Theorem nvvop 27464

Description: The vector space component of a normed complex vector space is an ordered pair of the underlying group and a scalar product. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nvvop.1	|- W = ( 1st ` U )
nvvop.2	|- G = ( +v ` U )
nvvop.4	|- S = ( .sOLD ` U )

Assertion

Ref	Expression
nvvop	|- ( U e. NrmCVec -> W = <. G , S >. )

Proof of Theorem nvvop

Step	Hyp	Ref	Expression
1		vcrel 27415	. . 3 |- Rel CVecOLD
2		nvss 27448	. . . . 5 |- NrmCVec C_ ( CVecOLD X. _V )
3		nvvop.1	. . . . . . . 8 |- W = ( 1st ` U )
4		eqid 2622	. . . . . . . 8 |- ( normCV ` U ) = ( normCV ` U )
5	3, 4	nvop2 27463	. . . . . . 7 |- ( U e. NrmCVec -> U = <. W , ( normCV ` U ) >. )
6	5	eleq1d 2686	. . . . . 6 |- ( U e. NrmCVec -> ( U e. NrmCVec <-> <. W , ( normCV ` U ) >. e. NrmCVec ) )
7	6	ibi 256	. . . . 5 |- ( U e. NrmCVec -> <. W , ( normCV ` U ) >. e. NrmCVec )
8	2, 7	sseldi 3601	. . . 4 |- ( U e. NrmCVec -> <. W , ( normCV ` U ) >. e. ( CVecOLD X. _V ) )
9		opelxp1 5150	. . . 4 |- ( <. W , ( normCV ` U ) >. e. ( CVecOLD X. _V ) -> W e. CVecOLD )
10	8, 9	syl 17	. . 3 |- ( U e. NrmCVec -> W e. CVecOLD )
11		1st2nd 7214	. . 3 |- ( ( Rel CVecOLD /\ W e. CVecOLD ) -> W = <. ( 1st ` W ) , ( 2nd ` W ) >. )
12	1, 10, 11	sylancr 695	. 2 |- ( U e. NrmCVec -> W = <. ( 1st ` W ) , ( 2nd ` W ) >. )
13		nvvop.2	. . . . 5 |- G = ( +v ` U )
14	13	vafval 27458	. . . 4 |- G = ( 1st ` ( 1st ` U ) )
15	3	fveq2i 6194	. . . 4 |- ( 1st ` W ) = ( 1st ` ( 1st ` U ) )
16	14, 15	eqtr4i 2647	. . 3 |- G = ( 1st ` W )
17		nvvop.4	. . . . 5 |- S = ( .sOLD ` U )
18	17	smfval 27460	. . . 4 |- S = ( 2nd ` ( 1st ` U ) )
19	3	fveq2i 6194	. . . 4 |- ( 2nd ` W ) = ( 2nd ` ( 1st ` U ) )
20	18, 19	eqtr4i 2647	. . 3 |- S = ( 2nd ` W )
21	16, 20	opeq12i 4407	. 2 |- <. G , S >. = <. ( 1st ` W ) , ( 2nd ` W ) >.
22	12, 21	syl6eqr 2674	1 |- ( U e. NrmCVec -> W = <. G , S >. )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   X. cxp 5112   Rel wrel 5119   ` cfv 5888   1stc1st 7166   2ndc2nd 7167   CVecOLDcvc 27413   NrmCVeccnv 27439   +vcpv 27440   .sOLDcns 27442   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-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-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-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  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-fo 5894  df-fv 5896  df-oprab 6654  df-1st 7168  df-2nd 7169  df-vc 27414  df-nv 27447  df-va 27450  df-sm 27452  df-nmcv 27455

This theorem is referenced by:  nvi  27469  nvvc  27470  nvop  27531