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Theorem phpar 30844

Description: The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
phpar.1	⊢ 𝑋 = (BaseSet‘𝑈)
phpar.2	⊢ 𝐺 = ( +_𝑣 ‘𝑈)
phpar.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
phpar.6	⊢ 𝑁 = (norm_CV‘𝑈)

**Assertion**
Ref	Expression
phpar	⊢ ((𝑈 ∈ CPreHil_OLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))

Proof of Theorem phpar Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		phpar.2	. . . . . 6 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
2	1	fvexi 6919	. . . . 5 ⊢ 𝐺 ∈ V
3		phpar.4	. . . . . 6 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
4	3	fvexi 6919	. . . . 5 ⊢ 𝑆 ∈ V
5		phpar.6	. . . . . 6 ⊢ 𝑁 = (norm_CV‘𝑈)
6	5	fvexi 6919	. . . . 5 ⊢ 𝑁 ∈ V
7	2, 4, 6	3pm3.2i 1339	. . . 4 ⊢ (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V)
8	1, 3, 5	phop 30838	. . . . . 6 ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
9	8	eleq1d 2825	. . . . 5 ⊢ (𝑈 ∈ CPreHil_OLD → (𝑈 ∈ CPreHil_OLD ↔ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHil_OLD))
10	9	ibi 267	. . . 4 ⊢ (𝑈 ∈ CPreHil_OLD → ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHil_OLD)
11		phpar.1	. . . . . . 7 ⊢ 𝑋 = (BaseSet‘𝑈)
12	11, 1	bafval 30624	. . . . . 6 ⊢ 𝑋 = ran 𝐺
13	12	isphg 30837	. . . . 5 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHil_OLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))))
14	13	simplbda 499	. . . 4 ⊢ (((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) ∧ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHil_OLD) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))
15	7, 10, 14	sylancr 587	. . 3 ⊢ (𝑈 ∈ CPreHil_OLD → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))
16	15	3ad2ant1 1133	. 2 ⊢ ((𝑈 ∈ CPreHil_OLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))
17		fvoveq1 7455	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺𝑦)) = (𝑁‘(𝐴𝐺𝑦)))
18	17	oveq1d 7447	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺𝑦))↑2) = ((𝑁‘(𝐴𝐺𝑦))↑2))
19		fvoveq1 7455	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺(-1𝑆𝑦))) = (𝑁‘(𝐴𝐺(-1𝑆𝑦))))
20	19	oveq1d 7447	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2) = ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2))
21	18, 20	oveq12d 7450	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2)))
22		fveq2 6905	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑁‘𝑥) = (𝑁‘𝐴))
23	22	oveq1d 7447	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑁‘𝑥)↑2) = ((𝑁‘𝐴)↑2))
24	23	oveq1d 7447	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2)))
25	24	oveq2d 7448	. . . . 5 ⊢ (𝑥 = 𝐴 → (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2))))
26	21, 25	eqeq12d 2752	. . . 4 ⊢ (𝑥 = 𝐴 → ((((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) ↔ (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2)))))
27		oveq2 7440	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
28	27	fveq2d 6909	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺𝑦)) = (𝑁‘(𝐴𝐺𝐵)))
29	28	oveq1d 7447	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺𝑦))↑2) = ((𝑁‘(𝐴𝐺𝐵))↑2))
30		oveq2 7440	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (-1𝑆𝑦) = (-1𝑆𝐵))
31	30	oveq2d 7448	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴𝐺(-1𝑆𝑦)) = (𝐴𝐺(-1𝑆𝐵)))
32	31	fveq2d 6909	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺(-1𝑆𝑦))) = (𝑁‘(𝐴𝐺(-1𝑆𝐵))))
33	32	oveq1d 7447	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2) = ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))
34	29, 33	oveq12d 7450	. . . . 5 ⊢ (𝑦 = 𝐵 → (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2)) = (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)))
35		fveq2 6905	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑁‘𝑦) = (𝑁‘𝐵))
36	35	oveq1d 7447	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝑁‘𝑦)↑2) = ((𝑁‘𝐵)↑2))
37	36	oveq2d 7448	. . . . . 6 ⊢ (𝑦 = 𝐵 → (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2)) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))
38	37	oveq2d 7448	. . . . 5 ⊢ (𝑦 = 𝐵 → (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2))) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))
39	34, 38	eqeq12d 2752	. . . 4 ⊢ (𝑦 = 𝐵 → ((((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2))) ↔ (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))))
40	26, 39	rspc2v 3632	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))))
41	40	3adant1 1130	. 2 ⊢ ((𝑈 ∈ CPreHil_OLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))))
42	16, 41	mpd 15	1 ⊢ ((𝑈 ∈ CPreHil_OLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 Vcvv 3479 ⟨cop 4631 ‘cfv 6560 (class class class)co 7432 1c1 11157 + caddc 11159 · cmul 11161 -cneg 11494 2c2 12322 ↑cexp 14103 NrmCVeccnv 30604 +_𝑣 cpv 30605 BaseSetcba 30606 ·_𝑠OLD cns 30607 norm_CVcnmcv 30610 CPreHil_OLDccphlo 30832

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-1st 8015 df-2nd 8016 df-vc 30579 df-nv 30612 df-va 30615 df-ba 30616 df-sm 30617 df-0v 30618 df-nmcv 30620 df-ph 30833

This theorem is referenced by: ip0i 30845 hlpar 30917

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