Theorem nvi 27469

Description: The properties of a normed complex vector space, which is a vector space accompanied by a norm. (Contributed by NM, 11-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nvi.1	|- X = ( BaseSet ` U )
nvi.2	|- G = ( +v ` U )
nvi.4	|- S = ( .sOLD ` U )
nvi.5	|- Z = ( 0vec ` U )
nvi.6	|- N = ( normCV ` U )

Assertion

Ref	Expression
nvi	|- ( U 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 ) ) ) ) )

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

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

Proof of Theorem nvi

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . 6 |- ( 1st ` U ) = ( 1st ` U )
2		nvi.6	. . . . . 6 |- N = ( normCV ` U )
3	1, 2	nvop2 27463	. . . . 5 |- ( U e. NrmCVec -> U = <. ( 1st ` U ) , N >. )
4		nvi.2	. . . . . . 7 |- G = ( +v ` U )
5		nvi.4	. . . . . . 7 |- S = ( .sOLD ` U )
6	1, 4, 5	nvvop 27464	. . . . . 6 |- ( U e. NrmCVec -> ( 1st ` U ) = <. G , S >. )
7	6	opeq1d 4408	. . . . 5 |- ( U e. NrmCVec -> <. ( 1st ` U ) , N >. = <. <. G , S >. , N >. )
8	3, 7	eqtrd 2656	. . . 4 |- ( U e. NrmCVec -> U = <. <. G , S >. , N >. )
9		id 22	. . . 4 |- ( U e. NrmCVec -> U e. NrmCVec )
10	8, 9	eqeltrrd 2702	. . 3 |- ( U e. NrmCVec -> <. <. G , S >. , N >. e. NrmCVec )
11		nvi.1	. . . . 5 |- X = ( BaseSet ` U )
12	11, 4	bafval 27459	. . . 4 |- X = ran G
13		eqid 2622	. . . 4 |- (GId ` G ) = (GId ` G )
14	12, 13	isnv 27467	. . 3 |- ( <. <. G , S >. , N >. e. NrmCVec <-> ( <. G , S >. e. CVecOLD /\ N : X --> RR /\ A. x e. X ( ( ( N ` x ) = 0 -> x = (GId ` G ) ) /\ 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 ) ) ) ) )
15	10, 14	sylib 208	. 2 |- ( U e. NrmCVec -> ( <. G , S >. e. CVecOLD /\ N : X --> RR /\ A. x e. X ( ( ( N ` x ) = 0 -> x = (GId ` G ) ) /\ 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		nvi.5	. . . . . . . 8 |- Z = ( 0vec ` U )
17	4, 16	0vfval 27461	. . . . . . 7 |- ( U e. NrmCVec -> Z = (GId ` G ) )
18	17	eqeq2d 2632	. . . . . 6 |- ( U e. NrmCVec -> ( x = Z <-> x = (GId ` G ) ) )
19	18	imbi2d 330	. . . . 5 |- ( U e. NrmCVec -> ( ( ( N ` x ) = 0 -> x = Z ) <-> ( ( N ` x ) = 0 -> x = (GId ` G ) ) ) )
20	19	3anbi1d 1403	. . . 4 |- ( U e. NrmCVec -> ( ( ( ( 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 ) ) ) <-> ( ( ( N ` x ) = 0 -> x = (GId ` G ) ) /\ 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 ) ) ) ) )
21	20	ralbidv 2986	. . 3 |- ( U e. NrmCVec -> ( 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 ) ) ) <-> A. x e. X ( ( ( N ` x ) = 0 -> x = (GId ` G ) ) /\ 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 ) ) ) ) )
22	21	3anbi3d 1405	. 2 |- ( U 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 ) ) ) ) <-> ( <. G , S >. e. CVecOLD /\ N : X --> RR /\ A. x e. X ( ( ( N ` x ) = 0 -> x = (GId ` G ) ) /\ 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 ) ) ) ) ) )
23	15, 22	mpbird 247	1 |- ( U 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 ) ) ) ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   <.cop 4183   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   <_ cle 10075   abscabs 13974  GIdcgi 27344   CVecOLDcvc 27413   NrmCVeccnv 27439   +vcpv 27440   BaseSetcba 27441   .sOLDcns 27442   0veccn0v 27443   norm_CVcnmcv 27445

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-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-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-1st 7168  df-2nd 7169  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-nmcv 27455

This theorem is referenced by:  nvvc  27470  nvf  27515  nvs  27518  nvz  27524  nvtri  27525