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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 24058 | . . . 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 24052 | . . . . . . . 8 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴 , 𝐵) = 0 ↔ (𝐵 , 𝐴) = 0)) |
11 | 4, 5, 6, 10 | syl3anc 1373 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 , 𝐵) = 0 ↔ (𝐵 , 𝐴) = 0)) |
12 | 11 | biimpa 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → (𝐵 , 𝐴) = 0) |
13 | 9, 12 | oveq12d 7209 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝐴 , 𝐵) + (𝐵 , 𝐴)) = (0 + 0)) |
14 | 00id 10972 | . . . . 5 ⊢ (0 + 0) = 0 | |
15 | 13, 14 | eqtrdi 2787 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝐴 , 𝐵) + (𝐵 , 𝐴)) = 0) |
16 | 15 | oveq2d 7207 | . . 3 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + 0)) |
17 | 2, 1 | cphipcl 24042 | . . . . . . 7 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 , 𝐴) ∈ ℂ) |
18 | 4, 5, 5, 17 | syl3anc 1373 | . . . . . 6 ⊢ (𝜑 → (𝐴 , 𝐴) ∈ ℂ) |
19 | 2, 1 | cphipcl 24042 | . . . . . . 7 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐵 , 𝐵) ∈ ℂ) |
20 | 4, 6, 6, 19 | syl3anc 1373 | . . . . . 6 ⊢ (𝜑 → (𝐵 , 𝐵) ∈ ℂ) |
21 | 18, 20 | addcld 10817 | . . . . 5 ⊢ (𝜑 → ((𝐴 , 𝐴) + (𝐵 , 𝐵)) ∈ ℂ) |
22 | 21 | addid1d 10997 | . . . 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 2775 | . 2 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝐴 + 𝐵) , (𝐴 + 𝐵)) = ((𝐴 , 𝐴) + (𝐵 , 𝐵))) |
25 | cphngp 24024 | . . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp) | |
26 | ngpgrp 23451 | . . . . . 6 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ Grp) | |
27 | 4, 25, 26 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ Grp) |
28 | 2, 3, 27, 5, 6 | grpcld 18332 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ 𝑉) |
29 | cphpyth.n | . . . . 5 ⊢ 𝑁 = (norm‘𝑊) | |
30 | 2, 1, 29 | nmsq 24045 | . . . 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 24045 | . . . . 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 24045 | . . . . 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 7209 | . . 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 2781 | 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 1543 ∈ wcel 2112 ‘cfv 6358 (class class class)co 7191 ℂcc 10692 0cc0 10694 + caddc 10697 2c2 11850 ↑cexp 13600 Basecbs 16666 +gcplusg 16749 ·𝑖cip 16754 Grpcgrp 18319 normcnm 23428 NrmGrpcngp 23429 ℂPreHilccph 24017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 ax-addf 10773 ax-mulf 10774 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-tpos 7946 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-rp 12552 df-fz 13061 df-seq 13540 df-exp 13601 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-starv 16764 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-0g 16900 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-grp 18322 df-minusg 18323 df-subg 18494 df-ghm 18574 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-cring 19519 df-oppr 19595 df-dvdsr 19613 df-unit 19614 df-rnghom 19689 df-drng 19723 df-subrg 19752 df-staf 19835 df-srng 19836 df-lmod 19855 df-lmhm 20013 df-lvec 20094 df-sra 20163 df-rgmod 20164 df-cnfld 20318 df-phl 20542 df-ngp 23435 df-nlm 23438 df-clm 23914 df-cph 24019 |
This theorem is referenced by: (None) |
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