Theorem nvvop 9560

Description: The vector space component of a normed complex vector space is an ordered pair of the underlying group and a scalar product.

Hypotheses

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

Assertion

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

Proof of Theorem nvvop

Step	Hyp	Ref	Expression
1		1st2nd 5048	. . 3 |- ((Rel (_V X. _V) /\ W e. (_V X. _V)) -> W = <.(1st` W), (2nd` W)>.)
2		relxp 4088	. . 3 |- Rel (_V X. _V)
3		nvvop.1	. . . . . . . . 9 |- W = (1st` U)
4		eqid 1884	. . . . . . . . 9 |- (norm` U) = (norm` U)
5	3, 4	nvop2 9559	. . . . . . . 8 |- (U e. NrmCVec -> U = <.W, (norm` U)>.)
6	5	eleq1d 1963	. . . . . . 7 |- (U e. NrmCVec -> (U e. NrmCVec <-> <.W, (norm` U)>. e. NrmCVec))
7	6	ibi 652	. . . . . 6 |- (U e. NrmCVec -> <.W, (norm` U)>. e. NrmCVec)
8		nvss 9544	. . . . . . 7 |- NrmCVec C_ ((_V X. _V) X. _V)
9	8	sseli 2617	. . . . . 6 |- (<.W, (norm` U)>. e. NrmCVec -> <.W, (norm` U)>. e. ((_V X. _V) X. _V))
10	7, 9	syl 12	. . . . 5 |- (U e. NrmCVec -> <.W, (norm` U)>. e. ((_V X. _V) X. _V))
11		fvex 4689	. . . . . 6 |- (norm` U) e. _V
12	11	opelxp 4036	. . . . 5 |- (<.W, (norm` U)>. e. ((_V X. _V) X. _V) <-> (W e. (_V X. _V) /\ (norm` U) e. _V))
13	10, 12	sylib 215	. . . 4 |- (U e. NrmCVec -> (W e. (_V X. _V) /\ (norm` U) e. _V))
14	13	simplld 348	. . 3 |- (U e. NrmCVec -> W e. (_V X. _V))
15	1, 2, 14	sylancr 526	. 2 |- (U e. NrmCVec -> W = <.(1st` W), (2nd` W)>.)
16		nvvop.2	. . . . 5 |- G = (+v` U)
17	16	vafval 9554	. . . 4 |- G = (1st` (1st` U))
18	3	fveq2i 4684	. . . 4 |- (1st` W) = (1st` (1st` U))
19	17, 18	eqtr4i 1911	. . 3 |- G = (1st` W)
20		nvvop.4	. . . . 5 |- S = (.s` U)
21	20	smfval 9556	. . . 4 |- S = (2nd` (1st` U))
22	3	fveq2i 4684	. . . 4 |- (2nd` W) = (2nd` (1st` U))
23	21, 22	eqtr4i 1911	. . 3 |- S = (2nd` W)
24	19, 23	opeq12i 3163	. 2 |- <.G, S>. = <.(1st` W), (2nd` W)>.
25	15, 24	syl6eqr 1946	1 |- (U e. NrmCVec -> W = <.G, S>.)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  <.cop 3046   X. cxp 3984  Rel wrel 3991  ` cfv 3998  1stc1st 5018  2ndc2nd 5019  NrmCVeccnv 9535  +vcpv 9536  .scns 9538  normcnm 9541

This theorem is referenced by:  nvvc 9566  nvop 9637  sspval 9721

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012  df-fv 4014  df-oprab 4887  df-1st 5020  df-2nd 5021  df-nv 9543  df-va 9546  df-sm 9548  df-nm 9551

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