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Theorem phpar2 30581
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 = (BaseSetβ€˜π‘ˆ)
isph.2 𝐺 = ( +𝑣 β€˜π‘ˆ)
isph.3 𝑀 = ( βˆ’π‘£ β€˜π‘ˆ)
isph.6 𝑁 = (normCVβ€˜π‘ˆ)
Assertion
Ref Expression
phpar2 ((π‘ˆ ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (((π‘β€˜(𝐴𝐺𝐡))↑2) + ((π‘β€˜(𝐴𝑀𝐡))↑2)) = (2 Β· (((π‘β€˜π΄)↑2) + ((π‘β€˜π΅)↑2))))

Proof of Theorem phpar2
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isph.1 . . . . 5 𝑋 = (BaseSetβ€˜π‘ˆ)
2 isph.2 . . . . 5 𝐺 = ( +𝑣 β€˜π‘ˆ)
3 isph.3 . . . . 5 𝑀 = ( βˆ’π‘£ β€˜π‘ˆ)
4 isph.6 . . . . 5 𝑁 = (normCVβ€˜π‘ˆ)
51, 2, 3, 4isph 30580 . . . 4 (π‘ˆ ∈ CPreHilOLD ↔ (π‘ˆ ∈ NrmCVec ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝑀𝑦))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2)))))
65simprbi 496 . . 3 (π‘ˆ ∈ CPreHilOLD β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝑀𝑦))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))))
763ad2ant1 1130 . 2 ((π‘ˆ ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝑀𝑦))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))))
8 fvoveq1 7427 . . . . . . 7 (π‘₯ = 𝐴 β†’ (π‘β€˜(π‘₯𝐺𝑦)) = (π‘β€˜(𝐴𝐺𝑦)))
98oveq1d 7419 . . . . . 6 (π‘₯ = 𝐴 β†’ ((π‘β€˜(π‘₯𝐺𝑦))↑2) = ((π‘β€˜(𝐴𝐺𝑦))↑2))
10 fvoveq1 7427 . . . . . . 7 (π‘₯ = 𝐴 β†’ (π‘β€˜(π‘₯𝑀𝑦)) = (π‘β€˜(𝐴𝑀𝑦)))
1110oveq1d 7419 . . . . . 6 (π‘₯ = 𝐴 β†’ ((π‘β€˜(π‘₯𝑀𝑦))↑2) = ((π‘β€˜(𝐴𝑀𝑦))↑2))
129, 11oveq12d 7422 . . . . 5 (π‘₯ = 𝐴 β†’ (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝑀𝑦))↑2)) = (((π‘β€˜(𝐴𝐺𝑦))↑2) + ((π‘β€˜(𝐴𝑀𝑦))↑2)))
13 fveq2 6884 . . . . . . . 8 (π‘₯ = 𝐴 β†’ (π‘β€˜π‘₯) = (π‘β€˜π΄))
1413oveq1d 7419 . . . . . . 7 (π‘₯ = 𝐴 β†’ ((π‘β€˜π‘₯)↑2) = ((π‘β€˜π΄)↑2))
1514oveq1d 7419 . . . . . 6 (π‘₯ = 𝐴 β†’ (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2)) = (((π‘β€˜π΄)↑2) + ((π‘β€˜π‘¦)↑2)))
1615oveq2d 7420 . . . . 5 (π‘₯ = 𝐴 β†’ (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))) = (2 Β· (((π‘β€˜π΄)↑2) + ((π‘β€˜π‘¦)↑2))))
1712, 16eqeq12d 2742 . . . 4 (π‘₯ = 𝐴 β†’ ((((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝑀𝑦))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))) ↔ (((π‘β€˜(𝐴𝐺𝑦))↑2) + ((π‘β€˜(𝐴𝑀𝑦))↑2)) = (2 Β· (((π‘β€˜π΄)↑2) + ((π‘β€˜π‘¦)↑2)))))
18 oveq2 7412 . . . . . . . 8 (𝑦 = 𝐡 β†’ (𝐴𝐺𝑦) = (𝐴𝐺𝐡))
1918fveq2d 6888 . . . . . . 7 (𝑦 = 𝐡 β†’ (π‘β€˜(𝐴𝐺𝑦)) = (π‘β€˜(𝐴𝐺𝐡)))
2019oveq1d 7419 . . . . . 6 (𝑦 = 𝐡 β†’ ((π‘β€˜(𝐴𝐺𝑦))↑2) = ((π‘β€˜(𝐴𝐺𝐡))↑2))
21 oveq2 7412 . . . . . . . 8 (𝑦 = 𝐡 β†’ (𝐴𝑀𝑦) = (𝐴𝑀𝐡))
2221fveq2d 6888 . . . . . . 7 (𝑦 = 𝐡 β†’ (π‘β€˜(𝐴𝑀𝑦)) = (π‘β€˜(𝐴𝑀𝐡)))
2322oveq1d 7419 . . . . . 6 (𝑦 = 𝐡 β†’ ((π‘β€˜(𝐴𝑀𝑦))↑2) = ((π‘β€˜(𝐴𝑀𝐡))↑2))
2420, 23oveq12d 7422 . . . . 5 (𝑦 = 𝐡 β†’ (((π‘β€˜(𝐴𝐺𝑦))↑2) + ((π‘β€˜(𝐴𝑀𝑦))↑2)) = (((π‘β€˜(𝐴𝐺𝐡))↑2) + ((π‘β€˜(𝐴𝑀𝐡))↑2)))
25 fveq2 6884 . . . . . . . 8 (𝑦 = 𝐡 β†’ (π‘β€˜π‘¦) = (π‘β€˜π΅))
2625oveq1d 7419 . . . . . . 7 (𝑦 = 𝐡 β†’ ((π‘β€˜π‘¦)↑2) = ((π‘β€˜π΅)↑2))
2726oveq2d 7420 . . . . . 6 (𝑦 = 𝐡 β†’ (((π‘β€˜π΄)↑2) + ((π‘β€˜π‘¦)↑2)) = (((π‘β€˜π΄)↑2) + ((π‘β€˜π΅)↑2)))
2827oveq2d 7420 . . . . 5 (𝑦 = 𝐡 β†’ (2 Β· (((π‘β€˜π΄)↑2) + ((π‘β€˜π‘¦)↑2))) = (2 Β· (((π‘β€˜π΄)↑2) + ((π‘β€˜π΅)↑2))))
2924, 28eqeq12d 2742 . . . 4 (𝑦 = 𝐡 β†’ ((((π‘β€˜(𝐴𝐺𝑦))↑2) + ((π‘β€˜(𝐴𝑀𝑦))↑2)) = (2 Β· (((π‘β€˜π΄)↑2) + ((π‘β€˜π‘¦)↑2))) ↔ (((π‘β€˜(𝐴𝐺𝐡))↑2) + ((π‘β€˜(𝐴𝑀𝐡))↑2)) = (2 Β· (((π‘β€˜π΄)↑2) + ((π‘β€˜π΅)↑2)))))
3017, 29rspc2v 3617 . . 3 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝑀𝑦))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))) β†’ (((π‘β€˜(𝐴𝐺𝐡))↑2) + ((π‘β€˜(𝐴𝑀𝐡))↑2)) = (2 Β· (((π‘β€˜π΄)↑2) + ((π‘β€˜π΅)↑2)))))
31303adant1 1127 . 2 ((π‘ˆ ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝑀𝑦))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))) β†’ (((π‘β€˜(𝐴𝐺𝐡))↑2) + ((π‘β€˜(𝐴𝑀𝐡))↑2)) = (2 Β· (((π‘β€˜π΄)↑2) + ((π‘β€˜π΅)↑2)))))
327, 31mpd 15 1 ((π‘ˆ ∈ CPreHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (((π‘β€˜(𝐴𝐺𝐡))↑2) + ((π‘β€˜(𝐴𝑀𝐡))↑2)) = (2 Β· (((π‘β€˜π΄)↑2) + ((π‘β€˜π΅)↑2))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  β€˜cfv 6536  (class class class)co 7404   + caddc 11112   Β· cmul 11114  2c2 12268  β†‘cexp 14030  NrmCVeccnv 30342   +𝑣 cpv 30343  BaseSetcba 30344   βˆ’π‘£ cnsb 30347  normCVcnmcv 30348  CPreHilOLDccphlo 30570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-po 5581  df-so 5582  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11251  df-mnf 11252  df-ltxr 11254  df-sub 11447  df-neg 11448  df-grpo 30251  df-gid 30252  df-ginv 30253  df-gdiv 30254  df-ablo 30303  df-vc 30317  df-nv 30350  df-va 30353  df-ba 30354  df-sm 30355  df-0v 30356  df-vs 30357  df-nmcv 30358  df-ph 30571
This theorem is referenced by:  minvecolem2  30633  hlpar2  30654
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