Theorem nvop 8305

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 8221	. . 3 |- Rel NrmCVec
2		1st2nd 4108	. . 3 |- ((Rel NrmCVec /\ U e. NrmCVec) -> U = <.(1st` U), (2nd` U)>.)
3	1, 2	mpan 695	. 2 |- (U e. NrmCVec -> U = <.(1st` U), (2nd` U)>.)
4		eqid 1475	. . . . 5 |- (1st` U) = (1st` U)
5		nvop.2	. . . . 5 |- G = (+v` U)
6		nvop.4	. . . . 5 |- S = (.s` U)
7	4, 5, 6	nvvop 8228	. . . 4 |- (U e. NrmCVec -> (1st` U) = <.G, S>.)
8	7	opeq1d 2493	. . 3 |- (U e. NrmCVec -> <.(1st` U), N>. = <.<.G, S>., N>.)
9		nvop.6	. . . . 5 |- N = (norm` U)
10	9	nmfval 8226	. . . 4 |- N = (2nd` U)
11	10	opeq2i 2491	. . 3 |- <.(1st` U), N>. = <.(1st` U), (2nd` U)>.
12	8, 11	syl5eqr 1521	. 2 |- (U e. NrmCVec -> <.(1st` U), (2nd` U)>. = <.<.G, S>., N>.)
13	3, 12	eqtrd 1507	1 |- (U e. NrmCVec -> U = <.<.G, S>., N>.)

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  <.cop 2411  Rel wrel 3175  ` cfv 3182  1stc1st 4077  2ndc2nd 4078  NrmCVeccnv 8203  +vcpv 8204  .scns 8206  normcnm 8209

This theorem is referenced by:  isph 8481  hilhh 9031

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-oprab 3966  df-1st 4079  df-2nd 4080  df-nv 8211  df-va 8214  df-sm 8216  df-nm 8219

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