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Mirrors > Home > MPE Home > Th. List > pythi | Structured version Visualization version GIF version |
Description: The Pythagorean theorem for an arbitrary complex inner product (pre-Hilbert) space 𝑈. The square of the norm of the sum of two orthogonal vectors (i.e. whose inner product is 0) is the sum of the squares of their norms. Problem 2 in [Kreyszig] p. 135. (Contributed by NM, 17-Apr-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pyth.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
pyth.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
pyth.6 | ⊢ 𝑁 = (normCV‘𝑈) |
pyth.7 | ⊢ 𝑃 = (·𝑖OLD‘𝑈) |
pythi.u | ⊢ 𝑈 ∈ CPreHilOLD |
pythi.a | ⊢ 𝐴 ∈ 𝑋 |
pythi.b | ⊢ 𝐵 ∈ 𝑋 |
Ref | Expression |
---|---|
pythi | ⊢ ((𝐴𝑃𝐵) = 0 → ((𝑁‘(𝐴𝐺𝐵))↑2) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pyth.1 | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
2 | pyth.2 | . . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
3 | pyth.7 | . . . 4 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
4 | pythi.u | . . . 4 ⊢ 𝑈 ∈ CPreHilOLD | |
5 | pythi.a | . . . 4 ⊢ 𝐴 ∈ 𝑋 | |
6 | pythi.b | . . . 4 ⊢ 𝐵 ∈ 𝑋 | |
7 | 1, 2, 3, 4, 5, 6, 5, 6 | ip2dii 30867 | . . 3 ⊢ ((𝐴𝐺𝐵)𝑃(𝐴𝐺𝐵)) = (((𝐴𝑃𝐴) + (𝐵𝑃𝐵)) + ((𝐴𝑃𝐵) + (𝐵𝑃𝐴))) |
8 | id 22 | . . . . . . 7 ⊢ ((𝐴𝑃𝐵) = 0 → (𝐴𝑃𝐵) = 0) | |
9 | 4 | phnvi 30839 | . . . . . . . . 9 ⊢ 𝑈 ∈ NrmCVec |
10 | 1, 3 | diporthcom 30739 | . . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑃𝐵) = 0 ↔ (𝐵𝑃𝐴) = 0)) |
11 | 9, 5, 6, 10 | mp3an 1461 | . . . . . . . 8 ⊢ ((𝐴𝑃𝐵) = 0 ↔ (𝐵𝑃𝐴) = 0) |
12 | 11 | biimpi 216 | . . . . . . 7 ⊢ ((𝐴𝑃𝐵) = 0 → (𝐵𝑃𝐴) = 0) |
13 | 8, 12 | oveq12d 7463 | . . . . . 6 ⊢ ((𝐴𝑃𝐵) = 0 → ((𝐴𝑃𝐵) + (𝐵𝑃𝐴)) = (0 + 0)) |
14 | 00id 11461 | . . . . . 6 ⊢ (0 + 0) = 0 | |
15 | 13, 14 | eqtrdi 2790 | . . . . 5 ⊢ ((𝐴𝑃𝐵) = 0 → ((𝐴𝑃𝐵) + (𝐵𝑃𝐴)) = 0) |
16 | 15 | oveq2d 7461 | . . . 4 ⊢ ((𝐴𝑃𝐵) = 0 → (((𝐴𝑃𝐴) + (𝐵𝑃𝐵)) + ((𝐴𝑃𝐵) + (𝐵𝑃𝐴))) = (((𝐴𝑃𝐴) + (𝐵𝑃𝐵)) + 0)) |
17 | 1, 3 | dipcl 30735 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴𝑃𝐴) ∈ ℂ) |
18 | 9, 5, 5, 17 | mp3an 1461 | . . . . . 6 ⊢ (𝐴𝑃𝐴) ∈ ℂ |
19 | 1, 3 | dipcl 30735 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝑃𝐵) ∈ ℂ) |
20 | 9, 6, 6, 19 | mp3an 1461 | . . . . . 6 ⊢ (𝐵𝑃𝐵) ∈ ℂ |
21 | 18, 20 | addcli 11292 | . . . . 5 ⊢ ((𝐴𝑃𝐴) + (𝐵𝑃𝐵)) ∈ ℂ |
22 | 21 | addridi 11473 | . . . 4 ⊢ (((𝐴𝑃𝐴) + (𝐵𝑃𝐵)) + 0) = ((𝐴𝑃𝐴) + (𝐵𝑃𝐵)) |
23 | 16, 22 | eqtrdi 2790 | . . 3 ⊢ ((𝐴𝑃𝐵) = 0 → (((𝐴𝑃𝐴) + (𝐵𝑃𝐵)) + ((𝐴𝑃𝐵) + (𝐵𝑃𝐴))) = ((𝐴𝑃𝐴) + (𝐵𝑃𝐵))) |
24 | 7, 23 | eqtrid 2786 | . 2 ⊢ ((𝐴𝑃𝐵) = 0 → ((𝐴𝐺𝐵)𝑃(𝐴𝐺𝐵)) = ((𝐴𝑃𝐴) + (𝐵𝑃𝐵))) |
25 | 1, 2 | nvgcl 30643 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋) |
26 | 9, 5, 6, 25 | mp3an 1461 | . . 3 ⊢ (𝐴𝐺𝐵) ∈ 𝑋 |
27 | pyth.6 | . . . 4 ⊢ 𝑁 = (normCV‘𝑈) | |
28 | 1, 27, 3 | ipidsq 30733 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺𝐵) ∈ 𝑋) → ((𝐴𝐺𝐵)𝑃(𝐴𝐺𝐵)) = ((𝑁‘(𝐴𝐺𝐵))↑2)) |
29 | 9, 26, 28 | mp2an 691 | . 2 ⊢ ((𝐴𝐺𝐵)𝑃(𝐴𝐺𝐵)) = ((𝑁‘(𝐴𝐺𝐵))↑2) |
30 | 1, 27, 3 | ipidsq 30733 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝑃𝐴) = ((𝑁‘𝐴)↑2)) |
31 | 9, 5, 30 | mp2an 691 | . . 3 ⊢ (𝐴𝑃𝐴) = ((𝑁‘𝐴)↑2) |
32 | 1, 27, 3 | ipidsq 30733 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝐵𝑃𝐵) = ((𝑁‘𝐵)↑2)) |
33 | 9, 6, 32 | mp2an 691 | . . 3 ⊢ (𝐵𝑃𝐵) = ((𝑁‘𝐵)↑2) |
34 | 31, 33 | oveq12i 7457 | . 2 ⊢ ((𝐴𝑃𝐴) + (𝐵𝑃𝐵)) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)) |
35 | 24, 29, 34 | 3eqtr3g 2797 | 1 ⊢ ((𝐴𝑃𝐵) = 0 → ((𝑁‘(𝐴𝐺𝐵))↑2) = (((𝑁‘𝐴)↑2) + ((𝑁‘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 ℂcc 11178 0cc0 11180 + caddc 11183 2c2 12344 ↑cexp 14108 NrmCVeccnv 30607 +𝑣 cpv 30608 BaseSetcba 30609 normCVcnmcv 30613 ·𝑖OLDcdip 30723 CPreHilOLDccphlo 30835 |
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-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-inf2 9706 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-addf 11259 ax-mulf 11260 |
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-int 4973 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-se 5655 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-isom 6581 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-sup 9507 df-oi 9575 df-card 10004 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-4 12354 df-n0 12550 df-z 12636 df-uz 12900 df-rp 13054 df-fz 13564 df-fzo 13708 df-seq 14049 df-exp 14109 df-hash 14376 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 df-clim 15530 df-sum 15731 df-grpo 30516 df-gid 30517 df-ginv 30518 df-ablo 30568 df-vc 30582 df-nv 30615 df-va 30618 df-ba 30619 df-sm 30620 df-0v 30621 df-nmcv 30623 df-dip 30724 df-ph 30836 |
This theorem is referenced by: (None) |
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