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Theorem normpythi 28339

Description: Analogy to Pythagorean theorem for orthogonal vectors. Remark 3.4(C) of [Beran] p. 98. (Contributed by NM, 17-Oct-1999.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
normsub.1	⊢ 𝐴 ∈ ℋ
normsub.2	⊢ 𝐵 ∈ ℋ

**Assertion**
Ref	Expression
normpythi	⊢ ((𝐴 ·_ih 𝐵) = 0 → ((norm_ℎ‘(𝐴 +_ℎ 𝐵))↑2) = (((norm_ℎ‘𝐴)↑2) + ((norm_ℎ‘𝐵)↑2)))

**Proof of Theorem normpythi**
Step	Hyp	Ref	Expression
1		normsub.1	. . . 4 ⊢ 𝐴 ∈ ℋ
2		normsub.2	. . . 4 ⊢ 𝐵 ∈ ℋ
3	1, 2, 1, 2	normlem8 28314	. . 3 ⊢ ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)))
4		id 22	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (𝐴 ·_ih 𝐵) = 0)
5		orthcom 28305	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ih 𝐵) = 0 ↔ (𝐵 ·_ih 𝐴) = 0))
6	1, 2, 5	mp2an 672	. . . . . . . 8 ⊢ ((𝐴 ·_ih 𝐵) = 0 ↔ (𝐵 ·_ih 𝐴) = 0)
7	6	biimpi 206	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (𝐵 ·_ih 𝐴) = 0)
8	4, 7	oveq12d 6811	. . . . . 6 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) = (0 + 0))
9		00id 10413	. . . . . 6 ⊢ (0 + 0) = 0
10	8, 9	syl6eq 2821	. . . . 5 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) = 0)
11	10	oveq2d 6809	. . . 4 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴))) = (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + 0))
12	1, 1	hicli 28278	. . . . . 6 ⊢ (𝐴 ·_ih 𝐴) ∈ ℂ
13	2, 2	hicli 28278	. . . . . 6 ⊢ (𝐵 ·_ih 𝐵) ∈ ℂ
14	12, 13	addcli 10246	. . . . 5 ⊢ ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) ∈ ℂ
15	14	addid1i 10425	. . . 4 ⊢ (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + 0) = ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵))
16	11, 15	syl6eq 2821	. . 3 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴))) = ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)))
17	3, 16	syl5eq 2817	. 2 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) = ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)))
18	1, 2	hvaddcli 28215	. . 3 ⊢ (𝐴 +_ℎ 𝐵) ∈ ℋ
19	18	normsqi 28329	. 2 ⊢ ((norm_ℎ‘(𝐴 +_ℎ 𝐵))↑2) = ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵))
20	1	normsqi 28329	. . 3 ⊢ ((norm_ℎ‘𝐴)↑2) = (𝐴 ·_ih 𝐴)
21	2	normsqi 28329	. . 3 ⊢ ((norm_ℎ‘𝐵)↑2) = (𝐵 ·_ih 𝐵)
22	20, 21	oveq12i 6805	. 2 ⊢ (((norm_ℎ‘𝐴)↑2) + ((norm_ℎ‘𝐵)↑2)) = ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵))
23	17, 19, 22	3eqtr4g 2830	1 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((norm_ℎ‘(𝐴 +_ℎ 𝐵))↑2) = (((norm_ℎ‘𝐴)↑2) + ((norm_ℎ‘𝐵)↑2)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 = wceq 1631 ∈ wcel 2145 ‘cfv 6031 (class class class)co 6793 0cc0 10138 + caddc 10141 2c2 11272 ↑cexp 13067 ℋchil 28116 +_ℎ cva 28117 ·_ih csp 28119 norm_ℎcno 28120

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 ax-hfvadd 28197 ax-hv0cl 28200 ax-hvmul0 28207 ax-hfi 28276 ax-his1 28279 ax-his2 28280 ax-his3 28281 ax-his4 28282

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-sup 8504 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-n0 11495 df-z 11580 df-uz 11889 df-rp 12036 df-seq 13009 df-exp 13068 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-hnorm 28165

This theorem is referenced by: normpyth 28342 pjopythi 28918

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