isph - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > isph		Structured version Visualization version GIF version

Theorem isph 27677

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 = (norm_CV‘𝑈)

**Assertion**
Ref	Expression
isph	⊢ (𝑈 ∈ CPreHil_OLD ↔ (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 27669	. 2 ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 ∈ NrmCVec)
2		isph.2	. . . . 5 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
3		eqid 2622	. . . . 5 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
4		isph.6	. . . . 5 ⊢ 𝑁 = (norm_CV‘𝑈)
5	2, 3, 4	nvop 27531	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, ( ·_𝑠OLD ‘𝑈)⟩, 𝑁⟩)
6		eleq1 2689	. . . . 5 ⊢ (𝑈 = ⟨⟨𝐺, ( ·_𝑠OLD ‘𝑈)⟩, 𝑁⟩ → (𝑈 ∈ CPreHil_OLD ↔ ⟨⟨𝐺, ( ·_𝑠OLD ‘𝑈)⟩, 𝑁⟩ ∈ CPreHil_OLD))
7		fvex 6201	. . . . . . . 8 ⊢ ( +_𝑣 ‘𝑈) ∈ V
8	2, 7	eqeltri 2697	. . . . . . 7 ⊢ 𝐺 ∈ V
9		fvex 6201	. . . . . . 7 ⊢ ( ·_𝑠OLD ‘𝑈) ∈ V
10		fvex 6201	. . . . . . . 8 ⊢ (norm_CV‘𝑈) ∈ V
11	4, 10	eqeltri 2697	. . . . . . 7 ⊢ 𝑁 ∈ V
12		isph.1	. . . . . . . . 9 ⊢ 𝑋 = (BaseSet‘𝑈)
13	12, 2	bafval 27459	. . . . . . . 8 ⊢ 𝑋 = ran 𝐺
14	13	isphg 27672	. . . . . . 7 ⊢ ((𝐺 ∈ V ∧ ( ·_𝑠OLD ‘𝑈) ∈ V ∧ 𝑁 ∈ V) → (⟨⟨𝐺, ( ·_𝑠OLD ‘𝑈)⟩, 𝑁⟩ ∈ CPreHil_OLD ↔ (⟨⟨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))))))
15	8, 9, 11, 14	mp3an 1424	. . . . . 6 ⊢ (⟨⟨𝐺, ( ·_𝑠OLD ‘𝑈)⟩, 𝑁⟩ ∈ CPreHil_OLD ↔ (⟨⟨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)))))
16		isph.3	. . . . . . . . . . . . . . . 16 ⊢ 𝑀 = ( −_𝑣 ‘𝑈)
17	12, 2, 3, 16	nvmval 27497	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝑀𝑦) = (𝑥𝐺(-1( ·_𝑠OLD ‘𝑈)𝑦)))
18	17	3expa 1265	. . . . . . . . . . . . . 14 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑥𝑀𝑦) = (𝑥𝐺(-1( ·_𝑠OLD ‘𝑈)𝑦)))
19	18	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (𝑁‘(𝑥𝑀𝑦)) = (𝑁‘(𝑥𝐺(-1( ·_𝑠OLD ‘𝑈)𝑦))))
20	19	oveq1d 6665	. . . . . . . . . . . 12 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → ((𝑁‘(𝑥𝑀𝑦))↑2) = ((𝑁‘(𝑥𝐺(-1( ·_𝑠OLD ‘𝑈)𝑦)))↑2))
21	20	oveq2d 6666	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) → (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1( ·_𝑠OLD ‘𝑈)𝑦)))↑2)))
22	21	eqeq1d 2624	. . . . . . . . . 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)))))
23	22	ralbidva 2985	. . . . . . . . 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)))))
24	23	ralbidva 2985	. . . . . . . 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)))))
25	24	pm5.32i 669	. . . . . . 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)))))
26		eleq1 2689	. . . . . . . 8 ⊢ (𝑈 = ⟨⟨𝐺, ( ·_𝑠OLD ‘𝑈)⟩, 𝑁⟩ → (𝑈 ∈ NrmCVec ↔ ⟨⟨𝐺, ( ·_𝑠OLD ‘𝑈)⟩, 𝑁⟩ ∈ NrmCVec))
27	26	anbi1d 741	. . . . . . 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))))))
28	25, 27	syl5rbb 273	. . . . . 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))))))
29	15, 28	syl5bb 272	. . . . 5 ⊢ (𝑈 = ⟨⟨𝐺, ( ·_𝑠OLD ‘𝑈)⟩, 𝑁⟩ → (⟨⟨𝐺, ( ·_𝑠OLD ‘𝑈)⟩, 𝑁⟩ ∈ CPreHil_OLD ↔ (𝑈 ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))))
30	6, 29	bitrd 268	. . . 4 ⊢ (𝑈 = ⟨⟨𝐺, ( ·_𝑠OLD ‘𝑈)⟩, 𝑁⟩ → (𝑈 ∈ CPreHil_OLD ↔ (𝑈 ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))))
31	5, 30	syl 17	. . 3 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ CPreHil_OLD ↔ (𝑈 ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))))
32	31	bianabs 924	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ CPreHil_OLD ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))
33	1, 32	biadan2 674	1 ⊢ (𝑈 ∈ CPreHil_OLD ↔ (𝑈 ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ⟨cop 4183 ‘cfv 5888 (class class class)co 6650 1c1 9937 + caddc 9939 · cmul 9941 -cneg 10267 2c2 11070 ↑cexp 12860 NrmCVeccnv 27439 +_𝑣 cpv 27440 BaseSetcba 27441 ·_𝑠OLD cns 27442 −_𝑣 cnsb 27444 norm_CVcnmcv 27445 CPreHil_OLDccphlo 27667

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-neg 10269 df-grpo 27347 df-gid 27348 df-ginv 27349 df-gdiv 27350 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-vs 27454 df-nmcv 27455 df-ph 27668

This theorem is referenced by: phpar2 27678 sspph 27710

W3C validator