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Theorem normpari 31083

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_ℎ‘(𝐴 −_ℎ 𝐵))↑2) + ((norm_ℎ‘(𝐴 +_ℎ 𝐵))↑2)) = ((2 · ((norm_ℎ‘𝐴)↑2)) + (2 · ((norm_ℎ‘𝐵)↑2)))

**Proof of Theorem normpari**
Step	Hyp	Ref	Expression
1		normpar.1	. . . . 5 ⊢ 𝐴 ∈ ℋ
2		normpar.2	. . . . 5 ⊢ 𝐵 ∈ ℋ
3	1, 2	hvsubcli 30950	. . . 4 ⊢ (𝐴 −_ℎ 𝐵) ∈ ℋ
4	3	normsqi 31061	. . 3 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2) = ((𝐴 −_ℎ 𝐵) ·_ih (𝐴 −_ℎ 𝐵))
5	1, 2	hvaddcli 30947	. . . 4 ⊢ (𝐴 +_ℎ 𝐵) ∈ ℋ
6	5	normsqi 31061	. . 3 ⊢ ((norm_ℎ‘(𝐴 +_ℎ 𝐵))↑2) = ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵))
7	4, 6	oveq12i 7399	. 2 ⊢ (((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2) + ((norm_ℎ‘(𝐴 +_ℎ 𝐵))↑2)) = (((𝐴 −_ℎ 𝐵) ·_ih (𝐴 −_ℎ 𝐵)) + ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)))
8	1	normsqi 31061	. . . . . 6 ⊢ ((norm_ℎ‘𝐴)↑2) = (𝐴 ·_ih 𝐴)
9	8	oveq2i 7398	. . . . 5 ⊢ (2 · ((norm_ℎ‘𝐴)↑2)) = (2 · (𝐴 ·_ih 𝐴))
10	1, 1	hicli 31010	. . . . . 6 ⊢ (𝐴 ·_ih 𝐴) ∈ ℂ
11	10	2timesi 12319	. . . . 5 ⊢ (2 · (𝐴 ·_ih 𝐴)) = ((𝐴 ·_ih 𝐴) + (𝐴 ·_ih 𝐴))
12	9, 11	eqtri 2752	. . . 4 ⊢ (2 · ((norm_ℎ‘𝐴)↑2)) = ((𝐴 ·_ih 𝐴) + (𝐴 ·_ih 𝐴))
13	2	normsqi 31061	. . . . . 6 ⊢ ((norm_ℎ‘𝐵)↑2) = (𝐵 ·_ih 𝐵)
14	13	oveq2i 7398	. . . . 5 ⊢ (2 · ((norm_ℎ‘𝐵)↑2)) = (2 · (𝐵 ·_ih 𝐵))
15	2, 2	hicli 31010	. . . . . 6 ⊢ (𝐵 ·_ih 𝐵) ∈ ℂ
16	15	2timesi 12319	. . . . 5 ⊢ (2 · (𝐵 ·_ih 𝐵)) = ((𝐵 ·_ih 𝐵) + (𝐵 ·_ih 𝐵))
17	14, 16	eqtri 2752	. . . 4 ⊢ (2 · ((norm_ℎ‘𝐵)↑2)) = ((𝐵 ·_ih 𝐵) + (𝐵 ·_ih 𝐵))
18	12, 17	oveq12i 7399	. . 3 ⊢ ((2 · ((norm_ℎ‘𝐴)↑2)) + (2 · ((norm_ℎ‘𝐵)↑2))) = (((𝐴 ·_ih 𝐴) + (𝐴 ·_ih 𝐴)) + ((𝐵 ·_ih 𝐵) + (𝐵 ·_ih 𝐵)))
19	1, 2, 1, 2	normlem9 31047	. . . . . 6 ⊢ ((𝐴 −_ℎ 𝐵) ·_ih (𝐴 −_ℎ 𝐵)) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) − ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)))
20	10, 15	addcli 11180	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) ∈ ℂ
21	1, 2	hicli 31010	. . . . . . . 8 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
22	2, 1	hicli 31010	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐴) ∈ ℂ
23	21, 22	addcli 11180	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) ∈ ℂ
24	20, 23	negsubi 11500	. . . . . 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)))
25	19, 24	eqtr4i 2755	. . . . 5 ⊢ ((𝐴 −_ℎ 𝐵) ·_ih (𝐴 −_ℎ 𝐵)) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + -((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)))
26	1, 2, 1, 2	normlem8 31046	. . . . 5 ⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)))
27	25, 26	oveq12i 7399	. . . 4 ⊢ (((𝐴 −_ℎ 𝐵) ·_ih (𝐴 −_ℎ 𝐵)) + ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 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))))
28	23	negcli 11490	. . . . 5 ⊢ -((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) ∈ ℂ
29	20, 28, 20, 23	add42i 11400	. . . 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))))
30	23	negidi 11491	. . . . . 6 ⊢ (((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) + -((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴))) = 0
31	30	oveq2i 7398	. . . . 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)
32	20, 20	addcli 11180	. . . . . 6 ⊢ (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵))) ∈ ℂ
33	32	addridi 11361	. . . . 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)))
34	10, 15, 10, 15	add4i 11399	. . . . 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)))
35	31, 33, 34	3eqtri 2756	. . . 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)))
36	27, 29, 35	3eqtri 2756	. . 3 ⊢ (((𝐴 −_ℎ 𝐵) ·_ih (𝐴 −_ℎ 𝐵)) + ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵))) = (((𝐴 ·_ih 𝐴) + (𝐴 ·_ih 𝐴)) + ((𝐵 ·_ih 𝐵) + (𝐵 ·_ih 𝐵)))
37	18, 36	eqtr4i 2755	. 2 ⊢ ((2 · ((norm_ℎ‘𝐴)↑2)) + (2 · ((norm_ℎ‘𝐵)↑2))) = (((𝐴 −_ℎ 𝐵) ·_ih (𝐴 −_ℎ 𝐵)) + ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)))
38	7, 37	eqtr4i 2755	1 ⊢ (((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2) + ((norm_ℎ‘(𝐴 +_ℎ 𝐵))↑2)) = ((2 · ((norm_ℎ‘𝐴)↑2)) + (2 · ((norm_ℎ‘𝐵)↑2)))

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2109 ‘cfv 6511 (class class class)co 7387 0cc0 11068 + caddc 11071 · cmul 11073 − cmin 11405 -cneg 11406 2c2 12241 ↑cexp 14026 ℋchba 30848 +_ℎ cva 30849 ·_ih csp 30851 norm_ℎcno 30852 −_ℎ cmv 30854

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-hfvadd 30929 ax-hv0cl 30932 ax-hfvmul 30934 ax-hvmul0 30939 ax-hfi 31008 ax-his1 31011 ax-his2 31012 ax-his3 31013 ax-his4 31014

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-en 8919 df-dom 8920 df-sdom 8921 df-sup 9393 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-z 12530 df-uz 12794 df-rp 12952 df-seq 13967 df-exp 14027 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-hnorm 30897 df-hvsub 30900

This theorem is referenced by: normpar 31084 normpar2i 31085

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