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Theorem normpythi 31232

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 31207	. . 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 31198	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·_ih 𝐵) = 0 ↔ (𝐵 ·_ih 𝐴) = 0))
6	1, 2, 5	mp2an 698	. . . . . . . 8 ⊢ ((𝐴 ·_ih 𝐵) = 0 ↔ (𝐵 ·_ih 𝐴) = 0)
7	6	biimpi 217	. . . . . . 7 ⊢ ((𝐴 ·_ih 𝐵) = 0 → (𝐵 ·_ih 𝐴) = 0)
8	4, 7	oveq12d 7375	. . . . . 6 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) = (0 + 0))
9		00id 11313	. . . . . 6 ⊢ (0 + 0) = 0
10	8, 9	eqtrdi 2790	. . . . 5 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((𝐴 ·_ih 𝐵) + (𝐵 ·_ih 𝐴)) = 0)
11	10	oveq2d 7373	. . . 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 31171	. . . . . 6 ⊢ (𝐴 ·_ih 𝐴) ∈ ℂ
13	2, 2	hicli 31171	. . . . . 6 ⊢ (𝐵 ·_ih 𝐵) ∈ ℂ
14	12, 13	addcli 11143	. . . . 5 ⊢ ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) ∈ ℂ
15	14	addridi 11325	. . . 4 ⊢ (((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)) + 0) = ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵))
16	11, 15	eqtrdi 2790	. . 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 2786	. 2 ⊢ ((𝐴 ·_ih 𝐵) = 0 → ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵)) = ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵)))
18	1, 2	hvaddcli 31108	. . 3 ⊢ (𝐴 +_ℎ 𝐵) ∈ ℋ
19	18	normsqi 31222	. 2 ⊢ ((norm_ℎ‘(𝐴 +_ℎ 𝐵))↑2) = ((𝐴 +_ℎ 𝐵) ·_ih (𝐴 +_ℎ 𝐵))
20	1	normsqi 31222	. . 3 ⊢ ((norm_ℎ‘𝐴)↑2) = (𝐴 ·_ih 𝐴)
21	2	normsqi 31222	. . 3 ⊢ ((norm_ℎ‘𝐵)↑2) = (𝐵 ·_ih 𝐵)
22	20, 21	oveq12i 7369	. 2 ⊢ (((norm_ℎ‘𝐴)↑2) + ((norm_ℎ‘𝐵)↑2)) = ((𝐴 ·_ih 𝐴) + (𝐵 ·_ih 𝐵))
23	17, 19, 22	3eqtr4g 2799	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 207 = wceq 1547 ∈ wcel 2119 ‘cfv 6486 (class class class)co 7357 0cc0 11030 + caddc 11033 2c2 12228 ↑cexp 14015 ℋchba 31009 +_ℎ cva 31010 ·_ih csp 31012 norm_ℎcno 31013

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-hfvadd 31090 ax-hv0cl 31093 ax-hvmul0 31100 ax-hfi 31169 ax-his1 31172 ax-his2 31173 ax-his3 31174 ax-his4 31175

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4840 df-iun 4924 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7808 df-2nd 7933 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-er 8634 df-en 8885 df-dom 8886 df-sdom 8887 df-sup 9346 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-div 11800 df-nn 12167 df-2 12236 df-3 12237 df-n0 12430 df-z 12517 df-uz 12781 df-rp 12935 df-seq 13956 df-exp 14016 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-hnorm 31058

This theorem is referenced by: normpyth 31235 pjopythi 31809

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