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Theorem normpari 27981
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 27848 . . . 4 (𝐴 𝐵) ∈ ℋ
43normsqi 27959 . . 3 ((norm‘(𝐴 𝐵))↑2) = ((𝐴 𝐵) ·ih (𝐴 𝐵))
51, 2hvaddcli 27845 . . . 4 (𝐴 + 𝐵) ∈ ℋ
65normsqi 27959 . . 3 ((norm‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) ·ih (𝐴 + 𝐵))
74, 6oveq12i 6647 . 2 (((norm‘(𝐴 𝐵))↑2) + ((norm‘(𝐴 + 𝐵))↑2)) = (((𝐴 𝐵) ·ih (𝐴 𝐵)) + ((𝐴 + 𝐵) ·ih (𝐴 + 𝐵)))
81normsqi 27959 . . . . . 6 ((norm𝐴)↑2) = (𝐴 ·ih 𝐴)
98oveq2i 6646 . . . . 5 (2 · ((norm𝐴)↑2)) = (2 · (𝐴 ·ih 𝐴))
101, 1hicli 27908 . . . . . 6 (𝐴 ·ih 𝐴) ∈ ℂ
11102timesi 11132 . . . . 5 (2 · (𝐴 ·ih 𝐴)) = ((𝐴 ·ih 𝐴) + (𝐴 ·ih 𝐴))
129, 11eqtri 2642 . . . 4 (2 · ((norm𝐴)↑2)) = ((𝐴 ·ih 𝐴) + (𝐴 ·ih 𝐴))
132normsqi 27959 . . . . . 6 ((norm𝐵)↑2) = (𝐵 ·ih 𝐵)
1413oveq2i 6646 . . . . 5 (2 · ((norm𝐵)↑2)) = (2 · (𝐵 ·ih 𝐵))
152, 2hicli 27908 . . . . . 6 (𝐵 ·ih 𝐵) ∈ ℂ
16152timesi 11132 . . . . 5 (2 · (𝐵 ·ih 𝐵)) = ((𝐵 ·ih 𝐵) + (𝐵 ·ih 𝐵))
1714, 16eqtri 2642 . . . 4 (2 · ((norm𝐵)↑2)) = ((𝐵 ·ih 𝐵) + (𝐵 ·ih 𝐵))
1812, 17oveq12i 6647 . . 3 ((2 · ((norm𝐴)↑2)) + (2 · ((norm𝐵)↑2))) = (((𝐴 ·ih 𝐴) + (𝐴 ·ih 𝐴)) + ((𝐵 ·ih 𝐵) + (𝐵 ·ih 𝐵)))
191, 2, 1, 2normlem9 27945 . . . . . 6 ((𝐴 𝐵) ·ih (𝐴 𝐵)) = (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) − ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)))
2010, 15addcli 10029 . . . . . . 7 ((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) ∈ ℂ
211, 2hicli 27908 . . . . . . . 8 (𝐴 ·ih 𝐵) ∈ ℂ
222, 1hicli 27908 . . . . . . . 8 (𝐵 ·ih 𝐴) ∈ ℂ
2321, 22addcli 10029 . . . . . . 7 ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)) ∈ ℂ
2420, 23negsubi 10344 . . . . . 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 2645 . . . . 5 ((𝐴 𝐵) ·ih (𝐴 𝐵)) = (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + -((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)))
261, 2, 1, 2normlem8 27944 . . . . 5 ((𝐴 + 𝐵) ·ih (𝐴 + 𝐵)) = (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)))
2725, 26oveq12i 6647 . . . 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 10334 . . . . 5 -((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)) ∈ ℂ
2920, 28, 20, 23add42i 10246 . . . 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 10335 . . . . . 6 (((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)) + -((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴))) = 0
3130oveq2i 6646 . . . . 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 10029 . . . . . 6 (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + ((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵))) ∈ ℂ
3332addid1i 10208 . . . . 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 10245 . . . . 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 2646 . . . 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 2646 . . 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 2645 . 2 ((2 · ((norm𝐴)↑2)) + (2 · ((norm𝐵)↑2))) = (((𝐴 𝐵) ·ih (𝐴 𝐵)) + ((𝐴 + 𝐵) ·ih (𝐴 + 𝐵)))
387, 37eqtr4i 2645 1 (((norm‘(𝐴 𝐵))↑2) + ((norm‘(𝐴 + 𝐵))↑2)) = ((2 · ((norm𝐴)↑2)) + (2 · ((norm𝐵)↑2)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1481  wcel 1988  cfv 5876  (class class class)co 6635  0cc0 9921   + caddc 9924   · cmul 9926  cmin 10251  -cneg 10252  2c2 11055  cexp 12843  chil 27746   + cva 27747   ·ih csp 27749  normcno 27750   cmv 27752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998  ax-pre-sup 9999  ax-hfvadd 27827  ax-hv0cl 27830  ax-hfvmul 27832  ax-hvmul0 27837  ax-hfi 27906  ax-his1 27909  ax-his2 27910  ax-his3 27911  ax-his4 27912
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-sup 8333  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-div 10670  df-nn 11006  df-2 11064  df-3 11065  df-n0 11278  df-z 11363  df-uz 11673  df-rp 11818  df-seq 12785  df-exp 12844  df-cj 13820  df-re 13821  df-im 13822  df-sqrt 13956  df-hnorm 27795  df-hvsub 27798
This theorem is referenced by:  normpar  27982  normpar2i  27983
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