Theorem isphg 9817

Description: The predicate "is a complex inner product space." An inner product space is a normed vector space whose norm satisfies the parallelogram law. The vector (group) addition operation is G, the scalar product is S, and the norm is N. An inner product space is also called a pre-Hilbert space.

Hypothesis

Ref	Expression
isphg.1	|- X = ran G

Assertion

Ref	Expression
isphg	|- ((G e. A /\ S e. B /\ N e. C) -> (<.<.G, S>., N>. e. CPreHil <-> (<.<.G, S>., N>. e. NrmCVec /\ A.x e. X A.y e. X (((N` (xGy))^2) + ((N` (xG(-u1Sy)))^2)) = (2 x. (((N` x)^2) + ((N` y)^2))))))

Distinct variable groups:   x,y,G   x,N,y   x,S,y   x,X,y

Proof of Theorem isphg

Step	Hyp	Ref	Expression
1		rneq 4186	. . . . . 6 |- (g = G -> ran g = ran G)
2		isphg.1	. . . . . 6 |- X = ran G
3	1, 2	syl6eqr 1946	. . . . 5 |- (g = G -> ran g = X)
4		opreq 4888	. . . . . . . . . 10 |- (g = G -> (xgy) = (xGy))
5	4	fveq2d 4685	. . . . . . . . 9 |- (g = G -> (n` (xgy)) = (n` (xGy)))
6	5	opreq1d 4897	. . . . . . . 8 |- (g = G -> ((n` (xgy))^2) = ((n` (xGy))^2))
7		opreq 4888	. . . . . . . . . 10 |- (g = G -> (xg(-u1sy)) = (xG(-u1sy)))
8	7	fveq2d 4685	. . . . . . . . 9 |- (g = G -> (n` (xg(-u1sy))) = (n` (xG(-u1sy))))
9	8	opreq1d 4897	. . . . . . . 8 |- (g = G -> ((n` (xg(-u1sy)))^2) = ((n` (xG(-u1sy)))^2))
10	6, 9	opreq12d 4900	. . . . . . 7 |- (g = G -> (((n` (xgy))^2) + ((n` (xg(-u1sy)))^2)) = (((n` (xGy))^2) + ((n` (xG(-u1sy)))^2)))
11	10	eqeq1d 1892	. . . . . 6 |- (g = G -> ((((n` (xgy))^2) + ((n` (xg(-u1sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2))) <-> (((n` (xGy))^2) + ((n` (xG(-u1sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2)))))
12	3, 11	raleqbidv 2274	. . . . 5 |- (g = G -> (A.y e. ran g(((n` (xgy))^2) + ((n` (xg(-u1sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2))) <-> A.y e. X (((n` (xGy))^2) + ((n` (xG(-u1sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2)))))
13	3, 12	raleqbidv 2274	. . . 4 |- (g = G -> (A.x e. ran gA.y e. ran g(((n` (xgy))^2) + ((n` (xg(-u1sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2))) <-> A.x e. X A.y e. X (((n` (xGy))^2) + ((n` (xG(-u1sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2)))))
14		opreq 4888	. . . . . . . . . 10 |- (s = S -> (-u1sy) = (-u1Sy))
15	14	opreq2d 4898	. . . . . . . . 9 |- (s = S -> (xG(-u1sy)) = (xG(-u1Sy)))
16	15	fveq2d 4685	. . . . . . . 8 |- (s = S -> (n` (xG(-u1sy))) = (n` (xG(-u1Sy))))
17	16	opreq1d 4897	. . . . . . 7 |- (s = S -> ((n` (xG(-u1sy)))^2) = ((n` (xG(-u1Sy)))^2))
18	17	opreq2d 4898	. . . . . 6 |- (s = S -> (((n` (xGy))^2) + ((n` (xG(-u1sy)))^2)) = (((n` (xGy))^2) + ((n` (xG(-u1Sy)))^2)))
19	18	eqeq1d 1892	. . . . 5 |- (s = S -> ((((n` (xGy))^2) + ((n` (xG(-u1sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2))) <-> (((n` (xGy))^2) + ((n` (xG(-u1Sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2)))))
20	19	2ralbidv 2140	. . . 4 |- (s = S -> (A.x e. X A.y e. X (((n` (xGy))^2) + ((n` (xG(-u1sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2))) <-> A.x e. X A.y e. X (((n` (xGy))^2) + ((n` (xG(-u1Sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2)))))
21		fveq1 4680	. . . . . . . . 9 |- (n = N -> (n` (xGy)) = (N` (xGy)))
22	21	opreq1d 4897	. . . . . . . 8 |- (n = N -> ((n` (xGy))^2) = ((N` (xGy))^2))
23		fveq1 4680	. . . . . . . . 9 |- (n = N -> (n` (xG(-u1Sy))) = (N` (xG(-u1Sy))))
24	23	opreq1d 4897	. . . . . . . 8 |- (n = N -> ((n` (xG(-u1Sy)))^2) = ((N` (xG(-u1Sy)))^2))
25	22, 24	opreq12d 4900	. . . . . . 7 |- (n = N -> (((n` (xGy))^2) + ((n` (xG(-u1Sy)))^2)) = (((N` (xGy))^2) + ((N` (xG(-u1Sy)))^2)))
26		fveq1 4680	. . . . . . . . . 10 |- (n = N -> (n` x) = (N` x))
27	26	opreq1d 4897	. . . . . . . . 9 |- (n = N -> ((n` x)^2) = ((N` x)^2))
28		fveq1 4680	. . . . . . . . . 10 |- (n = N -> (n` y) = (N` y))
29	28	opreq1d 4897	. . . . . . . . 9 |- (n = N -> ((n` y)^2) = ((N` y)^2))
30	27, 29	opreq12d 4900	. . . . . . . 8 |- (n = N -> (((n` x)^2) + ((n` y)^2)) = (((N` x)^2) + ((N` y)^2)))
31	30	opreq2d 4898	. . . . . . 7 |- (n = N -> (2 x. (((n` x)^2) + ((n` y)^2))) = (2 x. (((N` x)^2) + ((N` y)^2))))
32	25, 31	eqeq12d 1899	. . . . . 6 |- (n = N -> ((((n` (xGy))^2) + ((n` (xG(-u1Sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2))) <-> (((N` (xGy))^2) + ((N` (xG(-u1Sy)))^2)) = (2 x. (((N` x)^2) + ((N` y)^2)))))
33	32	ralbidv 2123	. . . . 5 |- (n = N -> (A.y e. X (((n` (xGy))^2) + ((n` (xG(-u1Sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2))) <-> A.y e. X (((N` (xGy))^2) + ((N` (xG(-u1Sy)))^2)) = (2 x. (((N` x)^2) + ((N` y)^2)))))
34	33	ralbidv 2123	. . . 4 |- (n = N -> (A.x e. X A.y e. X (((n` (xGy))^2) + ((n` (xG(-u1Sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2))) <-> A.x e. X A.y e. X (((N` (xGy))^2) + ((N` (xG(-u1Sy)))^2)) = (2 x. (((N` x)^2) + ((N` y)^2)))))
35	13, 20, 34	eloprabg 4936	. . 3 |- ((G e. A /\ S e. B /\ N e. C) -> (<.<.G, S>., N>. e. {<.<.g, s>., n>. | A.x e. ran gA.y e. ran g(((n` (xgy))^2) + ((n` (xg(-u1sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2)))} <-> A.x e. X A.y e. X (((N` (xGy))^2) + ((N` (xG(-u1Sy)))^2)) = (2 x. (((N` x)^2) + ((N` y)^2)))))
36	35	anbi2d 678	. 2 |- ((G e. A /\ S e. B /\ N e. C) -> ((<.<.G, S>., N>. e. NrmCVec /\ <.<.G, S>., N>. e. {<.<.g, s>., n>. | A.x e. ran gA.y e. ran g(((n` (xgy))^2) + ((n` (xg(-u1sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2)))}) <-> (<.<.G, S>., N>. e. NrmCVec /\ A.x e. X A.y e. X (((N` (xGy))^2) + ((N` (xG(-u1Sy)))^2)) = (2 x. (((N` x)^2) + ((N` y)^2))))))
37		df-ph 9813	. . . 4 |- CPreHil = (NrmCVec i^i {<.<.g, s>., n>. | A.x e. ran gA.y e. ran g(((n` (xgy))^2) + ((n` (xg(-u1sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2)))})
38	37	eleq2i 1961	. . 3 |- (<.<.G, S>., N>. e. CPreHil <-> <.<.G, S>., N>. e. (NrmCVec i^i {<.<.g, s>., n>. | A.x e. ran gA.y e. ran g(((n` (xgy))^2) + ((n` (xg(-u1sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2)))}))
39		elin 2786	. . 3 |- (<.<.G, S>., N>. e. (NrmCVec i^i {<.<.g, s>., n>. | A.x e. ran gA.y e. ran g(((n` (xgy))^2) + ((n` (xg(-u1sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2)))}) <-> (<.<.G, S>., N>. e. NrmCVec /\ <.<.G, S>., N>. e. {<.<.g, s>., n>. | A.x e. ran gA.y e. ran g(((n` (xgy))^2) + ((n` (xg(-u1sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2)))}))
40	38, 39	bitri 190	. 2 |- (<.<.G, S>., N>. e. CPreHil <-> (<.<.G, S>., N>. e. NrmCVec /\ <.<.G, S>., N>. e. {<.<.g, s>., n>. | A.x e. ran gA.y e. ran g(((n` (xgy))^2) + ((n` (xg(-u1sy)))^2)) = (2 x. (((n` x)^2) + ((n` y)^2)))}))
41	36, 40	syl5bb 591	1 |- ((G e. A /\ S e. B /\ N e. C) -> (<.<.G, S>., N>. e. CPreHil <-> (<.<.G, S>., N>. e. NrmCVec /\ A.x e. X A.y e. X (((N` (xGy))^2) + ((N` (xG(-u1Sy)))^2)) = (2 x. (((N` x)^2) + ((N` y)^2))))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105   i^i cin 2592  <.cop 3046  ran crn 3987  ` cfv 3998  (class class class)co 4884  {copab2 4885  1c1 6387   + caddc 6389   x. cmul 6391  -ucneg 6446  2c2 7145  ^cexp 7811  NrmCVeccnv 9535  CPreHilcphl 9812

This theorem is referenced by:  cnph 9819  isph 9822  phpar 9824  hhph 10678

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-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

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-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886  df-oprab 4887  df-ph 9813

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