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Theorem normpythi 30970

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 30945	. . 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 30936	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ih 𝐵) = 0 ↔ (𝐵 ·_ih 𝐴) = 0))
6	1, 2, 5	mp2an 690	. . . . . . . 8 ⊢ ((𝐴 ·_ih 𝐵) = 0 ↔ (𝐵 ·_ih 𝐴) = 0)
7	6	biimpi 215	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (𝐵 ·_ih 𝐴) = 0)
8	4, 7	oveq12d 7442	. . . . . 6 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) = (0 + 0))
9		00id 11425	. . . . . 6 ⊢ (0 + 0) = 0
10	8, 9	eqtrdi 2783	. . . . 5 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) = 0)
11	10	oveq2d 7440	. . . 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 30909	. . . . . 6 ⊢ (𝐴 ·_ih 𝐴) ∈ ℂ
13	2, 2	hicli 30909	. . . . . 6 ⊢ (𝐵 ·_ih 𝐵) ∈ ℂ
14	12, 13	addcli 11256	. . . . 5 ⊢ ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) ∈ ℂ
15	14	addridi 11437	. . . 4 ⊢ (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + 0) = ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵))
16	11, 15	eqtrdi 2783	. . 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 2779	. 2 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) = ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)))
18	1, 2	hvaddcli 30846	. . 3 ⊢ (𝐴 +_ℎ 𝐵) ∈ ℋ
19	18	normsqi 30960	. 2 ⊢ ((norm_ℎ‘(𝐴 +_ℎ 𝐵))↑2) = ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵))
20	1	normsqi 30960	. . 3 ⊢ ((norm_ℎ‘𝐴)↑2) = (𝐴 ·_ih 𝐴)
21	2	normsqi 30960	. . 3 ⊢ ((norm_ℎ‘𝐵)↑2) = (𝐵 ·_ih 𝐵)
22	20, 21	oveq12i 7436	. 2 ⊢ (((norm_ℎ‘𝐴)↑2) + ((norm_ℎ‘𝐵)↑2)) = ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵))
23	17, 19, 22	3eqtr4g 2792	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 205 = wceq 1533 ∈ wcel 2098 ‘cfv 6551 (class class class)co 7424 0cc0 11144 + caddc 11147 2c2 12303 ↑cexp 14064 ℋchba 30747 +_ℎ cva 30748 ·_ih csp 30750 norm_ℎcno 30751

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 ax-hfvadd 30828 ax-hv0cl 30831 ax-hvmul0 30838 ax-hfi 30907 ax-his1 30910 ax-his2 30911 ax-his3 30912 ax-his4 30913

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-sup 9471 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-n0 12509 df-z 12595 df-uz 12859 df-rp 13013 df-seq 14005 df-exp 14065 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-hnorm 30796

This theorem is referenced by: normpyth 30973 pjopythi 31547

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