Theorem nvop2 9559

Description: A normed complex vector space is an ordered pair of a vector space and a norm operation.

Hypotheses

Ref	Expression
nvop2.1	|- W = (1st` U)
nvop2.6	|- N = (norm` U)

Assertion

Ref	Expression
nvop2	|- (U e. NrmCVec -> U = <.W, N>.)

Proof of Theorem nvop2

Step	Hyp	Ref	Expression
1		nvrel 9553	. . 3 |- Rel NrmCVec
2		1st2nd 5048	. . 3 |- ((Rel NrmCVec /\ U e. NrmCVec) -> U = <.(1st` U), (2nd` U)>.)
3	1, 2	mpan 759	. 2 |- (U e. NrmCVec -> U = <.(1st` U), (2nd` U)>.)
4		nvop2.1	. . 3 |- W = (1st` U)
5		nvop2.6	. . . 4 |- N = (norm` U)
6	5	nmfval 9558	. . 3 |- N = (2nd` U)
7	4, 6	opeq12i 3163	. 2 |- <.W, N>. = <.(1st` U), (2nd` U)>.
8	3, 7	syl6eqr 1946	1 |- (U e. NrmCVec -> U = <.W, N>.)

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  <.cop 3046  Rel wrel 3991  ` cfv 3998  1stc1st 5018  2ndc2nd 5019  NrmCVeccnv 9535  normcnm 9541

This theorem is referenced by:  nvvop 9560  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-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-fv 4014  df-oprab 4887  df-1st 5020  df-2nd 5021  df-nv 9543  df-nm 9551

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