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Theorem normpari 28536
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 28403 . . . 4 (𝐴 𝐵) ∈ ℋ
43normsqi 28514 . . 3 ((norm‘(𝐴 𝐵))↑2) = ((𝐴 𝐵) ·ih (𝐴 𝐵))
51, 2hvaddcli 28400 . . . 4 (𝐴 + 𝐵) ∈ ℋ
65normsqi 28514 . . 3 ((norm‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) ·ih (𝐴 + 𝐵))
74, 6oveq12i 6890 . 2 (((norm‘(𝐴 𝐵))↑2) + ((norm‘(𝐴 + 𝐵))↑2)) = (((𝐴 𝐵) ·ih (𝐴 𝐵)) + ((𝐴 + 𝐵) ·ih (𝐴 + 𝐵)))
81normsqi 28514 . . . . . 6 ((norm𝐴)↑2) = (𝐴 ·ih 𝐴)
98oveq2i 6889 . . . . 5 (2 · ((norm𝐴)↑2)) = (2 · (𝐴 ·ih 𝐴))
101, 1hicli 28463 . . . . . 6 (𝐴 ·ih 𝐴) ∈ ℂ
11102timesi 11458 . . . . 5 (2 · (𝐴 ·ih 𝐴)) = ((𝐴 ·ih 𝐴) + (𝐴 ·ih 𝐴))
129, 11eqtri 2821 . . . 4 (2 · ((norm𝐴)↑2)) = ((𝐴 ·ih 𝐴) + (𝐴 ·ih 𝐴))
132normsqi 28514 . . . . . 6 ((norm𝐵)↑2) = (𝐵 ·ih 𝐵)
1413oveq2i 6889 . . . . 5 (2 · ((norm𝐵)↑2)) = (2 · (𝐵 ·ih 𝐵))
152, 2hicli 28463 . . . . . 6 (𝐵 ·ih 𝐵) ∈ ℂ
16152timesi 11458 . . . . 5 (2 · (𝐵 ·ih 𝐵)) = ((𝐵 ·ih 𝐵) + (𝐵 ·ih 𝐵))
1714, 16eqtri 2821 . . . 4 (2 · ((norm𝐵)↑2)) = ((𝐵 ·ih 𝐵) + (𝐵 ·ih 𝐵))
1812, 17oveq12i 6890 . . 3 ((2 · ((norm𝐴)↑2)) + (2 · ((norm𝐵)↑2))) = (((𝐴 ·ih 𝐴) + (𝐴 ·ih 𝐴)) + ((𝐵 ·ih 𝐵) + (𝐵 ·ih 𝐵)))
191, 2, 1, 2normlem9 28500 . . . . . 6 ((𝐴 𝐵) ·ih (𝐴 𝐵)) = (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) − ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)))
2010, 15addcli 10335 . . . . . . 7 ((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) ∈ ℂ
211, 2hicli 28463 . . . . . . . 8 (𝐴 ·ih 𝐵) ∈ ℂ
222, 1hicli 28463 . . . . . . . 8 (𝐵 ·ih 𝐴) ∈ ℂ
2321, 22addcli 10335 . . . . . . 7 ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)) ∈ ℂ
2420, 23negsubi 10651 . . . . . 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 2824 . . . . 5 ((𝐴 𝐵) ·ih (𝐴 𝐵)) = (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + -((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)))
261, 2, 1, 2normlem8 28499 . . . . 5 ((𝐴 + 𝐵) ·ih (𝐴 + 𝐵)) = (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)))
2725, 26oveq12i 6890 . . . 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 10641 . . . . 5 -((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)) ∈ ℂ
2920, 28, 20, 23add42i 10551 . . . 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 10642 . . . . . 6 (((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)) + -((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴))) = 0
3130oveq2i 6889 . . . . 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 10335 . . . . . 6 (((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) + ((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵))) ∈ ℂ
3332addid1i 10513 . . . . 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 10550 . . . . 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 2825 . . . 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 2825 . . 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 2824 . 2 ((2 · ((norm𝐴)↑2)) + (2 · ((norm𝐵)↑2))) = (((𝐴 𝐵) ·ih (𝐴 𝐵)) + ((𝐴 + 𝐵) ·ih (𝐴 + 𝐵)))
387, 37eqtr4i 2824 1 (((norm‘(𝐴 𝐵))↑2) + ((norm‘(𝐴 + 𝐵))↑2)) = ((2 · ((norm𝐴)↑2)) + (2 · ((norm𝐵)↑2)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  wcel 2157  cfv 6101  (class class class)co 6878  0cc0 10224   + caddc 10227   · cmul 10229  cmin 10556  -cneg 10557  2c2 11368  cexp 13114  chba 28301   + cva 28302   ·ih csp 28304  normcno 28305   cmv 28307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301  ax-pre-sup 10302  ax-hfvadd 28382  ax-hv0cl 28385  ax-hfvmul 28387  ax-hvmul0 28392  ax-hfi 28461  ax-his1 28464  ax-his2 28465  ax-his3 28466  ax-his4 28467
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-sup 8590  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-div 10977  df-nn 11313  df-2 11376  df-3 11377  df-n0 11581  df-z 11667  df-uz 11931  df-rp 12075  df-seq 13056  df-exp 13115  df-cj 14180  df-re 14181  df-im 14182  df-sqrt 14316  df-hnorm 28350  df-hvsub 28353
This theorem is referenced by:  normpar  28537  normpar2i  28538
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