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Theorem normpari 31636

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 31503	. . . 4 ⊢ (𝐴 −_ℎ 𝐵) ∈ ℋ
4	3	normsqi 31614	. . 3 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2) = ((𝐴 −_ℎ 𝐵) ·_ih (𝐴 −_ℎ 𝐵))
5	1, 2	hvaddcli 31500	. . . 4 ⊢ (𝐴 +_ℎ 𝐵) ∈ ℋ
6	5	normsqi 31614	. . 3 ⊢ ((norm_ℎ‘(𝐴 +_ℎ 𝐵))↑2) = ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵))
7	4, 6	oveq12i 7426	. 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 31614	. . . . . 6 ⊢ ((norm_ℎ‘𝐴)↑2) = (𝐴 ·_ih 𝐴)
9	8	oveq2i 7425	. . . . 5 ⊢ (2 · ((norm_ℎ‘𝐴)↑2)) = (2 · (𝐴 ·_ih 𝐴))
10	1, 1	hicli 31563	. . . . . 6 ⊢ (𝐴 ·_ih 𝐴) ∈ ℂ
11	10	2timesi 12403	. . . . 5 ⊢ (2 · (𝐴 ·_ih 𝐴)) = ((𝐴 ·_ih 𝐴) + (𝐴 ·_ih 𝐴))
12	9, 11	eqtri 2783	. . . 4 ⊢ (2 · ((norm_ℎ‘𝐴)↑2)) = ((𝐴 ·_ih 𝐴) + (𝐴 ·_ih 𝐴))
13	2	normsqi 31614	. . . . . 6 ⊢ ((norm_ℎ‘𝐵)↑2) = (𝐵 ·_ih 𝐵)
14	13	oveq2i 7425	. . . . 5 ⊢ (2 · ((norm_ℎ‘𝐵)↑2)) = (2 · (𝐵 ·_ih 𝐵))
15	2, 2	hicli 31563	. . . . . 6 ⊢ (𝐵 ·_ih 𝐵) ∈ ℂ
16	15	2timesi 12403	. . . . 5 ⊢ (2 · (𝐵 ·_ih 𝐵)) = ((𝐵 ·_ih 𝐵) + (𝐵 ·_ih 𝐵))
17	14, 16	eqtri 2783	. . . 4 ⊢ (2 · ((norm_ℎ‘𝐵)↑2)) = ((𝐵 ·_ih 𝐵) + (𝐵 ·_ih 𝐵))
18	12, 17	oveq12i 7426	. . 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 31600	. . . . . 6 ⊢ ((𝐴 −_ℎ 𝐵) ·_ih (𝐴 −_ℎ 𝐵)) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) − ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)))
20	10, 15	addcli 11240	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) ∈ ℂ
21	1, 2	hicli 31563	. . . . . . . 8 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
22	2, 1	hicli 31563	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐴) ∈ ℂ
23	21, 22	addcli 11240	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) ∈ ℂ
24	20, 23	negsubi 11561	. . . . . 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 2786	. . . . 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 31599	. . . . 5 ⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)))
27	25, 26	oveq12i 7426	. . . 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 11551	. . . . 5 ⊢ -((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) ∈ ℂ
29	20, 28, 20, 23	add42i 11461	. . . 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 11552	. . . . . 6 ⊢ (((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) + -((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴))) = 0
31	30	oveq2i 7425	. . . . 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 11240	. . . . . 6 ⊢ (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵))) ∈ ℂ
33	32	addridi 11422	. . . . 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 11460	. . . . 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 2787	. . . 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 2787	. . 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 2786	. 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 2786	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

This proof depends on syntax axioms: = wceq 1570 ∈ wcel 2145 ‘cfv 6533 (class class class)co 7414 0cc0 11125 + caddc 11128 · cmul 11130 − cmin 11466 -cneg 11467 2c2 12320 ↑cexp 14126 ℋchba 31401 +_ℎ cva 31402 ·_ih csp 31404 norm_ℎcno 31405 −_ℎ cmv 31407

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203 ax-hfvadd 31482 ax-hv0cl 31485 ax-hfvmul 31487 ax-hvmul0 31492 ax-hfi 31561 ax-his1 31564 ax-his2 31565 ax-his3 31566 ax-his4 31567

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-er 8697 df-en 8954 df-dom 8955 df-sdom 8956 df-sup 9413 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-n0 12530 df-z 12617 df-uz 12889 df-rp 13044 df-seq 14067 df-exp 14127 df-cj 15187 df-re 15188 df-im 15189 df-sqrt 15323 df-hnorm 31450 df-hvsub 31453

This theorem is used by: normpar 31637 normpar2i 31638

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