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Theorem normpari 31446

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 31313	. . . 4 ⊢ (𝐴 −_ℎ 𝐵) ∈ ℋ
4	3	normsqi 31424	. . 3 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2) = ((𝐴 −_ℎ 𝐵) ·_ih (𝐴 −_ℎ 𝐵))
5	1, 2	hvaddcli 31310	. . . 4 ⊢ (𝐴 +_ℎ 𝐵) ∈ ℋ
6	5	normsqi 31424	. . 3 ⊢ ((norm_ℎ‘(𝐴 +_ℎ 𝐵))↑2) = ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵))
7	4, 6	oveq12i 7423	. 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 31424	. . . . . 6 ⊢ ((norm_ℎ‘𝐴)↑2) = (𝐴 ·_ih 𝐴)
9	8	oveq2i 7422	. . . . 5 ⊢ (2 · ((norm_ℎ‘𝐴)↑2)) = (2 · (𝐴 ·_ih 𝐴))
10	1, 1	hicli 31373	. . . . . 6 ⊢ (𝐴 ·_ih 𝐴) ∈ ℂ
11	10	2timesi 12377	. . . . 5 ⊢ (2 · (𝐴 ·_ih 𝐴)) = ((𝐴 ·_ih 𝐴) + (𝐴 ·_ih 𝐴))
12	9, 11	eqtri 2792	. . . 4 ⊢ (2 · ((norm_ℎ‘𝐴)↑2)) = ((𝐴 ·_ih 𝐴) + (𝐴 ·_ih 𝐴))
13	2	normsqi 31424	. . . . . 6 ⊢ ((norm_ℎ‘𝐵)↑2) = (𝐵 ·_ih 𝐵)
14	13	oveq2i 7422	. . . . 5 ⊢ (2 · ((norm_ℎ‘𝐵)↑2)) = (2 · (𝐵 ·_ih 𝐵))
15	2, 2	hicli 31373	. . . . . 6 ⊢ (𝐵 ·_ih 𝐵) ∈ ℂ
16	15	2timesi 12377	. . . . 5 ⊢ (2 · (𝐵 ·_ih 𝐵)) = ((𝐵 ·_ih 𝐵) + (𝐵 ·_ih 𝐵))
17	14, 16	eqtri 2792	. . . 4 ⊢ (2 · ((norm_ℎ‘𝐵)↑2)) = ((𝐵 ·_ih 𝐵) + (𝐵 ·_ih 𝐵))
18	12, 17	oveq12i 7423	. . 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 31410	. . . . . 6 ⊢ ((𝐴 −_ℎ 𝐵) ·_ih (𝐴 −_ℎ 𝐵)) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) − ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)))
20	10, 15	addcli 11214	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) ∈ ℂ
21	1, 2	hicli 31373	. . . . . . . 8 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
22	2, 1	hicli 31373	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐴) ∈ ℂ
23	21, 22	addcli 11214	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) ∈ ℂ
24	20, 23	negsubi 11535	. . . . . 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 2795	. . . . 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 31409	. . . . 5 ⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)))
27	25, 26	oveq12i 7423	. . . 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 11525	. . . . 5 ⊢ -((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) ∈ ℂ
29	20, 28, 20, 23	add42i 11435	. . . 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 11526	. . . . . 6 ⊢ (((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) + -((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴))) = 0
31	30	oveq2i 7422	. . . . 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 11214	. . . . . 6 ⊢ (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵))) ∈ ℂ
33	32	addridi 11396	. . . . 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 11434	. . . . 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 2796	. . . 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 2796	. . 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 2795	. 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 2795	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 1567 ∈ wcel 2149 ‘cfv 6537 (class class class)co 7411 0cc0 11099 + caddc 11102 · cmul 11104 − cmin 11440 -cneg 11441 2c2 12294 ↑cexp 14096 ℋchba 31211 +_ℎ cva 31212 ·_ih csp 31214 norm_ℎcno 31215 −_ℎ cmv 31217

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177 ax-hfvadd 31292 ax-hv0cl 31295 ax-hfvmul 31297 ax-hvmul0 31302 ax-hfi 31371 ax-his1 31374 ax-his2 31375 ax-his3 31376 ax-his4 31377

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7862 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-er 8693 df-en 8943 df-dom 8944 df-sdom 8945 df-sup 9401 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-n0 12504 df-z 12591 df-uz 12862 df-rp 13016 df-seq 14037 df-exp 14097 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-hnorm 31260 df-hvsub 31263

This theorem is referenced by: normpar 31447 normpar2i 31448

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