Theorem nvss 27448

Description: Structure of the class of all normed complex vectors spaces. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.)

Assertion

Ref	Expression
nvss	|- NrmCVec C_ ( CVecOLD X. _V )

Proof of Theorem nvss

Dummy variables g s n w x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . . . . 7 |- ( w = <. g , s >. -> ( w e. CVecOLD <-> <. g , s >. e. CVecOLD ) )
2	1	biimpar 502	. . . . . 6 |- ( ( w = <. g , s >. /\ <. g , s >. e. CVecOLD ) -> w e. CVecOLD )
3	2	3ad2antr1 1226	. . . . 5 |- ( ( w = <. g , s >. /\ ( <. g , s >. e. CVecOLD /\ n : ran g --> RR /\ A. x e. ran g ( ( ( n ` x ) = 0 -> x = (GId ` g ) ) /\ A. y e. CC ( n ` ( y s x ) ) = ( ( abs ` y ) x. ( n ` x ) ) /\ A. y e. ran g ( n ` ( x g y ) ) <_ ( ( n ` x ) + ( n ` y ) ) ) ) ) -> w e. CVecOLD )
4	3	exlimivv 1860	. . . 4 |- ( E. g E. s ( w = <. g , s >. /\ ( <. g , s >. e. CVecOLD /\ n : ran g --> RR /\ A. x e. ran g ( ( ( n ` x ) = 0 -> x = (GId ` g ) ) /\ A. y e. CC ( n ` ( y s x ) ) = ( ( abs ` y ) x. ( n ` x ) ) /\ A. y e. ran g ( n ` ( x g y ) ) <_ ( ( n ` x ) + ( n ` y ) ) ) ) ) -> w e. CVecOLD )
5		vex 3203	. . . 4 |- n e. _V
6	4, 5	jctir 561	. . 3 |- ( E. g E. s ( w = <. g , s >. /\ ( <. g , s >. e. CVecOLD /\ n : ran g --> RR /\ A. x e. ran g ( ( ( n ` x ) = 0 -> x = (GId ` g ) ) /\ A. y e. CC ( n ` ( y s x ) ) = ( ( abs ` y ) x. ( n ` x ) ) /\ A. y e. ran g ( n ` ( x g y ) ) <_ ( ( n ` x ) + ( n ` y ) ) ) ) ) -> ( w e. CVecOLD /\ n e. _V ) )
7	6	ssopab2i 5003	. 2 |- { <. w , n >. | E. g E. s ( w = <. g , s >. /\ ( <. g , s >. e. CVecOLD /\ n : ran g --> RR /\ A. x e. ran g ( ( ( n ` x ) = 0 -> x = (GId ` g ) ) /\ A. y e. CC ( n ` ( y s x ) ) = ( ( abs ` y ) x. ( n ` x ) ) /\ A. y e. ran g ( n ` ( x g y ) ) <_ ( ( n ` x ) + ( n ` y ) ) ) ) ) } C_ { <. w , n >. | ( w e. CVecOLD /\ n e. _V ) }
8		df-nv 27447	. . 3 |- NrmCVec = { <. <. g , s >. , n >. | ( <. g , s >. e. CVecOLD /\ n : ran g --> RR /\ A. x e. ran g ( ( ( n ` x ) = 0 -> x = (GId ` g ) ) /\ A. y e. CC ( n ` ( y s x ) ) = ( ( abs ` y ) x. ( n ` x ) ) /\ A. y e. ran g ( n ` ( x g y ) ) <_ ( ( n ` x ) + ( n ` y ) ) ) ) }
9		dfoprab2 6701	. . 3 |- { <. <. g , s >. , n >. | ( <. g , s >. e. CVecOLD /\ n : ran g --> RR /\ A. x e. ran g ( ( ( n ` x ) = 0 -> x = (GId ` g ) ) /\ A. y e. CC ( n ` ( y s x ) ) = ( ( abs ` y ) x. ( n ` x ) ) /\ A. y e. ran g ( n ` ( x g y ) ) <_ ( ( n ` x ) + ( n ` y ) ) ) ) } = { <. w , n >. | E. g E. s ( w = <. g , s >. /\ ( <. g , s >. e. CVecOLD /\ n : ran g --> RR /\ A. x e. ran g ( ( ( n ` x ) = 0 -> x = (GId ` g ) ) /\ A. y e. CC ( n ` ( y s x ) ) = ( ( abs ` y ) x. ( n ` x ) ) /\ A. y e. ran g ( n ` ( x g y ) ) <_ ( ( n ` x ) + ( n ` y ) ) ) ) ) }
10	8, 9	eqtri 2644	. 2 |- NrmCVec = { <. w , n >. | E. g E. s ( w = <. g , s >. /\ ( <. g , s >. e. CVecOLD /\ n : ran g --> RR /\ A. x e. ran g ( ( ( n ` x ) = 0 -> x = (GId ` g ) ) /\ A. y e. CC ( n ` ( y s x ) ) = ( ( abs ` y ) x. ( n ` x ) ) /\ A. y e. ran g ( n ` ( x g y ) ) <_ ( ( n ` x ) + ( n ` y ) ) ) ) ) }
11		df-xp 5120	. 2 |- ( CVecOLD X. _V ) = { <. w , n >. | ( w e. CVecOLD /\ n e. _V ) }
12	7, 10, 11	3sstr4i 3644	1 |- NrmCVec C_ ( CVecOLD X. _V )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   <.cop 4183   class class class wbr 4653   {copab 4712   X. cxp 5112   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   {coprab 6651   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   <_ cle 10075   abscabs 13974  GIdcgi 27344   CVecOLDcvc 27413   NrmCVeccnv 27439

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-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-opab 4713  df-xp 5120  df-oprab 6654  df-nv 27447

This theorem is referenced by:  nvvcop  27449  nvrel  27457  nvvop  27464  nvex  27466