Theorem 0ofval 27642

Description: The zero operator between two normed complex vector spaces. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
0oval.1	|- X = ( BaseSet ` U )
0oval.6	|- Z = ( 0vec ` W )
0oval.0	|- O = ( U 0op W )

Assertion

Ref	Expression
0ofval	|- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> O = ( X X. { Z } ) )

Proof of Theorem 0ofval

Dummy variables w u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0oval.0	. 2 |- O = ( U 0op W )
2		fveq2 6191	. . . . 5 |- ( u = U -> ( BaseSet ` u ) = ( BaseSet ` U ) )
3		0oval.1	. . . . 5 |- X = ( BaseSet ` U )
4	2, 3	syl6eqr 2674	. . . 4 |- ( u = U -> ( BaseSet ` u ) = X )
5	4	xpeq1d 5138	. . 3 |- ( u = U -> ( ( BaseSet ` u ) X. { ( 0vec ` w ) } ) = ( X X. { ( 0vec ` w ) } ) )
6		fveq2 6191	. . . . . 6 |- ( w = W -> ( 0vec ` w ) = ( 0vec ` W ) )
7		0oval.6	. . . . . 6 |- Z = ( 0vec ` W )
8	6, 7	syl6eqr 2674	. . . . 5 |- ( w = W -> ( 0vec ` w ) = Z )
9	8	sneqd 4189	. . . 4 |- ( w = W -> { ( 0vec ` w ) } = { Z } )
10	9	xpeq2d 5139	. . 3 |- ( w = W -> ( X X. { ( 0vec ` w ) } ) = ( X X. { Z } ) )
11		df-0o 27602	. . 3 |- 0op = ( u e. NrmCVec , w e. NrmCVec |-> ( ( BaseSet ` u ) X. { ( 0vec ` w ) } ) )
12		fvex 6201	. . . . 5 |- ( BaseSet ` U ) e. _V
13	3, 12	eqeltri 2697	. . . 4 |- X e. _V
14		snex 4908	. . . 4 |- { Z } e. _V
15	13, 14	xpex 6962	. . 3 |- ( X X. { Z } ) e. _V
16	5, 10, 11, 15	ovmpt2 6796	. 2 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> ( U 0op W ) = ( X X. { Z } ) )
17	1, 16	syl5eq 2668	1 |- ( ( U e. NrmCVec /\ W e. NrmCVec ) -> O = ( X X. { Z } ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   X. cxp 5112   ` cfv 5888  (class class class)co 6650   NrmCVeccnv 27439   BaseSetcba 27441   0veccn0v 27443   0op c0o 27598

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-pw 4160  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  df-0o 27602

This theorem is referenced by:  0oval  27643  0oo  27644  lnon0  27653  blocni  27660  hh0oi  28762