Theorem isph 29085

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 29077	. 2 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec)
2		isph.2	. . . . 5 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
3		eqid 2738	. . . . 5 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈)
4		isph.6	. . . . 5 ⊢ 𝑁 = (normCV‘𝑈)
5	2, 3, 4	nvop 28939	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, ( ·𝑠OLD ‘𝑈)⟩, 𝑁⟩)
6		eleq1 2826	. . . . 5 ⊢ (𝑈 = ⟨⟨𝐺, ( ·𝑠OLD ‘𝑈)⟩, 𝑁⟩ → (𝑈 ∈ CPreHilOLD ↔ ⟨⟨𝐺, ( ·𝑠OLD ‘𝑈)⟩, 𝑁⟩ ∈ CPreHilOLD))
7	2	fvexi 6770	. . . . . . 7 ⊢ 𝐺 ∈ V
8		fvex 6769	. . . . . . 7 ⊢ ( ·𝑠OLD ‘𝑈) ∈ V
9	4	fvexi 6770	. . . . . . 7 ⊢ 𝑁 ∈ V
10		isph.1	. . . . . . . . 9 ⊢ 𝑋 = (BaseSet‘𝑈)
11	10, 2	bafval 28867	. . . . . . . 8 ⊢ 𝑋 = ran 𝐺
12	11	isphg 29080	. . . . . . 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 1459	. . . . . 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 28905	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝑀𝑦) = (𝑥𝐺(-1( ·𝑠OLD ‘𝑈)𝑦)))
16	15	3expa 1116	. . . . . . . . . . . . . 14 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑥𝑀𝑦) = (𝑥𝐺(-1( ·𝑠OLD ‘𝑈)𝑦)))
17	16	fveq2d 6760	. . . . . . . . . . . . 13 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑥𝑀𝑦)) = (𝑁‘(𝑥𝐺(-1( ·𝑠OLD ‘𝑈)𝑦))))
18	17	oveq1d 7270	. . . . . . . . . . . 12 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑁‘(𝑥𝑀𝑦))↑2) = ((𝑁‘(𝑥𝐺(-1( ·𝑠OLD ‘𝑈)𝑦)))↑2))
19	18	oveq2d 7271	. . . . . . . . . . 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 2740	. . . . . . . . . 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 3119	. . . . . . . . 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 3119	. . . . . . . 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 574	. . . . . . 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 2826	. . . . . . . 8 ⊢ (𝑈 = ⟨⟨𝐺, ( ·𝑠OLD ‘𝑈)⟩, 𝑁⟩ → (𝑈 ∈ NrmCVec ↔ ⟨⟨𝐺, ( ·𝑠OLD ‘𝑈)⟩, 𝑁⟩ ∈ NrmCVec))
25	24	anbi1d 629	. . . . . . 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	bitr2id 283	. . . . . 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 282	. . . . 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 278	. . . 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 541	. 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 818	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 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422  ⟨cop 4564  ‘cfv 6418  (class class class)co 7255  1c1 10803   + caddc 10805   · cmul 10807  -cneg 11136  2c2 11958  ↑cexp 13710  NrmCVeccnv 28847   +_𝑣 cpv 28848  BaseSetcba 28849   ·_𝑠OLD cns 28850   −_𝑣 cnsb 28852  norm_CVcnmcv 28853  CPreHil_OLDccphlo 29075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-po 5494  df-so 5495  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-ltxr 10945  df-sub 11137  df-neg 11138  df-grpo 28756  df-gid 28757  df-ginv 28758  df-gdiv 28759  df-ablo 28808  df-vc 28822  df-nv 28855  df-va 28858  df-ba 28859  df-sm 28860  df-0v 28861  df-vs 28862  df-nmcv 28863  df-ph 29076

This theorem is referenced by:  phpar2  29086