Theorem isnvi 27468

Description: Properties that determine a normed complex vector space. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
isnvi.5	|- X = ran G
isnvi.6	|- Z = (GId ` G )
isnvi.7	|- <. G , S >. e. CVecOLD
isnvi.8	|- N : X --> RR
isnvi.9	|- ( ( x e. X /\ ( N ` x ) = 0 ) -> x = Z )
isnvi.10	|- ( ( y e. CC /\ x e. X ) -> ( N ` ( y S x ) ) = ( ( abs ` y ) x. ( N ` x ) ) )
isnvi.11	|- ( ( x e. X /\ y e. X ) -> ( N ` ( x G y ) ) <_ ( ( N ` x ) + ( N ` y ) ) )
isnvi.12	|- U = <. <. G , S >. , N >.

Assertion

Ref	Expression
isnvi	|- U e. NrmCVec

Distinct variable groups:   x, y, G   x, N, y   x, S, y   x, X, y

Allowed substitution hints:   U( x, y)   Z( x, y)

Proof of Theorem isnvi

Step	Hyp	Ref	Expression
1		isnvi.12	. 2 |- U = <. <. G , S >. , N >.
2		isnvi.7	. . 3 |- <. G , S >. e. CVecOLD
3		isnvi.8	. . 3 |- N : X --> RR
4		isnvi.9	. . . . . 6 |- ( ( x e. X /\ ( N ` x ) = 0 ) -> x = Z )
5	4	ex 450	. . . . 5 |- ( x e. X -> ( ( N ` x ) = 0 -> x = Z ) )
6		isnvi.10	. . . . . . 7 |- ( ( y e. CC /\ x e. X ) -> ( N ` ( y S x ) ) = ( ( abs ` y ) x. ( N ` x ) ) )
7	6	ancoms 469	. . . . . 6 |- ( ( x e. X /\ y e. CC ) -> ( N ` ( y S x ) ) = ( ( abs ` y ) x. ( N ` x ) ) )
8	7	ralrimiva 2966	. . . . 5 |- ( x e. X -> A. y e. CC ( N ` ( y S x ) ) = ( ( abs ` y ) x. ( N ` x ) ) )
9		isnvi.11	. . . . . 6 |- ( ( x e. X /\ y e. X ) -> ( N ` ( x G y ) ) <_ ( ( N ` x ) + ( N ` y ) ) )
10	9	ralrimiva 2966	. . . . 5 |- ( x e. X -> A. y e. X ( N ` ( x G y ) ) <_ ( ( N ` x ) + ( N ` y ) ) )
11	5, 8, 10	3jca 1242	. . . 4 |- ( x e. X -> ( ( ( N ` x ) = 0 -> x = Z ) /\ A. y e. CC ( N ` ( y S x ) ) = ( ( abs ` y ) x. ( N ` x ) ) /\ A. y e. X ( N ` ( x G y ) ) <_ ( ( N ` x ) + ( N ` y ) ) ) )
12	11	rgen 2922	. . 3 |- A. x e. X ( ( ( N ` x ) = 0 -> x = Z ) /\ A. y e. CC ( N ` ( y S x ) ) = ( ( abs ` y ) x. ( N ` x ) ) /\ A. y e. X ( N ` ( x G y ) ) <_ ( ( N ` x ) + ( N ` y ) ) )
13		isnvi.5	. . . 4 |- X = ran G
14		isnvi.6	. . . 4 |- Z = (GId ` G )
15	13, 14	isnv 27467	. . 3 |- ( <. <. G , S >. , N >. e. NrmCVec <-> ( <. G , S >. e. CVecOLD /\ N : X --> RR /\ A. x e. X ( ( ( N ` x ) = 0 -> x = Z ) /\ A. y e. CC ( N ` ( y S x ) ) = ( ( abs ` y ) x. ( N ` x ) ) /\ A. y e. X ( N ` ( x G y ) ) <_ ( ( N ` x ) + ( N ` y ) ) ) ) )
16	2, 3, 12, 15	mpbir3an 1244	. 2 |- <. <. G , S >. , N >. e. NrmCVec
17	1, 16	eqeltri 2697	1 |- U e. NrmCVec

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   <.cop 4183   class class class wbr 4653   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-vc 27414  df-nv 27447

This theorem is referenced by:  cnnv  27532  hhnv  28022  hhssnv  28121