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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. This is Metamath 100 proof #4. (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 28613 | . . 3 ⊢ ((𝐴𝐺𝐵)𝑃(𝐴𝐺𝐵)) = (((𝐴𝑃𝐴) + (𝐵𝑃𝐵)) + ((𝐴𝑃𝐵) + (𝐵𝑃𝐴))) |
8 | id 22 | . . . . . . 7 ⊢ ((𝐴𝑃𝐵) = 0 → (𝐴𝑃𝐵) = 0) | |
9 | 4 | phnvi 28585 | . . . . . . . . 9 ⊢ 𝑈 ∈ NrmCVec |
10 | 1, 3 | diporthcom 28485 | . . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝑃𝐵) = 0 ↔ (𝐵𝑃𝐴) = 0)) |
11 | 9, 5, 6, 10 | mp3an 1455 | . . . . . . . 8 ⊢ ((𝐴𝑃𝐵) = 0 ↔ (𝐵𝑃𝐴) = 0) |
12 | 11 | biimpi 218 | . . . . . . 7 ⊢ ((𝐴𝑃𝐵) = 0 → (𝐵𝑃𝐴) = 0) |
13 | 8, 12 | oveq12d 7166 | . . . . . 6 ⊢ ((𝐴𝑃𝐵) = 0 → ((𝐴𝑃𝐵) + (𝐵𝑃𝐴)) = (0 + 0)) |
14 | 00id 10807 | . . . . . 6 ⊢ (0 + 0) = 0 | |
15 | 13, 14 | syl6eq 2870 | . . . . 5 ⊢ ((𝐴𝑃𝐵) = 0 → ((𝐴𝑃𝐵) + (𝐵𝑃𝐴)) = 0) |
16 | 15 | oveq2d 7164 | . . . 4 ⊢ ((𝐴𝑃𝐵) = 0 → (((𝐴𝑃𝐴) + (𝐵𝑃𝐵)) + ((𝐴𝑃𝐵) + (𝐵𝑃𝐴))) = (((𝐴𝑃𝐴) + (𝐵𝑃𝐵)) + 0)) |
17 | 1, 3 | dipcl 28481 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴𝑃𝐴) ∈ ℂ) |
18 | 9, 5, 5, 17 | mp3an 1455 | . . . . . 6 ⊢ (𝐴𝑃𝐴) ∈ ℂ |
19 | 1, 3 | dipcl 28481 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝑃𝐵) ∈ ℂ) |
20 | 9, 6, 6, 19 | mp3an 1455 | . . . . . 6 ⊢ (𝐵𝑃𝐵) ∈ ℂ |
21 | 18, 20 | addcli 10639 | . . . . 5 ⊢ ((𝐴𝑃𝐴) + (𝐵𝑃𝐵)) ∈ ℂ |
22 | 21 | addid1i 10819 | . . . 4 ⊢ (((𝐴𝑃𝐴) + (𝐵𝑃𝐵)) + 0) = ((𝐴𝑃𝐴) + (𝐵𝑃𝐵)) |
23 | 16, 22 | syl6eq 2870 | . . 3 ⊢ ((𝐴𝑃𝐵) = 0 → (((𝐴𝑃𝐴) + (𝐵𝑃𝐵)) + ((𝐴𝑃𝐵) + (𝐵𝑃𝐴))) = ((𝐴𝑃𝐴) + (𝐵𝑃𝐵))) |
24 | 7, 23 | syl5eq 2866 | . 2 ⊢ ((𝐴𝑃𝐵) = 0 → ((𝐴𝐺𝐵)𝑃(𝐴𝐺𝐵)) = ((𝐴𝑃𝐴) + (𝐵𝑃𝐵))) |
25 | 1, 2 | nvgcl 28389 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) ∈ 𝑋) |
26 | 9, 5, 6, 25 | mp3an 1455 | . . 3 ⊢ (𝐴𝐺𝐵) ∈ 𝑋 |
27 | pyth.6 | . . . 4 ⊢ 𝑁 = (normCV‘𝑈) | |
28 | 1, 27, 3 | ipidsq 28479 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝐺𝐵) ∈ 𝑋) → ((𝐴𝐺𝐵)𝑃(𝐴𝐺𝐵)) = ((𝑁‘(𝐴𝐺𝐵))↑2)) |
29 | 9, 26, 28 | mp2an 690 | . 2 ⊢ ((𝐴𝐺𝐵)𝑃(𝐴𝐺𝐵)) = ((𝑁‘(𝐴𝐺𝐵))↑2) |
30 | 1, 27, 3 | ipidsq 28479 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝑃𝐴) = ((𝑁‘𝐴)↑2)) |
31 | 9, 5, 30 | mp2an 690 | . . 3 ⊢ (𝐴𝑃𝐴) = ((𝑁‘𝐴)↑2) |
32 | 1, 27, 3 | ipidsq 28479 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋) → (𝐵𝑃𝐵) = ((𝑁‘𝐵)↑2)) |
33 | 9, 6, 32 | mp2an 690 | . . 3 ⊢ (𝐵𝑃𝐵) = ((𝑁‘𝐵)↑2) |
34 | 31, 33 | oveq12i 7160 | . 2 ⊢ ((𝐴𝑃𝐴) + (𝐵𝑃𝐵)) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)) |
35 | 24, 29, 34 | 3eqtr3g 2877 | 1 ⊢ ((𝐴𝑃𝐵) = 0 → ((𝑁‘(𝐴𝐺𝐵))↑2) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1531 ∈ wcel 2108 ‘cfv 6348 (class class class)co 7148 ℂcc 10527 0cc0 10529 + caddc 10532 2c2 11684 ↑cexp 13421 NrmCVeccnv 28353 +𝑣 cpv 28354 BaseSetcba 28355 normCVcnmcv 28359 ·𝑖OLDcdip 28469 CPreHilOLDccphlo 28581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-inf2 9096 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 ax-addf 10608 ax-mulf 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-sup 8898 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-n0 11890 df-z 11974 df-uz 12236 df-rp 12382 df-fz 12885 df-fzo 13026 df-seq 13362 df-exp 13422 df-hash 13683 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-grpo 28262 df-gid 28263 df-ginv 28264 df-ablo 28314 df-vc 28328 df-nv 28361 df-va 28364 df-ba 28365 df-sm 28366 df-0v 28367 df-nmcv 28369 df-dip 28470 df-ph 28582 |
This theorem is referenced by: (None) |
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