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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 24371 | . . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵) , (𝐴 + 𝐵)) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴)))) |
8 | 7 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝐴 + 𝐵) , (𝐴 + 𝐵)) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴)))) |
9 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → (𝐴 , 𝐵) = 0) | |
10 | 1, 2 | cphorthcom 24365 | . . . . . . . 8 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴 , 𝐵) = 0 ↔ (𝐵 , 𝐴) = 0)) |
11 | 4, 5, 6, 10 | syl3anc 1370 | . . . . . . 7 ⊢ (𝜑 → ((𝐴 , 𝐵) = 0 ↔ (𝐵 , 𝐴) = 0)) |
12 | 11 | biimpa 477 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → (𝐵 , 𝐴) = 0) |
13 | 9, 12 | oveq12d 7293 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝐴 , 𝐵) + (𝐵 , 𝐴)) = (0 + 0)) |
14 | 00id 11150 | . . . . 5 ⊢ (0 + 0) = 0 | |
15 | 13, 14 | eqtrdi 2794 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝐴 , 𝐵) + (𝐵 , 𝐴)) = 0) |
16 | 15 | oveq2d 7291 | . . 3 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + ((𝐴 , 𝐵) + (𝐵 , 𝐴))) = (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + 0)) |
17 | 2, 1 | cphipcl 24355 | . . . . . . 7 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 , 𝐴) ∈ ℂ) |
18 | 4, 5, 5, 17 | syl3anc 1370 | . . . . . 6 ⊢ (𝜑 → (𝐴 , 𝐴) ∈ ℂ) |
19 | 2, 1 | cphipcl 24355 | . . . . . . 7 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐵 , 𝐵) ∈ ℂ) |
20 | 4, 6, 6, 19 | syl3anc 1370 | . . . . . 6 ⊢ (𝜑 → (𝐵 , 𝐵) ∈ ℂ) |
21 | 18, 20 | addcld 10994 | . . . . 5 ⊢ (𝜑 → ((𝐴 , 𝐴) + (𝐵 , 𝐵)) ∈ ℂ) |
22 | 21 | addid1d 11175 | . . . 4 ⊢ (𝜑 → (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + 0) = ((𝐴 , 𝐴) + (𝐵 , 𝐵))) |
23 | 22 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → (((𝐴 , 𝐴) + (𝐵 , 𝐵)) + 0) = ((𝐴 , 𝐴) + (𝐵 , 𝐵))) |
24 | 8, 16, 23 | 3eqtrd 2782 | . 2 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝐴 + 𝐵) , (𝐴 + 𝐵)) = ((𝐴 , 𝐴) + (𝐵 , 𝐵))) |
25 | cphngp 24337 | . . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp) | |
26 | ngpgrp 23755 | . . . . . 6 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ Grp) | |
27 | 4, 25, 26 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ Grp) |
28 | 2, 3, 27, 5, 6 | grpcld 18590 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ 𝑉) |
29 | cphpyth.n | . . . . 5 ⊢ 𝑁 = (norm‘𝑊) | |
30 | 2, 1, 29 | nmsq 24358 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 + 𝐵) ∈ 𝑉) → ((𝑁‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) , (𝐴 + 𝐵))) |
31 | 4, 28, 30 | syl2anc 584 | . . 3 ⊢ (𝜑 → ((𝑁‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) , (𝐴 + 𝐵))) |
32 | 31 | adantr 481 | . 2 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → ((𝑁‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) , (𝐴 + 𝐵))) |
33 | 2, 1, 29 | nmsq 24358 | . . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴)) |
34 | 4, 5, 33 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴)) |
35 | 2, 1, 29 | nmsq 24358 | . . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐵 ∈ 𝑉) → ((𝑁‘𝐵)↑2) = (𝐵 , 𝐵)) |
36 | 4, 6, 35 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((𝑁‘𝐵)↑2) = (𝐵 , 𝐵)) |
37 | 34, 36 | oveq12d 7293 | . . 3 ⊢ (𝜑 → (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)) = ((𝐴 , 𝐴) + (𝐵 , 𝐵))) |
38 | 37 | adantr 481 | . 2 ⊢ ((𝜑 ∧ (𝐴 , 𝐵) = 0) → (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)) = ((𝐴 , 𝐴) + (𝐵 , 𝐵))) |
39 | 24, 32, 38 | 3eqtr4d 2788 | 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 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 0cc0 10871 + caddc 10874 2c2 12028 ↑cexp 13782 Basecbs 16912 +gcplusg 16962 ·𝑖cip 16967 Grpcgrp 18577 normcnm 23732 NrmGrpcngp 23733 ℂPreHilccph 24330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-rp 12731 df-fz 13240 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-0g 17152 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-grp 18580 df-minusg 18581 df-subg 18752 df-ghm 18832 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-rnghom 19959 df-drng 19993 df-subrg 20022 df-staf 20105 df-srng 20106 df-lmod 20125 df-lmhm 20284 df-lvec 20365 df-sra 20434 df-rgmod 20435 df-cnfld 20598 df-phl 20831 df-ngp 23739 df-nlm 23742 df-clm 24226 df-cph 24332 |
This theorem is referenced by: (None) |
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