g ) ) = ( U ` ( 1st ` F ) ) )
6		fveq2 6191	. . . 4 |- ( g = F -> ( 2nd ` g ) = ( 2nd ` F ) )
7	6	coeq2d 5284	. . 3 |- ( g = F -> ( U o. ( 2nd ` g ) ) = ( U o. ( 2nd ` F ) ) )
8	5, 7	opeq12d 4410	. 2 |- ( g = F -> <. ( U ` ( 1st ` g ) ) , ( U o. ( 2nd ` g ) ) >. = <. ( U ` ( 1st ` F ) ) , ( U o. ( 2nd ` F ) ) >. )
9		dvhvscaval.s	. . 3 |- .x. = ( s e. E , f e. ( T X. E ) |-> <. ( s ` ( 1st ` f ) ) , ( s o. ( 2nd ` f ) ) >. )
10	9	dvhvscacbv 36387	. 2 |- .x. = ( t e. E , g e. ( T X. E ) |-> <. ( t ` ( 1st ` g ) ) , ( t o. ( 2nd ` g ) ) >. )
11		opex 4932	. 2 |- <. ( U ` ( 1st ` F ) ) , ( U o. ( 2nd ` F ) ) >. e. _V
12	3, 8, 10, 11	ovmpt2 6796	1 |- ( ( U e. E /\ F e. ( T X. E ) ) -> ( U .x. F ) = <. ( U ` ( 1st ` F ) ) , ( U o. ( 2nd ` F ) ) >. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  dvhvsca  36390  dvhopspN  36404