MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  phpar2 Structured version   Visualization version   GIF version

Theorem phpar2 30646
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 30645 . . . 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 1131 . 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 7443 . . . . . . 7 (π‘₯ = 𝐴 β†’ (π‘β€˜(π‘₯𝐺𝑦)) = (π‘β€˜(𝐴𝐺𝑦)))
98oveq1d 7435 . . . . . 6 (π‘₯ = 𝐴 β†’ ((π‘β€˜(π‘₯𝐺𝑦))↑2) = ((π‘β€˜(𝐴𝐺𝑦))↑2))
10 fvoveq1 7443 . . . . . . 7 (π‘₯ = 𝐴 β†’ (π‘β€˜(π‘₯𝑀𝑦)) = (π‘β€˜(𝐴𝑀𝑦)))
1110oveq1d 7435 . . . . . 6 (π‘₯ = 𝐴 β†’ ((π‘β€˜(π‘₯𝑀𝑦))↑2) = ((π‘β€˜(𝐴𝑀𝑦))↑2))
129, 11oveq12d 7438 . . . . 5 (π‘₯ = 𝐴 β†’ (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝑀𝑦))↑2)) = (((π‘β€˜(𝐴𝐺𝑦))↑2) + ((π‘β€˜(𝐴𝑀𝑦))↑2)))
13 fveq2 6897 . . . . . . . 8 (π‘₯ = 𝐴 β†’ (π‘β€˜π‘₯) = (π‘β€˜π΄))
1413oveq1d 7435 . . . . . . 7 (π‘₯ = 𝐴 β†’ ((π‘β€˜π‘₯)↑2) = ((π‘β€˜π΄)↑2))
1514oveq1d 7435 . . . . . 6 (π‘₯ = 𝐴 β†’ (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2)) = (((π‘β€˜π΄)↑2) + ((π‘β€˜π‘¦)↑2)))
1615oveq2d 7436 . . . . 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 2744 . . . 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 7428 . . . . . . . 8 (𝑦 = 𝐡 β†’ (𝐴𝐺𝑦) = (𝐴𝐺𝐡))
1918fveq2d 6901 . . . . . . 7 (𝑦 = 𝐡 β†’ (π‘β€˜(𝐴𝐺𝑦)) = (π‘β€˜(𝐴𝐺𝐡)))
2019oveq1d 7435 . . . . . 6 (𝑦 = 𝐡 β†’ ((π‘β€˜(𝐴𝐺𝑦))↑2) = ((π‘β€˜(𝐴𝐺𝐡))↑2))
21 oveq2 7428 . . . . . . . 8 (𝑦 = 𝐡 β†’ (𝐴𝑀𝑦) = (𝐴𝑀𝐡))
2221fveq2d 6901 . . . . . . 7 (𝑦 = 𝐡 β†’ (π‘β€˜(𝐴𝑀𝑦)) = (π‘β€˜(𝐴𝑀𝐡)))
2322oveq1d 7435 . . . . . 6 (𝑦 = 𝐡 β†’ ((π‘β€˜(𝐴𝑀𝑦))↑2) = ((π‘β€˜(𝐴𝑀𝐡))↑2))
2420, 23oveq12d 7438 . . . . 5 (𝑦 = 𝐡 β†’ (((π‘β€˜(𝐴𝐺𝑦))↑2) + ((π‘β€˜(𝐴𝑀𝑦))↑2)) = (((π‘β€˜(𝐴𝐺𝐡))↑2) + ((π‘β€˜(𝐴𝑀𝐡))↑2)))
25 fveq2 6897 . . . . . . . 8 (𝑦 = 𝐡 β†’ (π‘β€˜π‘¦) = (π‘β€˜π΅))
2625oveq1d 7435 . . . . . . 7 (𝑦 = 𝐡 β†’ ((π‘β€˜π‘¦)↑2) = ((π‘β€˜π΅)↑2))
2726oveq2d 7436 . . . . . 6 (𝑦 = 𝐡 β†’ (((π‘β€˜π΄)↑2) + ((π‘β€˜π‘¦)↑2)) = (((π‘β€˜π΄)↑2) + ((π‘β€˜π΅)↑2)))
2827oveq2d 7436 . . . . 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 2744 . . . 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 3620 . . 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 1128 . 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 1085   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058  β€˜cfv 6548  (class class class)co 7420   + caddc 11142   Β· cmul 11144  2c2 12298  β†‘cexp 14059  NrmCVeccnv 30407   +𝑣 cpv 30408  BaseSetcba 30409   βˆ’π‘£ cnsb 30412  normCVcnmcv 30413  CPreHilOLDccphlo 30635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-po 5590  df-so 5591  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11281  df-mnf 11282  df-ltxr 11284  df-sub 11477  df-neg 11478  df-grpo 30316  df-gid 30317  df-ginv 30318  df-gdiv 30319  df-ablo 30368  df-vc 30382  df-nv 30415  df-va 30418  df-ba 30419  df-sm 30420  df-0v 30421  df-vs 30422  df-nmcv 30423  df-ph 30636
This theorem is referenced by:  minvecolem2  30698  hlpar2  30719
  Copyright terms: Public domain W3C validator