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Theorem phpar 28230
 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 = ( +𝑣𝑈)
phpar.4 𝑆 = ( ·𝑠OLD𝑈)
phpar.6 𝑁 = (normCV𝑈)
Assertion
Ref Expression
phpar ((𝑈 ∈ CPreHilOLD𝐴𝑋𝐵𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2))))

Proof of Theorem phpar
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 phpar.2 . . . . . 6 𝐺 = ( +𝑣𝑈)
21fvexi 6451 . . . . 5 𝐺 ∈ V
3 phpar.4 . . . . . 6 𝑆 = ( ·𝑠OLD𝑈)
43fvexi 6451 . . . . 5 𝑆 ∈ V
5 phpar.6 . . . . . 6 𝑁 = (normCV𝑈)
65fvexi 6451 . . . . 5 𝑁 ∈ V
72, 4, 63pm3.2i 1442 . . . 4 (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V)
81, 3, 5phop 28224 . . . . . 6 (𝑈 ∈ CPreHilOLD𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
98eleq1d 2891 . . . . 5 (𝑈 ∈ CPreHilOLD → (𝑈 ∈ CPreHilOLD ↔ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD))
109ibi 259 . . . 4 (𝑈 ∈ CPreHilOLD → ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD)
11 phpar.1 . . . . . . 7 𝑋 = (BaseSet‘𝑈)
1211, 1bafval 28010 . . . . . 6 𝑋 = ran 𝐺
1312isphg 28223 . . . . 5 ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) → (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD ↔ (⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ NrmCVec ∧ ∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))))))
1413simplbda 495 . . . 4 (((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) ∧ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD) → ∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))))
157, 10, 14sylancr 581 . . 3 (𝑈 ∈ CPreHilOLD → ∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))))
16153ad2ant1 1167 . 2 ((𝑈 ∈ CPreHilOLD𝐴𝑋𝐵𝑋) → ∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))))
17 fvoveq1 6933 . . . . . . 7 (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺𝑦)) = (𝑁‘(𝐴𝐺𝑦)))
1817oveq1d 6925 . . . . . 6 (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺𝑦))↑2) = ((𝑁‘(𝐴𝐺𝑦))↑2))
19 fvoveq1 6933 . . . . . . 7 (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺(-1𝑆𝑦))) = (𝑁‘(𝐴𝐺(-1𝑆𝑦))))
2019oveq1d 6925 . . . . . 6 (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2) = ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2))
2118, 20oveq12d 6928 . . . . 5 (𝑥 = 𝐴 → (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2)))
22 fveq2 6437 . . . . . . . 8 (𝑥 = 𝐴 → (𝑁𝑥) = (𝑁𝐴))
2322oveq1d 6925 . . . . . . 7 (𝑥 = 𝐴 → ((𝑁𝑥)↑2) = ((𝑁𝐴)↑2))
2423oveq1d 6925 . . . . . 6 (𝑥 = 𝐴 → (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2)) = (((𝑁𝐴)↑2) + ((𝑁𝑦)↑2)))
2524oveq2d 6926 . . . . 5 (𝑥 = 𝐴 → (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝑦)↑2))))
2621, 25eqeq12d 2840 . . . 4 (𝑥 = 𝐴 → ((((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))) ↔ (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝑦)↑2)))))
27 oveq2 6918 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
2827fveq2d 6441 . . . . . . 7 (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺𝑦)) = (𝑁‘(𝐴𝐺𝐵)))
2928oveq1d 6925 . . . . . 6 (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺𝑦))↑2) = ((𝑁‘(𝐴𝐺𝐵))↑2))
30 oveq2 6918 . . . . . . . . 9 (𝑦 = 𝐵 → (-1𝑆𝑦) = (-1𝑆𝐵))
3130oveq2d 6926 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴𝐺(-1𝑆𝑦)) = (𝐴𝐺(-1𝑆𝐵)))
3231fveq2d 6441 . . . . . . 7 (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺(-1𝑆𝑦))) = (𝑁‘(𝐴𝐺(-1𝑆𝐵))))
3332oveq1d 6925 . . . . . 6 (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2) = ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))
3429, 33oveq12d 6928 . . . . 5 (𝑦 = 𝐵 → (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2)) = (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)))
35 fveq2 6437 . . . . . . . 8 (𝑦 = 𝐵 → (𝑁𝑦) = (𝑁𝐵))
3635oveq1d 6925 . . . . . . 7 (𝑦 = 𝐵 → ((𝑁𝑦)↑2) = ((𝑁𝐵)↑2))
3736oveq2d 6926 . . . . . 6 (𝑦 = 𝐵 → (((𝑁𝐴)↑2) + ((𝑁𝑦)↑2)) = (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2)))
3837oveq2d 6926 . . . . 5 (𝑦 = 𝐵 → (2 · (((𝑁𝐴)↑2) + ((𝑁𝑦)↑2))) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2))))
3934, 38eqeq12d 2840 . . . 4 (𝑦 = 𝐵 → ((((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝑦)↑2))) ↔ (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2)))))
4026, 39rspc2v 3539 . . 3 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2)))))
41403adant1 1164 . 2 ((𝑈 ∈ CPreHilOLD𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2)))))
4216, 41mpd 15 1 ((𝑈 ∈ CPreHilOLD𝐴𝑋𝐵𝑋) → (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  ∀wral 3117  Vcvv 3414  ⟨cop 4405  ‘cfv 6127  (class class class)co 6910  1c1 10260   + caddc 10262   · cmul 10264  -cneg 10593  2c2 11413  ↑cexp 13161  NrmCVeccnv 27990   +𝑣 cpv 27991  BaseSetcba 27992   ·𝑠OLD cns 27993  normCVcnmcv 27996  CPreHilOLDccphlo 28218 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-1st 7433  df-2nd 7434  df-vc 27965  df-nv 27998  df-va 28001  df-ba 28002  df-sm 28003  df-0v 28004  df-nmcv 28006  df-ph 28219 This theorem is referenced by:  ip0i  28231  hlpar  28304
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