Theorem nvvop 8224

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 1478	. . . . . . . . 9 |- (norm` U) = (norm` U)
3	1, 2	nvop2 8223	. . . . . . . 8 |- (U e. NrmCVec -> U = <.W, (norm` U)>.)
4	3	eleq1d 1543	. . . . . . 7 |- (U e. NrmCVec -> (U e. NrmCVec <-> <.W, (norm` U)>. e. NrmCVec))
5	4	ibi 594	. . . . . 6 |- (U e. NrmCVec -> <.W, (norm` U)>. e. NrmCVec)
6		nvss 8208	. . . . . . 7 |- NrmCVec (_ ((V X. V) X. V)
7	6	sseli 2068	. . . . . 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 3738	. . . . . 6 |- (norm` U) e. V
10	9	opelxp 3220	. . . . 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 3261	. . . 4 |- Rel (V X. V)
14		1st2nd 4114	. . . 4 |- ((Rel (V X. V) /\ W e. (V X. V)) -> W = <.(1st` W), (2nd` W)>.)
15	13, 14	mpan 697	. . 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 8218	. . . 4 |- G = (1st` (1st` U))
19	1	fveq2i 3733	. . . 4 |- (1st` W) = (1st` (1st` U))
20	18, 19	eqtr4 1501	. . 3 |- G = (1st` W)
21		nvvop.4	. . . . 5 |- S = (.s` U)
22	21	smfval 8220	. . . 4 |- S = (2nd` (1st` U))
23	1	fveq2i 3733	. . . 4 |- (2nd` W) = (2nd` (1st` U))
24	22, 23	eqtr4 1501	. . 3 |- S = (2nd` W)
25	20, 24	opeq12i 2496	. 2 |- <.G, S>. = <.(1st` W), (2nd` W)>.
26	16, 25	syl6eqr 1528	1 |- (U e. NrmCVec -> W = <.G, S>.)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  Vcvv 1814  <.cop 2415   X. cxp 3174  Rel wrel 3181  ` cfv 3188  1stc1st 4083  2ndc2nd 4084  NrmCVeccnv 8199  +vcpv 8200  .scns 8202  normcnm 8205

This theorem is referenced by:  nvvc 8230  nvop 8301  sspval 8378

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202  df-fv 3204  df-oprab 3972  df-1st 4085  df-2nd 4086  df-nv 8207  df-va 8210  df-sm 8212  df-nm 8215

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