Theorem phpar 31223

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 = (normCV‘𝑈)

Assertion

Ref	Expression
phpar	⊢ ((𝑈 ∈ CPreHilOLD ∧ 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 6899	. . . . 5 ⊢ 𝐺 ∈ V
3		phpar.4	. . . . . 6 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
4	3	fvexi 6899	. . . . 5 ⊢ 𝑆 ∈ V
5		phpar.6	. . . . . 6 ⊢ 𝑁 = (normCV‘𝑈)
6	5	fvexi 6899	. . . . 5 ⊢ 𝑁 ∈ V
7	2, 4, 6	3pm3.2i 1358	. . . 4 ⊢ (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V)
8	1, 3, 5	phop 31217	. . . . . 6 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
9	8	eleq1d 2850	. . . . 5 ⊢ (𝑈 ∈ CPreHilOLD → (𝑈 ∈ CPreHilOLD ↔ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD))
10	9	ibi 270	. . . 4 ⊢ (𝑈 ∈ CPreHilOLD → ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD)
11		phpar.1	. . . . . . 7 ⊢ 𝑋 = (BaseSet‘𝑈)
12	11, 1	bafval 31003	. . . . . 6 ⊢ 𝑋 = ran 𝐺
13	12	isphg 31216	. . . . 5 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨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 505	. . . 4 ⊢ (((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) ∧ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))
15	7, 10, 14	sylancr 599	. . 3 ⊢ (𝑈 ∈ CPreHilOLD → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))
16	15	3ad2ant1 1151	. 2 ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))
17		fvoveq1 7439	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺𝑦)) = (𝑁‘(𝐴𝐺𝑦)))
18	17	oveq1d 7431	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺𝑦))↑2) = ((𝑁‘(𝐴𝐺𝑦))↑2))
19		fvoveq1 7439	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺(-1𝑆𝑦))) = (𝑁‘(𝐴𝐺(-1𝑆𝑦))))
20	19	oveq1d 7431	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2) = ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2))
21	18, 20	oveq12d 7434	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2)))
22		fveq2 6885	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑁‘𝑥) = (𝑁‘𝐴))
23	22	oveq1d 7431	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑁‘𝑥)↑2) = ((𝑁‘𝐴)↑2))
24	23	oveq1d 7431	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2)))
25	24	oveq2d 7432	. . . . 5 ⊢ (𝑥 = 𝐴 → (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2))))
26	21, 25	eqeq12d 2781	. . . 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 7424	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
28	27	fveq2d 6889	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺𝑦)) = (𝑁‘(𝐴𝐺𝐵)))
29	28	oveq1d 7431	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺𝑦))↑2) = ((𝑁‘(𝐴𝐺𝐵))↑2))
30		oveq2 7424	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (-1𝑆𝑦) = (-1𝑆𝐵))
31	30	oveq2d 7432	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴𝐺(-1𝑆𝑦)) = (𝐴𝐺(-1𝑆𝐵)))
32	31	fveq2d 6889	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺(-1𝑆𝑦))) = (𝑁‘(𝐴𝐺(-1𝑆𝐵))))
33	32	oveq1d 7431	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2) = ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))
34	29, 33	oveq12d 7434	. . . . 5 ⊢ (𝑦 = 𝐵 → (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2)) = (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)))
35		fveq2 6885	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑁‘𝑦) = (𝑁‘𝐵))
36	35	oveq1d 7431	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝑁‘𝑦)↑2) = ((𝑁‘𝐵)↑2))
37	36	oveq2d 7432	. . . . . 6 ⊢ (𝑦 = 𝐵 → (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2)) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))
38	37	oveq2d 7432	. . . . 5 ⊢ (𝑦 = 𝐵 → (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2))) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))
39	34, 38	eqeq12d 2781	. . . 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 3594	. . 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 1148	. 2 ⊢ ((𝑈 ∈ CPreHilOLD ∧ 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 16	1 ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081  Vcvv 3457  ⟨cop 4597  ‘cfv 6540  (class class class)co 7416  1c1 11112   + caddc 11114   · cmul 11116  -cneg 11453  2c2 12306  ↑cexp 14111  NrmCVeccnv 30983   +_𝑣 cpv 30984  BaseSetcba 30985   ·_𝑠OLD cns 30986  norm_CVcnmcv 30989  CPreHil_OLDccphlo 31211

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-oprab 7420  df-1st 7988  df-2nd 7989  df-vc 30958  df-nv 30991  df-va 30994  df-ba 30995  df-sm 30996  df-0v 30997  df-nmcv 30999  df-ph 31212

This theorem is used by:  ip0i  31224  hlpar  31296