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Theorem normpari 29049
 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 28916 . . . 4 (𝐴 𝐵) ∈ ℋ
43normsqi 29027 . . 3 ((norm‘(𝐴 𝐵))↑2) = ((𝐴 𝐵) ·ih (𝐴 𝐵))
51, 2hvaddcli 28913 . . . 4 (𝐴 + 𝐵) ∈ ℋ
65normsqi 29027 . . 3 ((norm‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) ·ih (𝐴 + 𝐵))
74, 6oveq12i 7168 . 2 (((norm‘(𝐴 𝐵))↑2) + ((norm‘(𝐴 + 𝐵))↑2)) = (((𝐴 𝐵) ·ih (𝐴 𝐵)) + ((𝐴 + 𝐵) ·ih (𝐴 + 𝐵)))
81normsqi 29027 . . . . . 6 ((norm𝐴)↑2) = (𝐴 ·ih 𝐴)
98oveq2i 7167 . . . . 5 (2 · ((norm𝐴)↑2)) = (2 · (𝐴 ·ih 𝐴))
101, 1hicli 28976 . . . . . 6 (𝐴 ·ih 𝐴) ∈ ℂ
11102timesi 11825 . . . . 5 (2 · (𝐴 ·ih 𝐴)) = ((𝐴 ·ih 𝐴) + (𝐴 ·ih 𝐴))
129, 11eqtri 2781 . . . 4 (2 · ((norm𝐴)↑2)) = ((𝐴 ·ih 𝐴) + (𝐴 ·ih 𝐴))
132normsqi 29027 . . . . . 6 ((norm𝐵)↑2) = (𝐵 ·ih 𝐵)
1413oveq2i 7167 . . . . 5 (2 · ((norm𝐵)↑2)) = (2 · (𝐵 ·ih 𝐵))
152, 2hicli 28976 . . . . . 6 (𝐵 ·ih 𝐵) ∈ ℂ
16152timesi 11825 . . . . 5 (2 · (𝐵 ·ih 𝐵)) = ((𝐵 ·ih 𝐵) + (𝐵 ·ih 𝐵))
1714, 16eqtri 2781 . . . 4 (2 · ((norm𝐵)↑2)) = ((𝐵 ·ih 𝐵) + (𝐵 ·ih 𝐵))
1812, 17oveq12i 7168 . . 3 ((2 · ((norm𝐴)↑2)) + (2 · ((norm𝐵)↑2))) = (((𝐴 ·ih 𝐴) + (𝐴 ·ih 𝐴)) + ((𝐵 ·ih 𝐵) + (𝐵 ·ih 𝐵)))
191, 2, 1, 2normlem9 29013 . . . . . 6 ((𝐴 𝐵) ·ih (𝐴 𝐵)) = (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) − ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)))
2010, 15addcli 10698 . . . . . . 7 ((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) ∈ ℂ
211, 2hicli 28976 . . . . . . . 8 (𝐴 ·ih 𝐵) ∈ ℂ
222, 1hicli 28976 . . . . . . . 8 (𝐵 ·ih 𝐴) ∈ ℂ
2321, 22addcli 10698 . . . . . . 7 ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)) ∈ ℂ
2420, 23negsubi 11015 . . . . . 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 2784 . . . . 5 ((𝐴 𝐵) ·ih (𝐴 𝐵)) = (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + -((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)))
261, 2, 1, 2normlem8 29012 . . . . 5 ((𝐴 + 𝐵) ·ih (𝐴 + 𝐵)) = (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)))
2725, 26oveq12i 7168 . . . 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 11005 . . . . 5 -((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)) ∈ ℂ
2920, 28, 20, 23add42i 10916 . . . 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 11006 . . . . . 6 (((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)) + -((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴))) = 0
3130oveq2i 7167 . . . . 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 10698 . . . . . 6 (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + ((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵))) ∈ ℂ
3332addid1i 10878 . . . . 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 10915 . . . . 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 2785 . . . 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 2785 . . 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 2784 . 2 ((2 · ((norm𝐴)↑2)) + (2 · ((norm𝐵)↑2))) = (((𝐴 𝐵) ·ih (𝐴 𝐵)) + ((𝐴 + 𝐵) ·ih (𝐴 + 𝐵)))
387, 37eqtr4i 2784 1 (((norm‘(𝐴 𝐵))↑2) + ((norm‘(𝐴 + 𝐵))↑2)) = ((2 · ((norm𝐴)↑2)) + (2 · ((norm𝐵)↑2)))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111  ‘cfv 6340  (class class class)co 7156  0cc0 10588   + caddc 10591   · cmul 10593   − cmin 10921  -cneg 10922  2c2 11742  ↑cexp 13492   ℋchba 28814   +ℎ cva 28815   ·ih csp 28817  normℎcno 28818   −ℎ cmv 28820 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665  ax-pre-sup 10666  ax-hfvadd 28895  ax-hv0cl 28898  ax-hfvmul 28900  ax-hvmul0 28905  ax-hfi 28974  ax-his1 28977  ax-his2 28978  ax-his3 28979  ax-his4 28980 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-er 8305  df-en 8541  df-dom 8542  df-sdom 8543  df-sup 8952  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-div 11349  df-nn 11688  df-2 11750  df-3 11751  df-n0 11948  df-z 12034  df-uz 12296  df-rp 12444  df-seq 13432  df-exp 13493  df-cj 14519  df-re 14520  df-im 14521  df-sqrt 14655  df-hnorm 28863  df-hvsub 28866 This theorem is referenced by:  normpar  29050  normpar2i  29051
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