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Theorem normpythi 31219

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 31194	. . 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 31185	. . . . . . . . 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 7376	. . . . . 6 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) = (0 + 0))
9		00id 11310	. . . . . 6 ⊢ (0 + 0) = 0
10	8, 9	eqtrdi 2787	. . . . 5 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) = 0)
11	10	oveq2d 7374	. . . 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 31158	. . . . . 6 ⊢ (𝐴 ·_ih 𝐴) ∈ ℂ
13	2, 2	hicli 31158	. . . . . 6 ⊢ (𝐵 ·_ih 𝐵) ∈ ℂ
14	12, 13	addcli 11140	. . . . 5 ⊢ ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) ∈ ℂ
15	14	addridi 11322	. . . 4 ⊢ (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + 0) = ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵))
16	11, 15	eqtrdi 2787	. . 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 2783	. 2 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) = ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)))
18	1, 2	hvaddcli 31095	. . 3 ⊢ (𝐴 +_ℎ 𝐵) ∈ ℋ
19	18	normsqi 31209	. 2 ⊢ ((norm_ℎ‘(𝐴 +_ℎ 𝐵))↑2) = ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵))
20	1	normsqi 31209	. . 3 ⊢ ((norm_ℎ‘𝐴)↑2) = (𝐴 ·_ih 𝐴)
21	2	normsqi 31209	. . 3 ⊢ ((norm_ℎ‘𝐵)↑2) = (𝐵 ·_ih 𝐵)
22	20, 21	oveq12i 7370	. 2 ⊢ (((norm_ℎ‘𝐴)↑2) + ((norm_ℎ‘𝐵)↑2)) = ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵))
23	17, 19, 22	3eqtr4g 2796	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 1541 ∈ wcel 2113 ‘cfv 6492 (class class class)co 7358 0cc0 11028 + caddc 11031 2c2 12202 ↑cexp 13986 ℋchba 30996 +_ℎ cva 30997 ·_ih csp 30999 norm_ℎcno 31000

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-hfvadd 31077 ax-hv0cl 31080 ax-hvmul0 31087 ax-hfi 31156 ax-his1 31159 ax-his2 31160 ax-his3 31161 ax-his4 31162

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-en 8886 df-dom 8887 df-sdom 8888 df-sup 9347 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-n0 12404 df-z 12491 df-uz 12754 df-rp 12908 df-seq 13927 df-exp 13987 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-hnorm 31045

This theorem is referenced by: normpyth 31222 pjopythi 31796

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