Theorem nvvop 8166

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		nvvop.1	. . . . . . . . 9 |- W = (1st` U)
2		eqid 1468	. . . . . . . . 9 |- (norm` U) = (norm` U)
3	1, 2	nvop2 8165	. . . . . . . 8 |- (U e. NrmCVec -> U = <.W, (norm` U)>.)
4	3	eleq1d 1532	. . . . . . 7 |- (U e. NrmCVec -> (U e. NrmCVec <-> <.W, (norm` U)>. e. NrmCVec))
5	4	ibi 590	. . . . . 6 |- (U e. NrmCVec -> <.W, (norm` U)>. e. NrmCVec)
6		nvss 8150	. . . . . . 7 |- NrmCVec (_ ((V X. V) X. V)
7	6	sseli 2055	. . . . . 6 |- (<.W, (norm` U)>. e. NrmCVec -> <.W, (norm` U)>. e. ((V X. V) X. V))
8	5, 7	syl 10	. . . . 5 |- (U e. NrmCVec -> <.W, (norm` U)>. e. ((V X. V) X. V))
9		fvex 3717	. . . . . 6 |- (norm` U) e. V
10	9	opelxp 3204	. . . . 5 |- (<.W, (norm` U)>. e. ((V X. V) X. V) <-> (W e. (V X. V) /\ (norm` U) e. V))
11	8, 10	sylib 198	. . . 4 |- (U e. NrmCVec -> (W e. (V X. V) /\ (norm` U) e. V))
12	11	pm3.26d 321	. . 3 |- (U e. NrmCVec -> W e. (V X. V))
13		relxp 3245	. . . 4 |- Rel (V X. V)
14		1st2nd 4092	. . . 4 |- ((Rel (V X. V) /\ W e. (V X. V)) -> W = <.(1st` W), (2nd` W)>.)
15	13, 14	mpan 693	. . 3 |- (W e. (V X. V) -> W = <.(1st` W), (2nd` W)>.)
16	12, 15	syl 10	. 2 |- (U e. NrmCVec -> W = <.(1st` W), (2nd` W)>.)
17		nvvop.2	. . . . 5 |- G = (+v` U)
18	17	vafval 8160	. . . 4 |- G = (1st` (1st` U))
19	1	fveq2i 3712	. . . 4 |- (1st` W) = (1st` (1st` U))
20	18, 19	eqtr4 1490	. . 3 |- G = (1st` W)
21		nvvop.4	. . . . 5 |- S = (.s` U)
22	21	smfval 8162	. . . 4 |- S = (2nd` (1st` U))
23	1	fveq2i 3712	. . . 4 |- (2nd` W) = (2nd` (1st` U))
24	22, 23	eqtr4 1490	. . 3 |- S = (2nd` W)
25	20, 24	opeq12i 2483	. 2 |- <.G, S>. = <.(1st` W), (2nd` W)>.
26	16, 25	syl6eqr 1517	1 |- (U e. NrmCVec -> W = <.G, S>.)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  Vcvv 1802  <.cop 2401   X. cxp 3158  Rel wrel 3165  ` cfv 3172  1stc1st 4061  2ndc2nd 4062  NrmCVeccnv 8141  +vcpv 8142  .scns 8144  normcnm 8147

This theorem is referenced by:  nvvc 8174  nvop 8244  sspval 8316

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fo 3186  df-fv 3188  df-oprab 3951  df-1st 4063  df-2nd 4064  df-nv 8149  df-va 8152  df-sm 8154  df-nm 8157

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