Theorem phop 8473

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

Hypotheses

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

Assertion

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

Proof of Theorem phop

Step	Hyp	Ref	Expression
1		phrel 8470	. . 3 |- Rel CPreHil
2		1st2nd 4114	. . 3 |- ((Rel CPreHil /\ U e. CPreHil) -> U = <.(1st` U), (2nd` U)>.)
3	1, 2	mpan 697	. 2 |- (U e. CPreHil -> U = <.(1st` U), (2nd` U)>.)
4		phnv 8469	. . . . 5 |- (U e. CPreHil -> U e. NrmCVec)
5		eqid 1478	. . . . . 6 |- (1st` U) = (1st` U)
6	5	nvvc 8230	. . . . 5 |- (U e. NrmCVec -> (1st` U) e. CVec)
7		vcrel 8162	. . . . . . 7 |- Rel CVec
8		1st2nd 4114	. . . . . . 7 |- ((Rel CVec /\ (1st` U) e. CVec) -> (1st` U) = <.(1st` (1st` U)), (2nd` (1st` U))>.)
9	7, 8	mpan 697	. . . . . 6 |- ((1st` U) e. CVec -> (1st` U) = <.(1st` (1st` U)), (2nd` (1st` U))>.)
10		phop.2	. . . . . . . 8 |- G = (+v` U)
11	10	vafval 8218	. . . . . . 7 |- G = (1st` (1st` U))
12		phop.4	. . . . . . . 8 |- S = (.s` U)
13	12	smfval 8220	. . . . . . 7 |- S = (2nd` (1st` U))
14	11, 13	opeq12i 2496	. . . . . 6 |- <.G, S>. = <.(1st` (1st` U)), (2nd` (1st` U))>.
15	9, 14	syl6eqr 1528	. . . . 5 |- ((1st` U) e. CVec -> (1st` U) = <.G, S>.)
16	4, 6, 15	3syl 20	. . . 4 |- (U e. CPreHil -> (1st` U) = <.G, S>.)
17	16	opeq1d 2497	. . 3 |- (U e. CPreHil -> <.(1st` U), N>. = <.<.G, S>., N>.)
18		phop.6	. . . . 5 |- N = (norm` U)
19	18	nmfval 8222	. . . 4 |- N = (2nd` U)
20	19	opeq2i 2495	. . 3 |- <.(1st` U), N>. = <.(1st` U), (2nd` U)>.
21	17, 20	syl5eqr 1524	. 2 |- (U e. CPreHil -> <.(1st` U), (2nd` U)>. = <.<.G, S>., N>.)
22	3, 21	eqtrd 1510	1 |- (U e. CPreHil -> U = <.<.G, S>., N>.)

Syntax hints:   -> wi 3   = wceq 958   e. wcel 960  <.cop 2415  Rel wrel 3181  ` cfv 3188  1stc1st 4083  2ndc2nd 4084  CVeccvc 8160  NrmCVeccnv 8199  +vcpv 8200  .scns 8202  normcnm 8205  CPreHilcphl 8467

This theorem is referenced by:  phpar 8479

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-rab 1655  df-v 1815  df-sbc 1945  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-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-grp 8034  df-gid 8035  df-vc 8161  df-nv 8207  df-va 8210  df-ba 8211  df-sm 8212  df-0v 8213  df-nm 8215  df-ph 8468

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