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Theorem normpari 31186

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 31053	. . . 4 ⊢ (𝐴 −_ℎ 𝐵) ∈ ℋ
4	3	normsqi 31164	. . 3 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2) = ((𝐴 −_ℎ 𝐵) ·_ih (𝐴 −_ℎ 𝐵))
5	1, 2	hvaddcli 31050	. . . 4 ⊢ (𝐴 +_ℎ 𝐵) ∈ ℋ
6	5	normsqi 31164	. . 3 ⊢ ((norm_ℎ‘(𝐴 +_ℎ 𝐵))↑2) = ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵))
7	4, 6	oveq12i 7460	. 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 31164	. . . . . 6 ⊢ ((norm_ℎ‘𝐴)↑2) = (𝐴 ·_ih 𝐴)
9	8	oveq2i 7459	. . . . 5 ⊢ (2 · ((norm_ℎ‘𝐴)↑2)) = (2 · (𝐴 ·_ih 𝐴))
10	1, 1	hicli 31113	. . . . . 6 ⊢ (𝐴 ·_ih 𝐴) ∈ ℂ
11	10	2timesi 12431	. . . . 5 ⊢ (2 · (𝐴 ·_ih 𝐴)) = ((𝐴 ·_ih 𝐴) + (𝐴 ·_ih 𝐴))
12	9, 11	eqtri 2768	. . . 4 ⊢ (2 · ((norm_ℎ‘𝐴)↑2)) = ((𝐴 ·_ih 𝐴) + (𝐴 ·_ih 𝐴))
13	2	normsqi 31164	. . . . . 6 ⊢ ((norm_ℎ‘𝐵)↑2) = (𝐵 ·_ih 𝐵)
14	13	oveq2i 7459	. . . . 5 ⊢ (2 · ((norm_ℎ‘𝐵)↑2)) = (2 · (𝐵 ·_ih 𝐵))
15	2, 2	hicli 31113	. . . . . 6 ⊢ (𝐵 ·_ih 𝐵) ∈ ℂ
16	15	2timesi 12431	. . . . 5 ⊢ (2 · (𝐵 ·_ih 𝐵)) = ((𝐵 ·_ih 𝐵) + (𝐵 ·_ih 𝐵))
17	14, 16	eqtri 2768	. . . 4 ⊢ (2 · ((norm_ℎ‘𝐵)↑2)) = ((𝐵 ·_ih 𝐵) + (𝐵 ·_ih 𝐵))
18	12, 17	oveq12i 7460	. . 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 31150	. . . . . 6 ⊢ ((𝐴 −_ℎ 𝐵) ·_ih (𝐴 −_ℎ 𝐵)) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) − ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)))
20	10, 15	addcli 11296	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) ∈ ℂ
21	1, 2	hicli 31113	. . . . . . . 8 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
22	2, 1	hicli 31113	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐴) ∈ ℂ
23	21, 22	addcli 11296	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) ∈ ℂ
24	20, 23	negsubi 11614	. . . . . 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 2771	. . . . 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 31149	. . . . 5 ⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)))
27	25, 26	oveq12i 7460	. . . 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 11604	. . . . 5 ⊢ -((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) ∈ ℂ
29	20, 28, 20, 23	add42i 11515	. . . 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 11605	. . . . . 6 ⊢ (((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) + -((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴))) = 0
31	30	oveq2i 7459	. . . . 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 11296	. . . . . 6 ⊢ (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵))) ∈ ℂ
33	32	addridi 11477	. . . . 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 11514	. . . . 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 2772	. . . 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 2772	. . 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 2771	. 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 2771	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 1537 ∈ wcel 2108 ‘cfv 6573 (class class class)co 7448 0cc0 11184 + caddc 11187 · cmul 11189 − cmin 11520 -cneg 11521 2c2 12348 ↑cexp 14112 ℋchba 30951 +_ℎ cva 30952 ·_ih csp 30954 norm_ℎcno 30955 −_ℎ cmv 30957

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-hfvadd 31032 ax-hv0cl 31035 ax-hfvmul 31037 ax-hvmul0 31042 ax-hfi 31111 ax-his1 31114 ax-his2 31115 ax-his3 31116 ax-his4 31117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-hnorm 31000 df-hvsub 31003

This theorem is referenced by: normpar 31187 normpar2i 31188

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