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Theorem phpar 28019
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 6345 . . . . 5 𝐺 ∈ V
3 phpar.4 . . . . . 6 𝑆 = ( ·𝑠OLD𝑈)
43fvexi 6345 . . . . 5 𝑆 ∈ V
5 phpar.6 . . . . . 6 𝑁 = (normCV𝑈)
65fvexi 6345 . . . . 5 𝑁 ∈ V
72, 4, 63pm3.2i 1423 . . . 4 (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V)
81, 3, 5phop 28013 . . . . . 6 (𝑈 ∈ CPreHilOLD𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
98eleq1d 2835 . . . . 5 (𝑈 ∈ CPreHilOLD → (𝑈 ∈ CPreHilOLD ↔ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD))
109ibi 256 . . . 4 (𝑈 ∈ CPreHilOLD → ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD)
11 phpar.1 . . . . . . 7 𝑋 = (BaseSet‘𝑈)
1211, 1bafval 27799 . . . . . 6 𝑋 = ran 𝐺
1312isphg 28012 . . . . 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 487 . . . 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 575 . . 3 (𝑈 ∈ CPreHilOLD → ∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))))
16153ad2ant1 1127 . 2 ((𝑈 ∈ CPreHilOLD𝐴𝑋𝐵𝑋) → ∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))))
17 fvoveq1 6819 . . . . . . 7 (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺𝑦)) = (𝑁‘(𝐴𝐺𝑦)))
1817oveq1d 6811 . . . . . 6 (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺𝑦))↑2) = ((𝑁‘(𝐴𝐺𝑦))↑2))
19 fvoveq1 6819 . . . . . . 7 (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺(-1𝑆𝑦))) = (𝑁‘(𝐴𝐺(-1𝑆𝑦))))
2019oveq1d 6811 . . . . . 6 (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2) = ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2))
2118, 20oveq12d 6814 . . . . 5 (𝑥 = 𝐴 → (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2)))
22 fveq2 6333 . . . . . . . 8 (𝑥 = 𝐴 → (𝑁𝑥) = (𝑁𝐴))
2322oveq1d 6811 . . . . . . 7 (𝑥 = 𝐴 → ((𝑁𝑥)↑2) = ((𝑁𝐴)↑2))
2423oveq1d 6811 . . . . . 6 (𝑥 = 𝐴 → (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2)) = (((𝑁𝐴)↑2) + ((𝑁𝑦)↑2)))
2524oveq2d 6812 . . . . 5 (𝑥 = 𝐴 → (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝑦)↑2))))
2621, 25eqeq12d 2786 . . . 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 6804 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
2827fveq2d 6337 . . . . . . 7 (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺𝑦)) = (𝑁‘(𝐴𝐺𝐵)))
2928oveq1d 6811 . . . . . 6 (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺𝑦))↑2) = ((𝑁‘(𝐴𝐺𝐵))↑2))
30 oveq2 6804 . . . . . . . . 9 (𝑦 = 𝐵 → (-1𝑆𝑦) = (-1𝑆𝐵))
3130oveq2d 6812 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴𝐺(-1𝑆𝑦)) = (𝐴𝐺(-1𝑆𝐵)))
3231fveq2d 6337 . . . . . . 7 (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺(-1𝑆𝑦))) = (𝑁‘(𝐴𝐺(-1𝑆𝐵))))
3332oveq1d 6811 . . . . . 6 (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2) = ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))
3429, 33oveq12d 6814 . . . . 5 (𝑦 = 𝐵 → (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2)) = (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)))
35 fveq2 6333 . . . . . . . 8 (𝑦 = 𝐵 → (𝑁𝑦) = (𝑁𝐵))
3635oveq1d 6811 . . . . . . 7 (𝑦 = 𝐵 → ((𝑁𝑦)↑2) = ((𝑁𝐵)↑2))
3736oveq2d 6812 . . . . . 6 (𝑦 = 𝐵 → (((𝑁𝐴)↑2) + ((𝑁𝑦)↑2)) = (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2)))
3837oveq2d 6812 . . . . 5 (𝑦 = 𝐵 → (2 · (((𝑁𝐴)↑2) + ((𝑁𝑦)↑2))) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2))))
3934, 38eqeq12d 2786 . . . 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 3472 . . 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 1124 . 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 1071   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  cop 4323  cfv 6030  (class class class)co 6796  1c1 10143   + caddc 10145   · cmul 10147  -cneg 10473  2c2 11276  cexp 13067  NrmCVeccnv 27779   +𝑣 cpv 27780  BaseSetcba 27781   ·𝑠OLD cns 27782  normCVcnmcv 27785  CPreHilOLDccphlo 28007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-1st 7319  df-2nd 7320  df-vc 27754  df-nv 27787  df-va 27790  df-ba 27791  df-sm 27792  df-0v 27793  df-nmcv 27795  df-ph 28008
This theorem is referenced by:  ip0i  28020  hlpar  28093
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