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Mirrors > Home > HSE Home > Th. List > normpythi | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
normsub.1 | ⊢ 𝐴 ∈ ℋ |
normsub.2 | ⊢ 𝐵 ∈ ℋ |
Ref | Expression |
---|---|
normpythi | ⊢ ((𝐴 ·ih 𝐵) = 0 → ((normℎ‘(𝐴 +ℎ 𝐵))↑2) = (((normℎ‘𝐴)↑2) + ((normℎ‘𝐵)↑2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | normsub.1 | . . . 4 ⊢ 𝐴 ∈ ℋ | |
2 | normsub.2 | . . . 4 ⊢ 𝐵 ∈ ℋ | |
3 | 1, 2, 1, 2 | normlem8 31140 | . . 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 31131 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ih 𝐵) = 0 ↔ (𝐵 ·ih 𝐴) = 0)) | |
6 | 1, 2, 5 | mp2an 691 | . . . . . . . 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 7463 | . . . . . 6 ⊢ ((𝐴 ·ih 𝐵) = 0 → ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)) = (0 + 0)) |
9 | 00id 11461 | . . . . . 6 ⊢ (0 + 0) = 0 | |
10 | 8, 9 | eqtrdi 2790 | . . . . 5 ⊢ ((𝐴 ·ih 𝐵) = 0 → ((𝐴 ·ih 𝐵) + (𝐵 ·ih 𝐴)) = 0) |
11 | 10 | oveq2d 7461 | . . . 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 31104 | . . . . . 6 ⊢ (𝐴 ·ih 𝐴) ∈ ℂ |
13 | 2, 2 | hicli 31104 | . . . . . 6 ⊢ (𝐵 ·ih 𝐵) ∈ ℂ |
14 | 12, 13 | addcli 11292 | . . . . 5 ⊢ ((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) ∈ ℂ |
15 | 14 | addridi 11473 | . . . 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 31041 | . . 3 ⊢ (𝐴 +ℎ 𝐵) ∈ ℋ |
19 | 18 | normsqi 31155 | . 2 ⊢ ((normℎ‘(𝐴 +ℎ 𝐵))↑2) = ((𝐴 +ℎ 𝐵) ·ih (𝐴 +ℎ 𝐵)) |
20 | 1 | normsqi 31155 | . . 3 ⊢ ((normℎ‘𝐴)↑2) = (𝐴 ·ih 𝐴) |
21 | 2 | normsqi 31155 | . . 3 ⊢ ((normℎ‘𝐵)↑2) = (𝐵 ·ih 𝐵) |
22 | 20, 21 | oveq12i 7457 | . 2 ⊢ (((normℎ‘𝐴)↑2) + ((normℎ‘𝐵)↑2)) = ((𝐴 ·ih 𝐴) + (𝐵 ·ih 𝐵)) |
23 | 17, 19, 22 | 3eqtr4g 2799 | 1 ⊢ ((𝐴 ·ih 𝐵) = 0 → ((normℎ‘(𝐴 +ℎ 𝐵))↑2) = (((normℎ‘𝐴)↑2) + ((normℎ‘𝐵)↑2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2103 ‘cfv 6572 (class class class)co 7445 0cc0 11180 + caddc 11183 2c2 12344 ↑cexp 14108 ℋchba 30942 +ℎ cva 30943 ·ih csp 30945 normℎcno 30946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-pre-sup 11258 ax-hfvadd 31023 ax-hv0cl 31026 ax-hvmul0 31033 ax-hfi 31102 ax-his1 31105 ax-his2 31106 ax-his3 31107 ax-his4 31108 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-sup 9507 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-div 11944 df-nn 12290 df-2 12352 df-3 12353 df-n0 12550 df-z 12636 df-uz 12900 df-rp 13054 df-seq 14049 df-exp 14109 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-hnorm 30991 |
This theorem is referenced by: normpyth 31168 pjopythi 31742 |
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