Theorem nvop 9637

Description: A complex inner product space in terms of ordered pair components.

Hypotheses

Ref	Expression
nvop.2	|- G = (+v` U)
nvop.4	|- S = (.s` U)
nvop.6	|- N = (norm` U)

Assertion

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

Proof of Theorem nvop

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		eqid 1884	. . . . 5 |- (1st` U) = (1st` U)
5		nvop.2	. . . . 5 |- G = (+v` U)
6		nvop.4	. . . . 5 |- S = (.s` U)
7	4, 5, 6	nvvop 9560	. . . 4 |- (U e. NrmCVec -> (1st` U) = <.G, S>.)
8	7	opeq1d 3164	. . 3 |- (U e. NrmCVec -> <.(1st` U), N>. = <.<.G, S>., N>.)
9		nvop.6	. . . . 5 |- N = (norm` U)
10	9	nmfval 9558	. . . 4 |- N = (2nd` U)
11	10	opeq2i 3162	. . 3 |- <.(1st` U), N>. = <.(1st` U), (2nd` U)>.
12	8, 11	syl5eqr 1942	. 2 |- (U e. NrmCVec -> <.(1st` U), (2nd` U)>. = <.<.G, S>., N>.)
13	3, 12	eqtrd 1925	1 |- (U e. NrmCVec -> U = <.<.G, S>., N>.)

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

This theorem is referenced by:  isph 9822  hilhhi 10664

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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