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Mirrors > Home > MPE Home > Th. List > cphpyth | Structured version Visualization version GIF version |
Description: The pythagorean theorem for a subcomplex pre-Hilbert space. 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.) (Revised by SN, 22-Sep-2024.) |
Ref | Expression |
---|---|
cphpyth.v | ⊢ 𝑉 = (Base‘𝑊) |
cphpyth.h | ⊢ , = (·𝑖‘𝑊) |
cphpyth.p | ⊢ + = (+g‘𝑊) |
cphpyth.n | ⊢ 𝑁 = (norm‘𝑊) |
cphpyth.w | ⊢ (𝜑 → 𝑊 ∈ ℂPreHil) |
cphpyth.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
cphpyth.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
Ref | Expression |
---|---|
cphpyth | ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝑁‘(𝐴 + 𝐵))↑2) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cphpyth.h | . . . . 5 ⊢ , = (·𝑖‘𝑊) | |
2 | cphpyth.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
3 | cphpyth.p | . . . . 5 ⊢ + = (+g‘𝑊) | |
4 | cphpyth.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ ℂPreHil) | |
5 | cphpyth.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
6 | cphpyth.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
7 | 1, 2, 3, 4, 5, 6, 5, 6 | cph2di 25253 | . . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵) , (𝐴 + 𝐵)) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴)))) |
8 | 7 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝐴 + 𝐵) , (𝐴 + 𝐵)) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴)))) |
9 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → (𝐴 , 𝐵) = 0) | |
10 | 1, 2 | cphorthcom 25247 | . . . . . . . 8 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴 , 𝐵) = 0 ↔ (𝐵 , 𝐴) = 0)) |
11 | 4, 5, 6, 10 | syl3anc 1371 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 , 𝐵) = 0 ↔ (𝐵 , 𝐴) = 0)) |
12 | 11 | biimpa 476 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → (𝐵 , 𝐴) = 0) |
13 | 9, 12 | oveq12d 7463 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝐴 , 𝐵) + (𝐵 , 𝐴)) = (0 + 0)) |
14 | 00id 11461 | . . . . 5 ⊢ (0 + 0) = 0 | |
15 | 13, 14 | eqtrdi 2790 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝐴 , 𝐵) + (𝐵 , 𝐴)) = 0) |
16 | 15 | oveq2d 7461 | . . 3 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + 0)) |
17 | 2, 1 | cphipcl 25237 | . . . . . . 7 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 , 𝐴) ∈ ℂ) |
18 | 4, 5, 5, 17 | syl3anc 1371 | . . . . . 6 ⊢ (𝜑 → (𝐴 , 𝐴) ∈ ℂ) |
19 | 2, 1 | cphipcl 25237 | . . . . . . 7 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐵 , 𝐵) ∈ ℂ) |
20 | 4, 6, 6, 19 | syl3anc 1371 | . . . . . 6 ⊢ (𝜑 → (𝐵 , 𝐵) ∈ ℂ) |
21 | 18, 20 | addcld 11305 | . . . . 5 ⊢ (𝜑 → ((𝐴 , 𝐴) + (𝐵 , 𝐵)) ∈ ℂ) |
22 | 21 | addridd 11486 | . . . 4 ⊢ (𝜑 → (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + 0) = ((𝐴 , 𝐴) + (𝐵 , 𝐵))) |
23 | 22 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + 0) = ((𝐴 , 𝐴) + (𝐵 , 𝐵))) |
24 | 8, 16, 23 | 3eqtrd 2778 | . 2 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝐴 + 𝐵) , (𝐴 + 𝐵)) = ((𝐴 , 𝐴) + (𝐵 , 𝐵))) |
25 | cphngp 25219 | . . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp) | |
26 | ngpgrp 24626 | . . . . . 6 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ Grp) | |
27 | 4, 25, 26 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ Grp) |
28 | 2, 3, 27, 5, 6 | grpcld 18982 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ 𝑉) |
29 | cphpyth.n | . . . . 5 ⊢ 𝑁 = (norm‘𝑊) | |
30 | 2, 1, 29 | nmsq 25240 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 + 𝐵) ∈ 𝑉) → ((𝑁‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) , (𝐴 + 𝐵))) |
31 | 4, 28, 30 | syl2anc 583 | . . 3 ⊢ (𝜑 → ((𝑁‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) , (𝐴 + 𝐵))) |
32 | 31 | adantr 480 | . 2 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝑁‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) , (𝐴 + 𝐵))) |
33 | 2, 1, 29 | nmsq 25240 | . . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴)) |
34 | 4, 5, 33 | syl2anc 583 | . . . 4 ⊢ (𝜑 → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴)) |
35 | 2, 1, 29 | nmsq 25240 | . . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐵 ∈ 𝑉) → ((𝑁‘𝐵)↑2) = (𝐵 , 𝐵)) |
36 | 4, 6, 35 | syl2anc 583 | . . . 4 ⊢ (𝜑 → ((𝑁‘𝐵)↑2) = (𝐵 , 𝐵)) |
37 | 34, 36 | oveq12d 7463 | . . 3 ⊢ (𝜑 → (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)) = ((𝐴 , 𝐴) + (𝐵 , 𝐵))) |
38 | 37 | adantr 480 | . 2 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)) = ((𝐴 , 𝐴) + (𝐵 , 𝐵))) |
39 | 24, 32, 38 | 3eqtr4d 2784 | 1 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝑁‘(𝐴 + 𝐵))↑2) = (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2103 ‘cfv 6572 (class class class)co 7445 ℂcc 11178 0cc0 11180 + caddc 11183 2c2 12344 ↑cexp 14108 Basecbs 17253 +gcplusg 17306 ·𝑖cip 17311 Grpcgrp 18968 normcnm 24603 NrmGrpcngp 24604 ℂPreHilccph 25212 |
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-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-tp 4653 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-1st 8026 df-2nd 8027 df-tpos 8263 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-er 8759 df-map 8882 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 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-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-z 12636 df-dec 12755 df-uz 12900 df-rp 13054 df-fz 13564 df-seq 14049 df-exp 14109 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-starv 17321 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-0g 17496 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-mhm 18813 df-grp 18971 df-minusg 18972 df-subg 19158 df-ghm 19248 df-cmn 19819 df-abl 19820 df-mgp 20157 df-rng 20175 df-ur 20204 df-ring 20257 df-cring 20258 df-oppr 20355 df-dvdsr 20378 df-unit 20379 df-rhm 20493 df-subrg 20592 df-drng 20748 df-staf 20857 df-srng 20858 df-lmod 20877 df-lmhm 21039 df-lvec 21120 df-sra 21190 df-rgmod 21191 df-cnfld 21383 df-phl 21662 df-ngp 24610 df-nlm 24613 df-clm 25108 df-cph 25214 |
This theorem is referenced by: (None) |
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