Theorem phpar2 31112

Description: The parallelogram law for an inner product space. (Contributed by NM, 2-Apr-2007.) (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
phpar2	⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))

Proof of Theorem phpar2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isph.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
2		isph.2	. . . . 5 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
3		isph.3	. . . . 5 ⊢ 𝑀 = ( −𝑣 ‘𝑈)
4		isph.6	. . . . 5 ⊢ 𝑁 = (normCV‘𝑈)
5	1, 2, 3, 4	isph 31111	. . . 4 ⊢ (𝑈 ∈ CPreHilOLD ↔ (𝑈 ∈ NrmCVec ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)))))
6	5	simprbi 502	. . 3 ⊢ (𝑈 ∈ CPreHilOLD → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))
7	6	3ad2ant1 1149	. 2 ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))))
8		fvoveq1 7431	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺𝑦)) = (𝑁‘(𝐴𝐺𝑦)))
9	8	oveq1d 7423	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺𝑦))↑2) = ((𝑁‘(𝐴𝐺𝑦))↑2))
10		fvoveq1 7431	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝑀𝑦)) = (𝑁‘(𝐴𝑀𝑦)))
11	10	oveq1d 7423	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝑀𝑦))↑2) = ((𝑁‘(𝐴𝑀𝑦))↑2))
12	9, 11	oveq12d 7426	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝑀𝑦))↑2)))
13		fveq2 6879	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑁‘𝑥) = (𝑁‘𝐴))
14	13	oveq1d 7423	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑁‘𝑥)↑2) = ((𝑁‘𝐴)↑2))
15	14	oveq1d 7423	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2)) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2)))
16	15	oveq2d 7424	. . . . 5 ⊢ (𝑥 = 𝐴 → (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2))))
17	12, 16	eqeq12d 2785	. . . 4 ⊢ (𝑥 = 𝐴 → ((((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) ↔ (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝑀𝑦))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2)))))
18		oveq2 7416	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
19	18	fveq2d 6883	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺𝑦)) = (𝑁‘(𝐴𝐺𝐵)))
20	19	oveq1d 7423	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺𝑦))↑2) = ((𝑁‘(𝐴𝐺𝐵))↑2))
21		oveq2 7416	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴𝑀𝑦) = (𝐴𝑀𝐵))
22	21	fveq2d 6883	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝑀𝑦)) = (𝑁‘(𝐴𝑀𝐵)))
23	22	oveq1d 7423	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝑀𝑦))↑2) = ((𝑁‘(𝐴𝑀𝐵))↑2))
24	20, 23	oveq12d 7426	. . . . 5 ⊢ (𝑦 = 𝐵 → (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝑀𝑦))↑2)) = (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)))
25		fveq2 6879	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑁‘𝑦) = (𝑁‘𝐵))
26	25	oveq1d 7423	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝑁‘𝑦)↑2) = ((𝑁‘𝐵)↑2))
27	26	oveq2d 7424	. . . . . 6 ⊢ (𝑦 = 𝐵 → (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2)) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))
28	27	oveq2d 7424	. . . . 5 ⊢ (𝑦 = 𝐵 → (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2))) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))
29	24, 28	eqeq12d 2785	. . . 4 ⊢ (𝑦 = 𝐵 → ((((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝑀𝑦))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝑦)↑2))) ↔ (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))))
30	17, 29	rspc2v 3601	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))))
31	30	3adant1 1146	. 2 ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝑀𝑦))↑2)) = (2 · (((𝑁‘𝑥)↑2) + ((𝑁‘𝑦)↑2))) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))))
32	7, 31	mpd 16	1 ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝑀𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))))

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ‘cfv 6534  (class class class)co 7408   + caddc 11099   · cmul 11101  2c2 12291  ↑cexp 14093  NrmCVeccnv 30873   +_𝑣 cpv 30874  BaseSetcba 30875   −_𝑣 cnsb 30878  norm_CVcnmcv 30879  CPreHil_OLDccphlo 31101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-po 5567  df-so 5568  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7982  df-2nd 7983  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11241  df-mnf 11242  df-ltxr 11244  df-sub 11439  df-neg 11440  df-grpo 30782  df-gid 30783  df-ginv 30784  df-gdiv 30785  df-ablo 30834  df-vc 30848  df-nv 30881  df-va 30884  df-ba 30885  df-sm 30886  df-0v 30887  df-vs 30888  df-nmcv 30889  df-ph 31102

This theorem is referenced by:  minvecolem2  31164  hlpar2  31185