Theorem isph 28605

Description: The predicate "is an inner product space." (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
isph.1	⊢ 𝑋 = (BaseSet‘𝑈)
isph.2	⊢ 𝐺 = ( +𝑣 ‘𝑈)
isph.3	⊢ 𝑀 = ( −𝑣 ‘𝑈)
isph.6	⊢ 𝑁 = (normCV‘𝑈)

Assertion

Ref	Expression
isph	⊢ (𝑈 ∈ CPreHilOLD ↔ (𝑈 ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))

Distinct variable groups:   𝑥,𝑦,𝐺   𝑥,𝑀,𝑦   𝑥,𝑁,𝑦   𝑥,𝑈,𝑦   𝑥,𝑋,𝑦

Proof of Theorem isph

Step	Hyp	Ref	Expression
1		phnv 28597	. 2 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec)
2		isph.2	. . . . 5 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
3		eqid 2798	. . . . 5 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
4		isph.6	. . . . 5 ⊢ 𝑁 = (normCV‘𝑈)
5	2, 3, 4	nvop 28459	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, ( ·𝑠OLD ‘𝑈)⟩, 𝑁⟩)
6		eleq1 2877	. . . . 5 ⊢ (𝑈 = ⟨⟨𝐺, ( ·𝑠OLD ‘𝑈)⟩, 𝑁⟩ → (𝑈 ∈ CPreHilOLD ↔ ⟨⟨𝐺, ( ·𝑠OLD ‘𝑈)⟩, 𝑁⟩ ∈ CPreHilOLD))
7	2	fvexi 6659	. . . . . . 7 ⊢ 𝐺 ∈ V
8		fvex 6658	. . . . . . 7 ⊢ ( ·𝑠OLD ‘𝑈) ∈ V
9	4	fvexi 6659	. . . . . . 7 ⊢ 𝑁 ∈ V
10		isph.1	. . . . . . . . 9 ⊢ 𝑋 = (BaseSet‘𝑈)
11	10, 2	bafval 28387	. . . . . . . 8 ⊢ 𝑋 = ran 𝐺
12	11	isphg 28600	. . . . . . 7 ⊢ ((𝐺 ∈ V ∧ ( ·𝑠OLD ‘𝑈) ∈ V ∧ 𝑁 ∈ V) → (⟨⟨𝐺, ( ·𝑠OLD ‘𝑈)⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, ( ·𝑠OLD ‘𝑈)⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1( ·𝑠OLD ‘𝑈)𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))))
13	7, 8, 9, 12	mp3an 1458	. . . . . 6 ⊢ (⟨⟨𝐺, ( ·𝑠OLD ‘𝑈)⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, ( ·𝑠OLD ‘𝑈)⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1( ·𝑠OLD ‘𝑈)𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))
14		isph.3	. . . . . . . . . . . . . . . 16 ⊢ 𝑀 = ( −𝑣 ‘𝑈)
15	10, 2, 3, 14	nvmval 28425	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝑀𝑦) = (𝑥𝐺(-1( ·𝑠OLD ‘𝑈)𝑦)))
16	15	3expa 1115	. . . . . . . . . . . . . 14 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑥𝑀𝑦) = (𝑥𝐺(-1( ·𝑠OLD ‘𝑈)𝑦)))
17	16	fveq2d 6649	. . . . . . . . . . . . 13 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑥𝑀𝑦)) = (𝑁‘(𝑥𝐺(-1( ·𝑠OLD ‘𝑈)𝑦))))
18	17	oveq1d 7150	. . . . . . . . . . . 12 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑁‘(𝑥𝑀𝑦))↑2) = ((𝑁‘(𝑥𝐺(-1( ·𝑠OLD ‘𝑈)𝑦)))↑2))
19	18	oveq2d 7151	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1( ·𝑠OLD ‘𝑈)𝑦)))↑2)))
20	19	eqeq1d 2800	. . . . . . . . . 10 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) ↔ (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1( ·𝑠OLD ‘𝑈)𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))
21	20	ralbidva 3161	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) → (∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) ↔ ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1( ·𝑠OLD ‘𝑈)𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))
22	21	ralbidva 3161	. . . . . . . 8 ⊢ (𝑈 ∈ NrmCVec → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1( ·𝑠OLD ‘𝑈)𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))
23	22	pm5.32i 578	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))) ↔ (𝑈 ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1( ·𝑠OLD ‘𝑈)𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))
24		eleq1 2877	. . . . . . . 8 ⊢ (𝑈 = ⟨⟨𝐺, ( ·𝑠OLD ‘𝑈)⟩, 𝑁⟩ → (𝑈 ∈ NrmCVec ↔ ⟨⟨𝐺, ( ·𝑠OLD ‘𝑈)⟩, 𝑁⟩ ∈ NrmCVec))
25	24	anbi1d 632	. . . . . . 7 ⊢ (𝑈 = ⟨⟨𝐺, ( ·𝑠OLD ‘𝑈)⟩, 𝑁⟩ → ((𝑈 ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1( ·𝑠OLD ‘𝑈)𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))) ↔ (⟨⟨𝐺, ( ·𝑠OLD ‘𝑈)⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1( ·𝑠OLD ‘𝑈)𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))))
26	23, 25	syl5rbb 287	. . . . . 6 ⊢ (𝑈 = ⟨⟨𝐺, ( ·𝑠OLD ‘𝑈)⟩, 𝑁⟩ → ((⟨⟨𝐺, ( ·𝑠OLD ‘𝑈)⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1( ·𝑠OLD ‘𝑈)𝑦)))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))) ↔ (𝑈 ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))))
27	13, 26	syl5bb 286	. . . . 5 ⊢ (𝑈 = ⟨⟨𝐺, ( ·𝑠OLD ‘𝑈)⟩, 𝑁⟩ → (⟨⟨𝐺, ( ·𝑠OLD ‘𝑈)⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (𝑈 ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))))
28	6, 27	bitrd 282	. . . 4 ⊢ (𝑈 = ⟨⟨𝐺, ( ·𝑠OLD ‘𝑈)⟩, 𝑁⟩ → (𝑈 ∈ CPreHilOLD ↔ (𝑈 ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))))
29	5, 28	syl 17	. . 3 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ CPreHilOLD ↔ (𝑈 ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))))
30	29	bianabs 545	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ CPreHilOLD ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))
31	1, 30	biadanii 821	1 ⊢ (𝑈 ∈ CPreHilOLD ↔ (𝑈 ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441  ⟨cop 4531  ‘cfv 6324  (class class class)co 7135  1c1 10527   + caddc 10529   · cmul 10531  -cneg 10860  2c2 11680  ↑cexp 13425  NrmCVeccnv 28367   +_𝑣 cpv 28368  BaseSetcba 28369   ·_𝑠OLD cns 28370   −_𝑣 cnsb 28372  norm_CVcnmcv 28373  CPreHil_OLDccphlo 28595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-po 5438  df-so 5439  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-ltxr 10669  df-sub 10861  df-neg 10862  df-grpo 28276  df-gid 28277  df-ginv 28278  df-gdiv 28279  df-ablo 28328  df-vc 28342  df-nv 28375  df-va 28378  df-ba 28379  df-sm 28380  df-0v 28381  df-vs 28382  df-nmcv 28383  df-ph 28596

This theorem is referenced by:  phpar2  28606