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Theorem phpar 29808
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 = (BaseSetβ€˜π‘ˆ)
phpar.2 𝐺 = ( +𝑣 β€˜π‘ˆ)
phpar.4 𝑆 = ( ·𝑠OLD β€˜π‘ˆ)
phpar.6 𝑁 = (normCVβ€˜π‘ˆ)
Assertion
Ref Expression
phpar ((π‘ˆ ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (((π‘β€˜(𝐴𝐺𝐡))↑2) + ((π‘β€˜(𝐴𝐺(-1𝑆𝐡)))↑2)) = (2 Β· (((π‘β€˜π΄)↑2) + ((π‘β€˜π΅)↑2))))

Proof of Theorem phpar
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 phpar.2 . . . . . 6 𝐺 = ( +𝑣 β€˜π‘ˆ)
21fvexi 6861 . . . . 5 𝐺 ∈ V
3 phpar.4 . . . . . 6 𝑆 = ( ·𝑠OLD β€˜π‘ˆ)
43fvexi 6861 . . . . 5 𝑆 ∈ V
5 phpar.6 . . . . . 6 𝑁 = (normCVβ€˜π‘ˆ)
65fvexi 6861 . . . . 5 𝑁 ∈ V
72, 4, 63pm3.2i 1340 . . . 4 (𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V)
81, 3, 5phop 29802 . . . . . 6 (π‘ˆ ∈ CPreHilOLD β†’ π‘ˆ = ⟨⟨𝐺, π‘†βŸ©, π‘βŸ©)
98eleq1d 2823 . . . . 5 (π‘ˆ ∈ CPreHilOLD β†’ (π‘ˆ ∈ CPreHilOLD ↔ ⟨⟨𝐺, π‘†βŸ©, π‘βŸ© ∈ CPreHilOLD))
109ibi 267 . . . 4 (π‘ˆ ∈ CPreHilOLD β†’ ⟨⟨𝐺, π‘†βŸ©, π‘βŸ© ∈ CPreHilOLD)
11 phpar.1 . . . . . . 7 𝑋 = (BaseSetβ€˜π‘ˆ)
1211, 1bafval 29588 . . . . . 6 𝑋 = ran 𝐺
1312isphg 29801 . . . . 5 ((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) β†’ (⟨⟨𝐺, π‘†βŸ©, π‘βŸ© ∈ CPreHilOLD ↔ (⟨⟨𝐺, π‘†βŸ©, π‘βŸ© ∈ NrmCVec ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝐺(-1𝑆𝑦)))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))))))
1413simplbda 501 . . . 4 (((𝐺 ∈ V ∧ 𝑆 ∈ V ∧ 𝑁 ∈ V) ∧ ⟨⟨𝐺, π‘†βŸ©, π‘βŸ© ∈ CPreHilOLD) β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝐺(-1𝑆𝑦)))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))))
157, 10, 14sylancr 588 . . 3 (π‘ˆ ∈ CPreHilOLD β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝐺(-1𝑆𝑦)))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))))
16153ad2ant1 1134 . 2 ((π‘ˆ ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝐺(-1𝑆𝑦)))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))))
17 fvoveq1 7385 . . . . . . 7 (π‘₯ = 𝐴 β†’ (π‘β€˜(π‘₯𝐺𝑦)) = (π‘β€˜(𝐴𝐺𝑦)))
1817oveq1d 7377 . . . . . 6 (π‘₯ = 𝐴 β†’ ((π‘β€˜(π‘₯𝐺𝑦))↑2) = ((π‘β€˜(𝐴𝐺𝑦))↑2))
19 fvoveq1 7385 . . . . . . 7 (π‘₯ = 𝐴 β†’ (π‘β€˜(π‘₯𝐺(-1𝑆𝑦))) = (π‘β€˜(𝐴𝐺(-1𝑆𝑦))))
2019oveq1d 7377 . . . . . 6 (π‘₯ = 𝐴 β†’ ((π‘β€˜(π‘₯𝐺(-1𝑆𝑦)))↑2) = ((π‘β€˜(𝐴𝐺(-1𝑆𝑦)))↑2))
2118, 20oveq12d 7380 . . . . 5 (π‘₯ = 𝐴 β†’ (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝐺(-1𝑆𝑦)))↑2)) = (((π‘β€˜(𝐴𝐺𝑦))↑2) + ((π‘β€˜(𝐴𝐺(-1𝑆𝑦)))↑2)))
22 fveq2 6847 . . . . . . . 8 (π‘₯ = 𝐴 β†’ (π‘β€˜π‘₯) = (π‘β€˜π΄))
2322oveq1d 7377 . . . . . . 7 (π‘₯ = 𝐴 β†’ ((π‘β€˜π‘₯)↑2) = ((π‘β€˜π΄)↑2))
2423oveq1d 7377 . . . . . 6 (π‘₯ = 𝐴 β†’ (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2)) = (((π‘β€˜π΄)↑2) + ((π‘β€˜π‘¦)↑2)))
2524oveq2d 7378 . . . . 5 (π‘₯ = 𝐴 β†’ (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))) = (2 Β· (((π‘β€˜π΄)↑2) + ((π‘β€˜π‘¦)↑2))))
2621, 25eqeq12d 2753 . . . 4 (π‘₯ = 𝐴 β†’ ((((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝐺(-1𝑆𝑦)))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))) ↔ (((π‘β€˜(𝐴𝐺𝑦))↑2) + ((π‘β€˜(𝐴𝐺(-1𝑆𝑦)))↑2)) = (2 Β· (((π‘β€˜π΄)↑2) + ((π‘β€˜π‘¦)↑2)))))
27 oveq2 7370 . . . . . . . 8 (𝑦 = 𝐡 β†’ (𝐴𝐺𝑦) = (𝐴𝐺𝐡))
2827fveq2d 6851 . . . . . . 7 (𝑦 = 𝐡 β†’ (π‘β€˜(𝐴𝐺𝑦)) = (π‘β€˜(𝐴𝐺𝐡)))
2928oveq1d 7377 . . . . . 6 (𝑦 = 𝐡 β†’ ((π‘β€˜(𝐴𝐺𝑦))↑2) = ((π‘β€˜(𝐴𝐺𝐡))↑2))
30 oveq2 7370 . . . . . . . . 9 (𝑦 = 𝐡 β†’ (-1𝑆𝑦) = (-1𝑆𝐡))
3130oveq2d 7378 . . . . . . . 8 (𝑦 = 𝐡 β†’ (𝐴𝐺(-1𝑆𝑦)) = (𝐴𝐺(-1𝑆𝐡)))
3231fveq2d 6851 . . . . . . 7 (𝑦 = 𝐡 β†’ (π‘β€˜(𝐴𝐺(-1𝑆𝑦))) = (π‘β€˜(𝐴𝐺(-1𝑆𝐡))))
3332oveq1d 7377 . . . . . 6 (𝑦 = 𝐡 β†’ ((π‘β€˜(𝐴𝐺(-1𝑆𝑦)))↑2) = ((π‘β€˜(𝐴𝐺(-1𝑆𝐡)))↑2))
3429, 33oveq12d 7380 . . . . 5 (𝑦 = 𝐡 β†’ (((π‘β€˜(𝐴𝐺𝑦))↑2) + ((π‘β€˜(𝐴𝐺(-1𝑆𝑦)))↑2)) = (((π‘β€˜(𝐴𝐺𝐡))↑2) + ((π‘β€˜(𝐴𝐺(-1𝑆𝐡)))↑2)))
35 fveq2 6847 . . . . . . . 8 (𝑦 = 𝐡 β†’ (π‘β€˜π‘¦) = (π‘β€˜π΅))
3635oveq1d 7377 . . . . . . 7 (𝑦 = 𝐡 β†’ ((π‘β€˜π‘¦)↑2) = ((π‘β€˜π΅)↑2))
3736oveq2d 7378 . . . . . 6 (𝑦 = 𝐡 β†’ (((π‘β€˜π΄)↑2) + ((π‘β€˜π‘¦)↑2)) = (((π‘β€˜π΄)↑2) + ((π‘β€˜π΅)↑2)))
3837oveq2d 7378 . . . . 5 (𝑦 = 𝐡 β†’ (2 Β· (((π‘β€˜π΄)↑2) + ((π‘β€˜π‘¦)↑2))) = (2 Β· (((π‘β€˜π΄)↑2) + ((π‘β€˜π΅)↑2))))
3934, 38eqeq12d 2753 . . . 4 (𝑦 = 𝐡 β†’ ((((π‘β€˜(𝐴𝐺𝑦))↑2) + ((π‘β€˜(𝐴𝐺(-1𝑆𝑦)))↑2)) = (2 Β· (((π‘β€˜π΄)↑2) + ((π‘β€˜π‘¦)↑2))) ↔ (((π‘β€˜(𝐴𝐺𝐡))↑2) + ((π‘β€˜(𝐴𝐺(-1𝑆𝐡)))↑2)) = (2 Β· (((π‘β€˜π΄)↑2) + ((π‘β€˜π΅)↑2)))))
4026, 39rspc2v 3593 . . 3 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝐺(-1𝑆𝑦)))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))) β†’ (((π‘β€˜(𝐴𝐺𝐡))↑2) + ((π‘β€˜(𝐴𝐺(-1𝑆𝐡)))↑2)) = (2 Β· (((π‘β€˜π΄)↑2) + ((π‘β€˜π΅)↑2)))))
41403adant1 1131 . 2 ((π‘ˆ ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝐺(-1𝑆𝑦)))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))) β†’ (((π‘β€˜(𝐴𝐺𝐡))↑2) + ((π‘β€˜(𝐴𝐺(-1𝑆𝐡)))↑2)) = (2 Β· (((π‘β€˜π΄)↑2) + ((π‘β€˜π΅)↑2)))))
4216, 41mpd 15 1 ((π‘ˆ ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (((π‘β€˜(𝐴𝐺𝐡))↑2) + ((π‘β€˜(𝐴𝐺(-1𝑆𝐡)))↑2)) = (2 Β· (((π‘β€˜π΄)↑2) + ((π‘β€˜π΅)↑2))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  Vcvv 3448  βŸ¨cop 4597  β€˜cfv 6501  (class class class)co 7362  1c1 11059   + caddc 11061   Β· cmul 11063  -cneg 11393  2c2 12215  β†‘cexp 13974  NrmCVeccnv 29568   +𝑣 cpv 29569  BaseSetcba 29570   ·𝑠OLD cns 29571  normCVcnmcv 29574  CPreHilOLDccphlo 29796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-1st 7926  df-2nd 7927  df-vc 29543  df-nv 29576  df-va 29579  df-ba 29580  df-sm 29581  df-0v 29582  df-nmcv 29584  df-ph 29797
This theorem is referenced by:  ip0i  29809  hlpar  29881
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