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

Theorem phpar 28605
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 6673 . . . . 5 𝐺 ∈ V
3 phpar.4 . . . . . 6 𝑆 = ( ·𝑠OLD𝑈)
43fvexi 6673 . . . . 5 𝑆 ∈ V
5 phpar.6 . . . . . 6 𝑁 = (normCV𝑈)
65fvexi 6673 . . . . 5 𝑁 ∈ V
72, 4, 63pm3.2i 1336 . . . 4 (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V)
81, 3, 5phop 28599 . . . . . 6 (𝑈 ∈ CPreHilOLD𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
98eleq1d 2900 . . . . 5 (𝑈 ∈ CPreHilOLD → (𝑈 ∈ CPreHilOLD ↔ ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD))
109ibi 270 . . . 4 (𝑈 ∈ CPreHilOLD → ⟨⟨𝐺, 𝑆⟩, 𝑁⟩ ∈ CPreHilOLD)
11 phpar.1 . . . . . . 7 𝑋 = (BaseSet‘𝑈)
1211, 1bafval 28385 . . . . . 6 𝑋 = ran 𝐺
1312isphg 28598 . . . . 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 503 . . . 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 590 . . 3 (𝑈 ∈ CPreHilOLD → ∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))))
16153ad2ant1 1130 . 2 ((𝑈 ∈ CPreHilOLD𝐴𝑋𝐵𝑋) → ∀𝑥𝑋𝑦𝑋 (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (2 · (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2))))
17 fvoveq1 7169 . . . . . . 7 (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺𝑦)) = (𝑁‘(𝐴𝐺𝑦)))
1817oveq1d 7161 . . . . . 6 (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺𝑦))↑2) = ((𝑁‘(𝐴𝐺𝑦))↑2))
19 fvoveq1 7169 . . . . . . 7 (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺(-1𝑆𝑦))) = (𝑁‘(𝐴𝐺(-1𝑆𝑦))))
2019oveq1d 7161 . . . . . 6 (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2) = ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2))
2118, 20oveq12d 7164 . . . . 5 (𝑥 = 𝐴 → (((𝑁‘(𝑥𝐺𝑦))↑2) + ((𝑁‘(𝑥𝐺(-1𝑆𝑦)))↑2)) = (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2)))
22 fveq2 6659 . . . . . . . 8 (𝑥 = 𝐴 → (𝑁𝑥) = (𝑁𝐴))
2322oveq1d 7161 . . . . . . 7 (𝑥 = 𝐴 → ((𝑁𝑥)↑2) = ((𝑁𝐴)↑2))
2423oveq1d 7161 . . . . . 6 (𝑥 = 𝐴 → (((𝑁𝑥)↑2) + ((𝑁𝑦)↑2)) = (((𝑁𝐴)↑2) + ((𝑁𝑦)↑2)))
2524oveq2d 7162 . . . . 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 7154 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
2827fveq2d 6663 . . . . . . 7 (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺𝑦)) = (𝑁‘(𝐴𝐺𝐵)))
2928oveq1d 7161 . . . . . 6 (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺𝑦))↑2) = ((𝑁‘(𝐴𝐺𝐵))↑2))
30 oveq2 7154 . . . . . . . . 9 (𝑦 = 𝐵 → (-1𝑆𝑦) = (-1𝑆𝐵))
3130oveq2d 7162 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴𝐺(-1𝑆𝑦)) = (𝐴𝐺(-1𝑆𝐵)))
3231fveq2d 6663 . . . . . . 7 (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺(-1𝑆𝑦))) = (𝑁‘(𝐴𝐺(-1𝑆𝐵))))
3332oveq1d 7161 . . . . . 6 (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2) = ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2))
3429, 33oveq12d 7164 . . . . 5 (𝑦 = 𝐵 → (((𝑁‘(𝐴𝐺𝑦))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝑦)))↑2)) = (((𝑁‘(𝐴𝐺𝐵))↑2) + ((𝑁‘(𝐴𝐺(-1𝑆𝐵)))↑2)))
35 fveq2 6659 . . . . . . . 8 (𝑦 = 𝐵 → (𝑁𝑦) = (𝑁𝐵))
3635oveq1d 7161 . . . . . . 7 (𝑦 = 𝐵 → ((𝑁𝑦)↑2) = ((𝑁𝐵)↑2))
3736oveq2d 7162 . . . . . 6 (𝑦 = 𝐵 → (((𝑁𝐴)↑2) + ((𝑁𝑦)↑2)) = (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2)))
3837oveq2d 7162 . . . . 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 3619 . . 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 1127 . 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 1084   = wceq 1538  wcel 2115  wral 3133  Vcvv 3480  cop 4556  cfv 6344  (class class class)co 7146  1c1 10532   + caddc 10534   · cmul 10536  -cneg 10865  2c2 11687  cexp 13432  NrmCVeccnv 28365   +𝑣 cpv 28366  BaseSetcba 28367   ·𝑠OLD cns 28368  normCVcnmcv 28371  CPreHilOLDccphlo 28593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7149  df-oprab 7150  df-1st 7681  df-2nd 7682  df-vc 28340  df-nv 28373  df-va 28376  df-ba 28377  df-sm 28378  df-0v 28379  df-nmcv 28381  df-ph 28594
This theorem is referenced by:  ip0i  28606  hlpar  28678
  Copyright terms: Public domain W3C validator