Theorem isnv 27467

Description: The predicate "is a normed complex vector space." (Contributed by NM, 5-Jun-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
isnv.1	|- X = ran G
isnv.2	|- Z = (GId ` G )

Assertion

Ref	Expression
isnv	|- ( <. <. 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 ) ) ) ) )

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

Allowed substitution hints:   Z( x, y)

Proof of Theorem isnv

Step	Hyp	Ref	Expression
1		nvex 27466	. 2 |- ( <. <. G , S >. , N >. e. NrmCVec -> ( G e. _V /\ S e. _V /\ N e. _V ) )
2		vcex 27433	. . . . 5 |- ( <. G , S >. e. CVecOLD -> ( G e. _V /\ S e. _V ) )
3	2	adantr 481	. . . 4 |- ( ( <. G , S >. e. CVecOLD /\ N : X --> RR ) -> ( G e. _V /\ S e. _V ) )
4		isnv.1	. . . . . . 7 |- X = ran G
5	2	simpld 475	. . . . . . . 8 |- ( <. G , S >. e. CVecOLD -> G e. _V )
6		rnexg 7098	. . . . . . . 8 |- ( G e. _V -> ran G e. _V )
7	5, 6	syl 17	. . . . . . 7 |- ( <. G , S >. e. CVecOLD -> ran G e. _V )
8	4, 7	syl5eqel 2705	. . . . . 6 |- ( <. G , S >. e. CVecOLD -> X e. _V )
9		fex 6490	. . . . . 6 |- ( ( N : X --> RR /\ X e. _V ) -> N e. _V )
10	8, 9	sylan2 491	. . . . 5 |- ( ( N : X --> RR /\ <. G , S >. e. CVecOLD ) -> N e. _V )
11	10	ancoms 469	. . . 4 |- ( ( <. G , S >. e. CVecOLD /\ N : X --> RR ) -> N e. _V )
12		df-3an 1039	. . . 4 |- ( ( G e. _V /\ S e. _V /\ N e. _V ) <-> ( ( G e. _V /\ S e. _V ) /\ N e. _V ) )
13	3, 11, 12	sylanbrc 698	. . 3 |- ( ( <. G , S >. e. CVecOLD /\ N : X --> RR ) -> ( G e. _V /\ S e. _V /\ N e. _V ) )
14	13	3adant3 1081	. 2 |- ( ( <. 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 e. _V /\ S e. _V /\ N e. _V ) )
15		isnv.2	. . 3 |- Z = (GId ` G )
16	4, 15	isnvlem 27465	. 2 |- ( ( G e. _V /\ S e. _V /\ N e. _V ) -> ( <. <. 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 ) ) ) ) ) )
17	1, 14, 16	pm5.21nii 368	1 |- ( <. <. 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 ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   <.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:  isnvi  27468  nvi  27469