Theorem phpar 9824

Description: The parallelogram law for an inner product space.

Hypotheses

Ref	Expression
phpar.1	|- X = (BaseSet` U)
phpar.2	|- G = (+v` U)
phpar.4	|- S = (.s` U)
phpar.6	|- N = (norm` U)

Assertion

Ref	Expression
phpar	|- ((U e. CPreHil /\ A e. X /\ B e. X) -> (((N` (AGB))^2) + ((N` (AG(-u1SB)))^2)) = (2 x. (((N` A)^2) + ((N` B)^2))))

Proof of Theorem phpar

Step	Hyp	Ref	Expression
1		phpar.2	. . . . . . . 8 |- G = (+v` U)
2	1	vafval 9554	. . . . . . 7 |- G = (1st` (1st` U))
3		fvex 4689	. . . . . . 7 |- (1st` (1st` U)) e. _V
4	2, 3	eqeltri 1967	. . . . . 6 |- G e. _V
5		phpar.4	. . . . . . . 8 |- S = (.s` U)
6	5	smfval 9556	. . . . . . 7 |- S = (2nd` (1st` U))
7		fvex 4689	. . . . . . 7 |- (2nd` (1st` U)) e. _V
8	6, 7	eqeltri 1967	. . . . . 6 |- S e. _V
9		phpar.6	. . . . . . . 8 |- N = (norm` U)
10	9	nmfval 9558	. . . . . . 7 |- N = (2nd` U)
11		fvex 4689	. . . . . . 7 |- (2nd` U) e. _V
12	10, 11	eqeltri 1967	. . . . . 6 |- N e. _V
13	4, 8, 12	3pm3.2i 1048	. . . . 5 |- (G e. _V /\ S e. _V /\ N e. _V)
14	13	a1i 8	. . . 4 |- (U e. CPreHil -> (G e. _V /\ S e. _V /\ N e. _V))
15	1, 5, 9	phop 9818	. . . . . 6 |- (U e. CPreHil -> U = <.<.G, S>., N>.)
16	15	eleq1d 1963	. . . . 5 |- (U e. CPreHil -> (U e. CPreHil <-> <.<.G, S>., N>. e. CPreHil))
17	16	ibi 652	. . . 4 |- (U e. CPreHil -> <.<.G, S>., N>. e. CPreHil)
18		phpar.1	. . . . . . 7 |- X = (BaseSet` U)
19	18, 1	bafval 9555	. . . . . 6 |- X = ran G
20	19	isphg 9817	. . . . 5 |- ((G e. _V /\ S e. _V /\ N e. _V) -> (<.<.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))))))
21	20	simplbda 465	. . . 4 |- (((G e. _V /\ S e. _V /\ N e. _V) /\ <.<.G, S>., N>. e. CPreHil) -> A.x e. X A.y e. X (((N` (xGy))^2) + ((N` (xG(-u1Sy)))^2)) = (2 x. (((N` x)^2) + ((N` y)^2))))
22	14, 17, 21	syl11anc 524	. . 3 |- (U e. CPreHil -> A.x e. X A.y e. X (((N` (xGy))^2) + ((N` (xG(-u1Sy)))^2)) = (2 x. (((N` x)^2) + ((N` y)^2))))
23	22	3ad2ant1 897	. 2 |- ((U e. CPreHil /\ A e. X /\ B e. X) -> A.x e. X A.y e. X (((N` (xGy))^2) + ((N` (xG(-u1Sy)))^2)) = (2 x. (((N` x)^2) + ((N` y)^2))))
24		opreq1 4889	. . . . . . . 8 |- (x = A -> (xGy) = (AGy))
25	24	fveq2d 4685	. . . . . . 7 |- (x = A -> (N` (xGy)) = (N` (AGy)))
26	25	opreq1d 4897	. . . . . 6 |- (x = A -> ((N` (xGy))^2) = ((N` (AGy))^2))
27		opreq1 4889	. . . . . . . 8 |- (x = A -> (xG(-u1Sy)) = (AG(-u1Sy)))
28	27	fveq2d 4685	. . . . . . 7 |- (x = A -> (N` (xG(-u1Sy))) = (N` (AG(-u1Sy))))
29	28	opreq1d 4897	. . . . . 6 |- (x = A -> ((N` (xG(-u1Sy)))^2) = ((N` (AG(-u1Sy)))^2))
30	26, 29	opreq12d 4900	. . . . 5 |- (x = A -> (((N` (xGy))^2) + ((N` (xG(-u1Sy)))^2)) = (((N` (AGy))^2) + ((N` (AG(-u1Sy)))^2)))
31		fveq2 4681	. . . . . . . 8 |- (x = A -> (N` x) = (N` A))
32	31	opreq1d 4897	. . . . . . 7 |- (x = A -> ((N` x)^2) = ((N` A)^2))
33	32	opreq1d 4897	. . . . . 6 |- (x = A -> (((N` x)^2) + ((N` y)^2)) = (((N` A)^2) + ((N` y)^2)))
34	33	opreq2d 4898	. . . . 5 |- (x = A -> (2 x. (((N` x)^2) + ((N` y)^2))) = (2 x. (((N` A)^2) + ((N` y)^2))))
35	30, 34	eqeq12d 1899	. . . 4 |- (x = A -> ((((N` (xGy))^2) + ((N` (xG(-u1Sy)))^2)) = (2 x. (((N` x)^2) + ((N` y)^2))) <-> (((N` (AGy))^2) + ((N` (AG(-u1Sy)))^2)) = (2 x. (((N` A)^2) + ((N` y)^2)))))
36		opreq2 4890	. . . . . . . 8 |- (y = B -> (AGy) = (AGB))
37	36	fveq2d 4685	. . . . . . 7 |- (y = B -> (N` (AGy)) = (N` (AGB)))
38	37	opreq1d 4897	. . . . . 6 |- (y = B -> ((N` (AGy))^2) = ((N` (AGB))^2))
39		opreq2 4890	. . . . . . . . 9 |- (y = B -> (-u1Sy) = (-u1SB))
40	39	opreq2d 4898	. . . . . . . 8 |- (y = B -> (AG(-u1Sy)) = (AG(-u1SB)))
41	40	fveq2d 4685	. . . . . . 7 |- (y = B -> (N` (AG(-u1Sy))) = (N` (AG(-u1SB))))
42	41	opreq1d 4897	. . . . . 6 |- (y = B -> ((N` (AG(-u1Sy)))^2) = ((N` (AG(-u1SB)))^2))
43	38, 42	opreq12d 4900	. . . . 5 |- (y = B -> (((N` (AGy))^2) + ((N` (AG(-u1Sy)))^2)) = (((N` (AGB))^2) + ((N` (AG(-u1SB)))^2)))
44		fveq2 4681	. . . . . . . 8 |- (y = B -> (N` y) = (N` B))
45	44	opreq1d 4897	. . . . . . 7 |- (y = B -> ((N` y)^2) = ((N` B)^2))
46	45	opreq2d 4898	. . . . . 6 |- (y = B -> (((N` A)^2) + ((N` y)^2)) = (((N` A)^2) + ((N` B)^2)))
47	46	opreq2d 4898	. . . . 5 |- (y = B -> (2 x. (((N` A)^2) + ((N` y)^2))) = (2 x. (((N` A)^2) + ((N` B)^2))))
48	43, 47	eqeq12d 1899	. . . 4 |- (y = B -> ((((N` (AGy))^2) + ((N` (AG(-u1Sy)))^2)) = (2 x. (((N` A)^2) + ((N` y)^2))) <-> (((N` (AGB))^2) + ((N` (AG(-u1SB)))^2)) = (2 x. (((N` A)^2) + ((N` B)^2)))))
49	35, 48	rcla42v 2384	. . 3 |- ((A e. X /\ B e. X) -> (A.x e. X A.y e. X (((N` (xGy))^2) + ((N` (xG(-u1Sy)))^2)) = (2 x. (((N` x)^2) + ((N` y)^2))) -> (((N` (AGB))^2) + ((N` (AG(-u1SB)))^2)) = (2 x. (((N` A)^2) + ((N` B)^2)))))
50	49	3adant1 894	. 2 |- ((U e. CPreHil /\ A e. X /\ B e. X) -> (A.x e. X A.y e. X (((N` (xGy))^2) + ((N` (xG(-u1Sy)))^2)) = (2 x. (((N` x)^2) + ((N` y)^2))) -> (((N` (AGB))^2) + ((N` (AG(-u1SB)))^2)) = (2 x. (((N` A)^2) + ((N` B)^2)))))
51	23, 50	mpd 29	1 |- ((U e. CPreHil /\ A e. X /\ B e. X) -> (((N` (AGB))^2) + ((N` (AG(-u1SB)))^2)) = (2 x. (((N` A)^2) + ((N` B)^2))))

Syntax hints:   -> wi 3   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  <.cop 3046  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  1c1 6387   + caddc 6389   x. cmul 6391  -ucneg 6446  2c2 7145  ^cexp 7811  NrmCVeccnv 9535  +vcpv 9536  BaseSetcba 9537  .scns 9538  normcnm 9541  CPreHilcphl 9812

This theorem is referenced by:  ip0i 9825  hlpar 9946

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-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  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-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-gid 9317  df-vc 9497  df-nv 9543  df-va 9546  df-ba 9547  df-sm 9548  df-0v 9549  df-nm 9551  df-ph 9813

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