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Theorem normpari 28351
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 28218 . . . 4 (𝐴 𝐵) ∈ ℋ
43normsqi 28329 . . 3 ((norm‘(𝐴 𝐵))↑2) = ((𝐴 𝐵) ·ih (𝐴 𝐵))
51, 2hvaddcli 28215 . . . 4 (𝐴 + 𝐵) ∈ ℋ
65normsqi 28329 . . 3 ((norm‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) ·ih (𝐴 + 𝐵))
74, 6oveq12i 6805 . 2 (((norm‘(𝐴 𝐵))↑2) + ((norm‘(𝐴 + 𝐵))↑2)) = (((𝐴 𝐵) ·ih (𝐴 𝐵)) + ((𝐴 + 𝐵) ·ih (𝐴 + 𝐵)))
81normsqi 28329 . . . . . 6 ((norm𝐴)↑2) = (𝐴 ·ih 𝐴)
98oveq2i 6804 . . . . 5 (2 · ((norm𝐴)↑2)) = (2 · (𝐴 ·ih 𝐴))
101, 1hicli 28278 . . . . . 6 (𝐴 ·ih 𝐴) ∈ ℂ
11102timesi 11349 . . . . 5 (2 · (𝐴 ·ih 𝐴)) = ((𝐴 ·ih 𝐴) + (𝐴 ·ih 𝐴))
129, 11eqtri 2793 . . . 4 (2 · ((norm𝐴)↑2)) = ((𝐴 ·ih 𝐴) + (𝐴 ·ih 𝐴))
132normsqi 28329 . . . . . 6 ((norm𝐵)↑2) = (𝐵 ·ih 𝐵)
1413oveq2i 6804 . . . . 5 (2 · ((norm𝐵)↑2)) = (2 · (𝐵 ·ih 𝐵))
152, 2hicli 28278 . . . . . 6 (𝐵 ·ih 𝐵) ∈ ℂ
16152timesi 11349 . . . . 5 (2 · (𝐵 ·ih 𝐵)) = ((𝐵 ·ih 𝐵) + (𝐵 ·ih 𝐵))
1714, 16eqtri 2793 . . . 4 (2 · ((norm𝐵)↑2)) = ((𝐵 ·ih 𝐵) + (𝐵 ·ih 𝐵))
1812, 17oveq12i 6805 . . 3 ((2 · ((norm𝐴)↑2)) + (2 · ((norm𝐵)↑2))) = (((𝐴 ·ih 𝐴) + (𝐴 ·ih 𝐴)) + ((𝐵 ·ih 𝐵) + (𝐵 ·ih 𝐵)))
191, 2, 1, 2normlem9 28315 . . . . . 6 ((𝐴 𝐵) ·ih (𝐴 𝐵)) = (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) − ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)))
2010, 15addcli 10246 . . . . . . 7 ((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) ∈ ℂ
211, 2hicli 28278 . . . . . . . 8 (𝐴 ·ih 𝐵) ∈ ℂ
222, 1hicli 28278 . . . . . . . 8 (𝐵 ·ih 𝐴) ∈ ℂ
2321, 22addcli 10246 . . . . . . 7 ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)) ∈ ℂ
2420, 23negsubi 10561 . . . . . 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 2796 . . . . 5 ((𝐴 𝐵) ·ih (𝐴 𝐵)) = (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + -((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)))
261, 2, 1, 2normlem8 28314 . . . . 5 ((𝐴 + 𝐵) ·ih (𝐴 + 𝐵)) = (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)))
2725, 26oveq12i 6805 . . . 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 10551 . . . . 5 -((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)) ∈ ℂ
2920, 28, 20, 23add42i 10463 . . . 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 10552 . . . . . 6 (((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)) + -((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴))) = 0
3130oveq2i 6804 . . . . 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 10246 . . . . . 6 (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + ((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵))) ∈ ℂ
3332addid1i 10425 . . . . 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 10462 . . . . 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 2797 . . . 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 2797 . . 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 2796 . 2 ((2 · ((norm𝐴)↑2)) + (2 · ((norm𝐵)↑2))) = (((𝐴 𝐵) ·ih (𝐴 𝐵)) + ((𝐴 + 𝐵) ·ih (𝐴 + 𝐵)))
387, 37eqtr4i 2796 1 (((norm‘(𝐴 𝐵))↑2) + ((norm‘(𝐴 + 𝐵))↑2)) = ((2 · ((norm𝐴)↑2)) + (2 · ((norm𝐵)↑2)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wcel 2145  cfv 6031  (class class class)co 6793  0cc0 10138   + caddc 10141   · cmul 10143  cmin 10468  -cneg 10469  2c2 11272  cexp 13067  chil 28116   + cva 28117   ·ih csp 28119  normcno 28120   cmv 28122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215  ax-pre-sup 10216  ax-hfvadd 28197  ax-hv0cl 28200  ax-hfvmul 28202  ax-hvmul0 28207  ax-hfi 28276  ax-his1 28279  ax-his2 28280  ax-his3 28281  ax-his4 28282
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-sup 8504  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-nn 11223  df-2 11281  df-3 11282  df-n0 11495  df-z 11580  df-uz 11889  df-rp 12036  df-seq 13009  df-exp 13068  df-cj 14047  df-re 14048  df-im 14049  df-sqrt 14183  df-hnorm 28165  df-hvsub 28168
This theorem is referenced by:  normpar  28352  normpar2i  28353
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