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Theorem normpythi 31077

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 31052	. . 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 31043	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ih 𝐵) = 0 ↔ (𝐵 ·_ih 𝐴) = 0))
6	1, 2, 5	mp2an 692	. . . . . . . 8 ⊢ ((𝐴 ·_ih 𝐵) = 0 ↔ (𝐵 ·_ih 𝐴) = 0)
7	6	biimpi 216	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (𝐵 ·_ih 𝐴) = 0)
8	4, 7	oveq12d 7407	. . . . . 6 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) = (0 + 0))
9		00id 11355	. . . . . 6 ⊢ (0 + 0) = 0
10	8, 9	eqtrdi 2781	. . . . 5 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) = 0)
11	10	oveq2d 7405	. . . 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 31016	. . . . . 6 ⊢ (𝐴 ·_ih 𝐴) ∈ ℂ
13	2, 2	hicli 31016	. . . . . 6 ⊢ (𝐵 ·_ih 𝐵) ∈ ℂ
14	12, 13	addcli 11186	. . . . 5 ⊢ ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) ∈ ℂ
15	14	addridi 11367	. . . 4 ⊢ (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + 0) = ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵))
16	11, 15	eqtrdi 2781	. . 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	eqtrid 2777	. 2 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) = ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)))
18	1, 2	hvaddcli 30953	. . 3 ⊢ (𝐴 +_ℎ 𝐵) ∈ ℋ
19	18	normsqi 31067	. 2 ⊢ ((norm_ℎ‘(𝐴 +_ℎ 𝐵))↑2) = ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵))
20	1	normsqi 31067	. . 3 ⊢ ((norm_ℎ‘𝐴)↑2) = (𝐴 ·_ih 𝐴)
21	2	normsqi 31067	. . 3 ⊢ ((norm_ℎ‘𝐵)↑2) = (𝐵 ·_ih 𝐵)
22	20, 21	oveq12i 7401	. 2 ⊢ (((norm_ℎ‘𝐴)↑2) + ((norm_ℎ‘𝐵)↑2)) = ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵))
23	17, 19, 22	3eqtr4g 2790	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 206 = wceq 1540 ∈ wcel 2109 ‘cfv 6513 (class class class)co 7389 0cc0 11074 + caddc 11077 2c2 12242 ↑cexp 14032 ℋchba 30854 +_ℎ cva 30855 ·_ih csp 30857 norm_ℎcno 30858

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152 ax-hfvadd 30935 ax-hv0cl 30938 ax-hvmul0 30945 ax-hfi 31014 ax-his1 31017 ax-his2 31018 ax-his3 31019 ax-his4 31020

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-om 7845 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-er 8673 df-en 8921 df-dom 8922 df-sdom 8923 df-sup 9399 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12188 df-2 12250 df-3 12251 df-n0 12449 df-z 12536 df-uz 12800 df-rp 12958 df-seq 13973 df-exp 14033 df-cj 15071 df-re 15072 df-im 15073 df-sqrt 15207 df-hnorm 30903

This theorem is referenced by: normpyth 31080 pjopythi 31654

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