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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 23952 | . . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵) , (𝐴 + 𝐵)) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴)))) |
8 | 7 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝐴 + 𝐵) , (𝐴 + 𝐵)) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴)))) |
9 | simpr 488 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → (𝐴 , 𝐵) = 0) | |
10 | 1, 2 | cphorthcom 23946 | . . . . . . . 8 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴 , 𝐵) = 0 ↔ (𝐵 , 𝐴) = 0)) |
11 | 4, 5, 6, 10 | syl3anc 1372 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 , 𝐵) = 0 ↔ (𝐵 , 𝐴) = 0)) |
12 | 11 | biimpa 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → (𝐵 , 𝐴) = 0) |
13 | 9, 12 | oveq12d 7182 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝐴 , 𝐵) + (𝐵 , 𝐴)) = (0 + 0)) |
14 | 00id 10886 | . . . . 5 ⊢ (0 + 0) = 0 | |
15 | 13, 14 | eqtrdi 2789 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝐴 , 𝐵) + (𝐵 , 𝐴)) = 0) |
16 | 15 | oveq2d 7180 | . . 3 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + 0)) |
17 | 2, 1 | cphipcl 23936 | . . . . . . 7 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 , 𝐴) ∈ ℂ) |
18 | 4, 5, 5, 17 | syl3anc 1372 | . . . . . 6 ⊢ (𝜑 → (𝐴 , 𝐴) ∈ ℂ) |
19 | 2, 1 | cphipcl 23936 | . . . . . . 7 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐵 , 𝐵) ∈ ℂ) |
20 | 4, 6, 6, 19 | syl3anc 1372 | . . . . . 6 ⊢ (𝜑 → (𝐵 , 𝐵) ∈ ℂ) |
21 | 18, 20 | addcld 10731 | . . . . 5 ⊢ (𝜑 → ((𝐴 , 𝐴) + (𝐵 , 𝐵)) ∈ ℂ) |
22 | 21 | addid1d 10911 | . . . 4 ⊢ (𝜑 → (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + 0) = ((𝐴 , 𝐴) + (𝐵 , 𝐵))) |
23 | 22 | adantr 484 | . . 3 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + 0) = ((𝐴 , 𝐴) + (𝐵 , 𝐵))) |
24 | 8, 16, 23 | 3eqtrd 2777 | . 2 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝐴 + 𝐵) , (𝐴 + 𝐵)) = ((𝐴 , 𝐴) + (𝐵 , 𝐵))) |
25 | cphngp 23918 | . . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp) | |
26 | ngpgrp 23345 | . . . . . 6 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ Grp) | |
27 | 4, 25, 26 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ Grp) |
28 | 2, 3, 27, 5, 6 | grpcld 18225 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ 𝑉) |
29 | cphpyth.n | . . . . 5 ⊢ 𝑁 = (norm‘𝑊) | |
30 | 2, 1, 29 | nmsq 23939 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 + 𝐵) ∈ 𝑉) → ((𝑁‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) , (𝐴 + 𝐵))) |
31 | 4, 28, 30 | syl2anc 587 | . . 3 ⊢ (𝜑 → ((𝑁‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) , (𝐴 + 𝐵))) |
32 | 31 | adantr 484 | . 2 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝑁‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) , (𝐴 + 𝐵))) |
33 | 2, 1, 29 | nmsq 23939 | . . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴)) |
34 | 4, 5, 33 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴)) |
35 | 2, 1, 29 | nmsq 23939 | . . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐵 ∈ 𝑉) → ((𝑁‘𝐵)↑2) = (𝐵 , 𝐵)) |
36 | 4, 6, 35 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ((𝑁‘𝐵)↑2) = (𝐵 , 𝐵)) |
37 | 34, 36 | oveq12d 7182 | . . 3 ⊢ (𝜑 → (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)) = ((𝐴 , 𝐴) + (𝐵 , 𝐵))) |
38 | 37 | adantr 484 | . 2 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)) = ((𝐴 , 𝐴) + (𝐵 , 𝐵))) |
39 | 24, 32, 38 | 3eqtr4d 2783 | 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 209 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ‘cfv 6333 (class class class)co 7164 ℂcc 10606 0cc0 10608 + caddc 10611 2c2 11764 ↑cexp 13514 Basecbs 16579 +gcplusg 16661 ·𝑖cip 16666 Grpcgrp 18212 normcnm 23322 NrmGrpcngp 23323 ℂPreHilccph 23911 |
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 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-pre-sup 10686 ax-addf 10687 ax-mulf 10688 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-tpos 7914 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-map 8432 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-sup 8972 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-z 12056 df-dec 12173 df-uz 12318 df-rp 12466 df-fz 12975 df-seq 13454 df-exp 13515 df-cj 14541 df-re 14542 df-im 14543 df-sqrt 14677 df-abs 14678 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-starv 16676 df-sca 16677 df-vsca 16678 df-ip 16679 df-tset 16680 df-ple 16681 df-ds 16683 df-unif 16684 df-0g 16811 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-mhm 18065 df-grp 18215 df-minusg 18216 df-subg 18387 df-ghm 18467 df-cmn 19019 df-abl 19020 df-mgp 19352 df-ur 19364 df-ring 19411 df-cring 19412 df-oppr 19488 df-dvdsr 19506 df-unit 19507 df-rnghom 19582 df-drng 19616 df-subrg 19645 df-staf 19728 df-srng 19729 df-lmod 19748 df-lmhm 19906 df-lvec 19987 df-sra 20056 df-rgmod 20057 df-cnfld 20211 df-phl 20435 df-ngp 23329 df-nlm 23332 df-clm 23808 df-cph 23913 |
This theorem is referenced by: (None) |
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