normpari - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > normpari		Structured version Visualization version GIF version

Theorem normpari 27395

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 27262	. . . 4 ⊢ (𝐴 −_ℎ 𝐵) ∈ ℋ
4	3	normsqi 27373	. . 3 ⊢ ((norm_ℎ‘(𝐴 −_ℎ 𝐵))↑2) = ((𝐴 −_ℎ 𝐵) ·_ih (𝐴 −_ℎ 𝐵))
5	1, 2	hvaddcli 27259	. . . 4 ⊢ (𝐴 +_ℎ 𝐵) ∈ ℋ
6	5	normsqi 27373	. . 3 ⊢ ((norm_ℎ‘(𝐴 +_ℎ 𝐵))↑2) = ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵))
7	4, 6	oveq12i 6561	. 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 27373	. . . . . 6 ⊢ ((norm_ℎ‘𝐴)↑2) = (𝐴 ·_ih 𝐴)
9	8	oveq2i 6560	. . . . 5 ⊢ (2 · ((norm_ℎ‘𝐴)↑2)) = (2 · (𝐴 ·_ih 𝐴))
10	1, 1	hicli 27322	. . . . . 6 ⊢ (𝐴 ·_ih 𝐴) ∈ ℂ
11	10	2timesi 11024	. . . . 5 ⊢ (2 · (𝐴 ·_ih 𝐴)) = ((𝐴 ·_ih 𝐴) + (𝐴 ·_ih 𝐴))
12	9, 11	eqtri 2632	. . . 4 ⊢ (2 · ((norm_ℎ‘𝐴)↑2)) = ((𝐴 ·_ih 𝐴) + (𝐴 ·_ih 𝐴))
13	2	normsqi 27373	. . . . . 6 ⊢ ((norm_ℎ‘𝐵)↑2) = (𝐵 ·_ih 𝐵)
14	13	oveq2i 6560	. . . . 5 ⊢ (2 · ((norm_ℎ‘𝐵)↑2)) = (2 · (𝐵 ·_ih 𝐵))
15	2, 2	hicli 27322	. . . . . 6 ⊢ (𝐵 ·_ih 𝐵) ∈ ℂ
16	15	2timesi 11024	. . . . 5 ⊢ (2 · (𝐵 ·_ih 𝐵)) = ((𝐵 ·_ih 𝐵) + (𝐵 ·_ih 𝐵))
17	14, 16	eqtri 2632	. . . 4 ⊢ (2 · ((norm_ℎ‘𝐵)↑2)) = ((𝐵 ·_ih 𝐵) + (𝐵 ·_ih 𝐵))
18	12, 17	oveq12i 6561	. . 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 27359	. . . . . 6 ⊢ ((𝐴 −_ℎ 𝐵) ·_ih (𝐴 −_ℎ 𝐵)) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) − ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)))
20	10, 15	addcli 9923	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) ∈ ℂ
21	1, 2	hicli 27322	. . . . . . . 8 ⊢ (𝐴 ·_ih 𝐵) ∈ ℂ
22	2, 1	hicli 27322	. . . . . . . 8 ⊢ (𝐵 ·_ih 𝐴) ∈ ℂ
23	21, 22	addcli 9923	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) ∈ ℂ
24	20, 23	negsubi 10238	. . . . . 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 2635	. . . . 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 27358	. . . . 5 ⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)))
27	25, 26	oveq12i 6561	. . . 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 10228	. . . . 5 ⊢ -((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) ∈ ℂ
29	20, 28, 20, 23	add42i 10140	. . . 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 10229	. . . . . 6 ⊢ (((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) + -((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴))) = 0
31	30	oveq2i 6560	. . . . 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 9923	. . . . . 6 ⊢ (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵))) ∈ ℂ
33	32	addid1i 10102	. . . . 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 10139	. . . . 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 2636	. . . 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 2636	. . 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 2635	. 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 2635	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 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 0cc0 9815 + caddc 9818 · cmul 9820 − cmin 10145 -cneg 10146 2c2 10947 ↑cexp 12722 ℋchil 27160 +_ℎ cva 27161 ·_ih csp 27163 norm_ℎcno 27164 −_ℎ cmv 27166

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-hfvadd 27241 ax-hv0cl 27244 ax-hfvmul 27246 ax-hvmul0 27251 ax-hfi 27320 ax-his1 27323 ax-his2 27324 ax-his3 27325 ax-his4 27326

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-hnorm 27209 df-hvsub 27212

This theorem is referenced by: normpar 27396 normpar2i 27397

W3C validator