Theorem phop 8477

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 8474	. . 3 |- Rel CPreHil
2		1st2nd 4108	. . 3 |- ((Rel CPreHil /\ U e. CPreHil) -> U = <.(1st` U), (2nd` U)>.)
3	1, 2	mpan 695	. 2 |- (U e. CPreHil -> U = <.(1st` U), (2nd` U)>.)
4		phnv 8473	. . . . 5 |- (U e. CPreHil -> U e. NrmCVec)
5		eqid 1475	. . . . . 6 |- (1st` U) = (1st` U)
6	5	nvvc 8234	. . . . 5 |- (U e. NrmCVec -> (1st` U) e. CVec)
7		vcrel 8166	. . . . . . 7 |- Rel CVec
8		1st2nd 4108	. . . . . . 7 |- ((Rel CVec /\ (1st` U) e. CVec) -> (1st` U) = <.(1st` (1st` U)), (2nd` (1st` U))>.)
9	7, 8	mpan 695	. . . . . 6 |- ((1st` U) e. CVec -> (1st` U) = <.(1st` (1st` U)), (2nd` (1st` U))>.)
10		phop.2	. . . . . . . 8 |- G = (+v` U)
11	10	vafval 8222	. . . . . . 7 |- G = (1st` (1st` U))
12		phop.4	. . . . . . . 8 |- S = (.s` U)
13	12	smfval 8224	. . . . . . 7 |- S = (2nd` (1st` U))
14	11, 13	opeq12i 2492	. . . . . 6 |- <.G, S>. = <.(1st` (1st` U)), (2nd` (1st` U))>.
15	9, 14	syl6eqr 1525	. . . . 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 2493	. . 3 |- (U e. CPreHil -> <.(1st` U), N>. = <.<.G, S>., N>.)
18		phop.6	. . . . 5 |- N = (norm` U)
19	18	nmfval 8226	. . . 4 |- N = (2nd` U)
20	19	opeq2i 2491	. . 3 |- <.(1st` U), N>. = <.(1st` U), (2nd` U)>.
21	17, 20	syl5eqr 1521	. 2 |- (U e. CPreHil -> <.(1st` U), (2nd` U)>. = <.<.G, S>., N>.)
22	3, 21	eqtrd 1507	1 |- (U e. CPreHil -> U = <.<.G, S>., N>.)

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  <.cop 2411  Rel wrel 3175  ` cfv 3182  1stc1st 4077  2ndc2nd 4078  CVeccvc 8164  NrmCVeccnv 8203  +vcpv 8204  .scns 8206  normcnm 8209  CPreHilcphl 8471

This theorem is referenced by:  phpar 8483

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-rab 1652  df-v 1812  df-sbc 1942  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-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-grp 8037  df-gid 8038  df-vc 8165  df-nv 8211  df-va 8214  df-ba 8215  df-sm 8216  df-0v 8217  df-nm 8219  df-ph 8472

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