HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  normpari Structured version   Visualization version   GIF version

Theorem normpari 28858
Description: Parallelogram law for norms. Remark 3.4(B) of [Beran] p. 98. (Contributed by NM, 21-Aug-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
normpar.1 𝐴 ∈ ℋ
normpar.2 𝐵 ∈ ℋ
Assertion
Ref Expression
normpari (((norm‘(𝐴 𝐵))↑2) + ((norm‘(𝐴 + 𝐵))↑2)) = ((2 · ((norm𝐴)↑2)) + (2 · ((norm𝐵)↑2)))

Proof of Theorem normpari
StepHypRef Expression
1 normpar.1 . . . . 5 𝐴 ∈ ℋ
2 normpar.2 . . . . 5 𝐵 ∈ ℋ
31, 2hvsubcli 28725 . . . 4 (𝐴 𝐵) ∈ ℋ
43normsqi 28836 . . 3 ((norm‘(𝐴 𝐵))↑2) = ((𝐴 𝐵) ·ih (𝐴 𝐵))
51, 2hvaddcli 28722 . . . 4 (𝐴 + 𝐵) ∈ ℋ
65normsqi 28836 . . 3 ((norm‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) ·ih (𝐴 + 𝐵))
74, 6oveq12i 7157 . 2 (((norm‘(𝐴 𝐵))↑2) + ((norm‘(𝐴 + 𝐵))↑2)) = (((𝐴 𝐵) ·ih (𝐴 𝐵)) + ((𝐴 + 𝐵) ·ih (𝐴 + 𝐵)))
81normsqi 28836 . . . . . 6 ((norm𝐴)↑2) = (𝐴 ·ih 𝐴)
98oveq2i 7156 . . . . 5 (2 · ((norm𝐴)↑2)) = (2 · (𝐴 ·ih 𝐴))
101, 1hicli 28785 . . . . . 6 (𝐴 ·ih 𝐴) ∈ ℂ
11102timesi 11763 . . . . 5 (2 · (𝐴 ·ih 𝐴)) = ((𝐴 ·ih 𝐴) + (𝐴 ·ih 𝐴))
129, 11eqtri 2841 . . . 4 (2 · ((norm𝐴)↑2)) = ((𝐴 ·ih 𝐴) + (𝐴 ·ih 𝐴))
132normsqi 28836 . . . . . 6 ((norm𝐵)↑2) = (𝐵 ·ih 𝐵)
1413oveq2i 7156 . . . . 5 (2 · ((norm𝐵)↑2)) = (2 · (𝐵 ·ih 𝐵))
152, 2hicli 28785 . . . . . 6 (𝐵 ·ih 𝐵) ∈ ℂ
16152timesi 11763 . . . . 5 (2 · (𝐵 ·ih 𝐵)) = ((𝐵 ·ih 𝐵) + (𝐵 ·ih 𝐵))
1714, 16eqtri 2841 . . . 4 (2 · ((norm𝐵)↑2)) = ((𝐵 ·ih 𝐵) + (𝐵 ·ih 𝐵))
1812, 17oveq12i 7157 . . 3 ((2 · ((norm𝐴)↑2)) + (2 · ((norm𝐵)↑2))) = (((𝐴 ·ih 𝐴) + (𝐴 ·ih 𝐴)) + ((𝐵 ·ih 𝐵) + (𝐵 ·ih 𝐵)))
191, 2, 1, 2normlem9 28822 . . . . . 6 ((𝐴 𝐵) ·ih (𝐴 𝐵)) = (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) − ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)))
2010, 15addcli 10635 . . . . . . 7 ((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) ∈ ℂ
211, 2hicli 28785 . . . . . . . 8 (𝐴 ·ih 𝐵) ∈ ℂ
222, 1hicli 28785 . . . . . . . 8 (𝐵 ·ih 𝐴) ∈ ℂ
2321, 22addcli 10635 . . . . . . 7 ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)) ∈ ℂ
2420, 23negsubi 10952 . . . . . 6 (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + -((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴))) = (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) − ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)))
2519, 24eqtr4i 2844 . . . . 5 ((𝐴 𝐵) ·ih (𝐴 𝐵)) = (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + -((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)))
261, 2, 1, 2normlem8 28821 . . . . 5 ((𝐴 + 𝐵) ·ih (𝐴 + 𝐵)) = (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)))
2725, 26oveq12i 7157 . . . 4 (((𝐴 𝐵) ·ih (𝐴 𝐵)) + ((𝐴 + 𝐵) ·ih (𝐴 + 𝐵))) = ((((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + -((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴))) + (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴))))
2823negcli 10942 . . . . 5 -((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)) ∈ ℂ
2920, 28, 20, 23add42i 10853 . . . 4 ((((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + -((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴))) + (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)))) = ((((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + ((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵))) + (((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)) + -((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴))))
3023negidi 10943 . . . . . 6 (((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)) + -((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴))) = 0
3130oveq2i 7156 . . . . 5 ((((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + ((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵))) + (((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)) + -((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)))) = ((((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + ((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵))) + 0)
3220, 20addcli 10635 . . . . . 6 (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + ((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵))) ∈ ℂ
3332addid1i 10815 . . . . 5 ((((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + ((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵))) + 0) = (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + ((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)))
3410, 15, 10, 15add4i 10852 . . . . 5 (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + ((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵))) = (((𝐴 ·ih 𝐴) + (𝐴 ·ih 𝐴)) + ((𝐵 ·ih 𝐵) + (𝐵 ·ih 𝐵)))
3531, 33, 343eqtri 2845 . . . 4 ((((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + ((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵))) + (((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)) + -((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)))) = (((𝐴 ·ih 𝐴) + (𝐴 ·ih 𝐴)) + ((𝐵 ·ih 𝐵) + (𝐵 ·ih 𝐵)))
3627, 29, 353eqtri 2845 . . 3 (((𝐴 𝐵) ·ih (𝐴 𝐵)) + ((𝐴 + 𝐵) ·ih (𝐴 + 𝐵))) = (((𝐴 ·ih 𝐴) + (𝐴 ·ih 𝐴)) + ((𝐵 ·ih 𝐵) + (𝐵 ·ih 𝐵)))
3718, 36eqtr4i 2844 . 2 ((2 · ((norm𝐴)↑2)) + (2 · ((norm𝐵)↑2))) = (((𝐴 𝐵) ·ih (𝐴 𝐵)) + ((𝐴 + 𝐵) ·ih (𝐴 + 𝐵)))
387, 37eqtr4i 2844 1 (((norm‘(𝐴 𝐵))↑2) + ((norm‘(𝐴 + 𝐵))↑2)) = ((2 · ((norm𝐴)↑2)) + (2 · ((norm𝐵)↑2)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  cfv 6348  (class class class)co 7145  0cc0 10525   + caddc 10528   · cmul 10530  cmin 10858  -cneg 10859  2c2 11680  cexp 13417  chba 28623   + cva 28624   ·ih csp 28626  normcno 28627   cmv 28629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-hfvadd 28704  ax-hv0cl 28707  ax-hfvmul 28709  ax-hvmul0 28714  ax-hfi 28783  ax-his1 28786  ax-his2 28787  ax-his3 28788  ax-his4 28789
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-hnorm 28672  df-hvsub 28675
This theorem is referenced by:  normpar  28859  normpar2i  28860
  Copyright terms: Public domain W3C validator