![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 | β’ π = (Baseβπ) |
cphpyth.h | β’ , = (Β·πβπ) |
cphpyth.p | β’ + = (+gβπ) |
cphpyth.n | β’ π = (normβπ) |
cphpyth.w | β’ (π β π β βPreHil) |
cphpyth.a | β’ (π β π΄ β π) |
cphpyth.b | β’ (π β π΅ β π) |
Ref | Expression |
---|---|
cphpyth | β’ ((π β§ (π΄ , π΅) = 0) β ((πβ(π΄ + π΅))β2) = (((πβπ΄)β2) + ((πβπ΅)β2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cphpyth.h | . . . . 5 β’ , = (Β·πβπ) | |
2 | cphpyth.v | . . . . 5 β’ π = (Baseβπ) | |
3 | cphpyth.p | . . . . 5 β’ + = (+gβπ) | |
4 | cphpyth.w | . . . . 5 β’ (π β π β βPreHil) | |
5 | cphpyth.a | . . . . 5 β’ (π β π΄ β π) | |
6 | cphpyth.b | . . . . 5 β’ (π β π΅ β π) | |
7 | 1, 2, 3, 4, 5, 6, 5, 6 | cph2di 25155 | . . . 4 β’ (π β ((π΄ + π΅) , (π΄ + π΅)) = (((π΄ , π΄) + (π΅ , π΅)) + ((π΄ , π΅) + (π΅ , π΄)))) |
8 | 7 | adantr 479 | . . 3 β’ ((π β§ (π΄ , π΅) = 0) β ((π΄ + π΅) , (π΄ + π΅)) = (((π΄ , π΄) + (π΅ , π΅)) + ((π΄ , π΅) + (π΅ , π΄)))) |
9 | simpr 483 | . . . . . 6 β’ ((π β§ (π΄ , π΅) = 0) β (π΄ , π΅) = 0) | |
10 | 1, 2 | cphorthcom 25149 | . . . . . . . 8 β’ ((π β βPreHil β§ π΄ β π β§ π΅ β π) β ((π΄ , π΅) = 0 β (π΅ , π΄) = 0)) |
11 | 4, 5, 6, 10 | syl3anc 1368 | . . . . . . 7 β’ (π β ((π΄ , π΅) = 0 β (π΅ , π΄) = 0)) |
12 | 11 | biimpa 475 | . . . . . 6 β’ ((π β§ (π΄ , π΅) = 0) β (π΅ , π΄) = 0) |
13 | 9, 12 | oveq12d 7444 | . . . . 5 β’ ((π β§ (π΄ , π΅) = 0) β ((π΄ , π΅) + (π΅ , π΄)) = (0 + 0)) |
14 | 00id 11427 | . . . . 5 β’ (0 + 0) = 0 | |
15 | 13, 14 | eqtrdi 2784 | . . . 4 β’ ((π β§ (π΄ , π΅) = 0) β ((π΄ , π΅) + (π΅ , π΄)) = 0) |
16 | 15 | oveq2d 7442 | . . 3 β’ ((π β§ (π΄ , π΅) = 0) β (((π΄ , π΄) + (π΅ , π΅)) + ((π΄ , π΅) + (π΅ , π΄))) = (((π΄ , π΄) + (π΅ , π΅)) + 0)) |
17 | 2, 1 | cphipcl 25139 | . . . . . . 7 β’ ((π β βPreHil β§ π΄ β π β§ π΄ β π) β (π΄ , π΄) β β) |
18 | 4, 5, 5, 17 | syl3anc 1368 | . . . . . 6 β’ (π β (π΄ , π΄) β β) |
19 | 2, 1 | cphipcl 25139 | . . . . . . 7 β’ ((π β βPreHil β§ π΅ β π β§ π΅ β π) β (π΅ , π΅) β β) |
20 | 4, 6, 6, 19 | syl3anc 1368 | . . . . . 6 β’ (π β (π΅ , π΅) β β) |
21 | 18, 20 | addcld 11271 | . . . . 5 β’ (π β ((π΄ , π΄) + (π΅ , π΅)) β β) |
22 | 21 | addridd 11452 | . . . 4 β’ (π β (((π΄ , π΄) + (π΅ , π΅)) + 0) = ((π΄ , π΄) + (π΅ , π΅))) |
23 | 22 | adantr 479 | . . 3 β’ ((π β§ (π΄ , π΅) = 0) β (((π΄ , π΄) + (π΅ , π΅)) + 0) = ((π΄ , π΄) + (π΅ , π΅))) |
24 | 8, 16, 23 | 3eqtrd 2772 | . 2 β’ ((π β§ (π΄ , π΅) = 0) β ((π΄ + π΅) , (π΄ + π΅)) = ((π΄ , π΄) + (π΅ , π΅))) |
25 | cphngp 25121 | . . . . . 6 β’ (π β βPreHil β π β NrmGrp) | |
26 | ngpgrp 24528 | . . . . . 6 β’ (π β NrmGrp β π β Grp) | |
27 | 4, 25, 26 | 3syl 18 | . . . . 5 β’ (π β π β Grp) |
28 | 2, 3, 27, 5, 6 | grpcld 18911 | . . . 4 β’ (π β (π΄ + π΅) β π) |
29 | cphpyth.n | . . . . 5 β’ π = (normβπ) | |
30 | 2, 1, 29 | nmsq 25142 | . . . 4 β’ ((π β βPreHil β§ (π΄ + π΅) β π) β ((πβ(π΄ + π΅))β2) = ((π΄ + π΅) , (π΄ + π΅))) |
31 | 4, 28, 30 | syl2anc 582 | . . 3 β’ (π β ((πβ(π΄ + π΅))β2) = ((π΄ + π΅) , (π΄ + π΅))) |
32 | 31 | adantr 479 | . 2 β’ ((π β§ (π΄ , π΅) = 0) β ((πβ(π΄ + π΅))β2) = ((π΄ + π΅) , (π΄ + π΅))) |
33 | 2, 1, 29 | nmsq 25142 | . . . . 5 β’ ((π β βPreHil β§ π΄ β π) β ((πβπ΄)β2) = (π΄ , π΄)) |
34 | 4, 5, 33 | syl2anc 582 | . . . 4 β’ (π β ((πβπ΄)β2) = (π΄ , π΄)) |
35 | 2, 1, 29 | nmsq 25142 | . . . . 5 β’ ((π β βPreHil β§ π΅ β π) β ((πβπ΅)β2) = (π΅ , π΅)) |
36 | 4, 6, 35 | syl2anc 582 | . . . 4 β’ (π β ((πβπ΅)β2) = (π΅ , π΅)) |
37 | 34, 36 | oveq12d 7444 | . . 3 β’ (π β (((πβπ΄)β2) + ((πβπ΅)β2)) = ((π΄ , π΄) + (π΅ , π΅))) |
38 | 37 | adantr 479 | . 2 β’ ((π β§ (π΄ , π΅) = 0) β (((πβπ΄)β2) + ((πβπ΅)β2)) = ((π΄ , π΄) + (π΅ , π΅))) |
39 | 24, 32, 38 | 3eqtr4d 2778 | 1 β’ ((π β§ (π΄ , π΅) = 0) β ((πβ(π΄ + π΅))β2) = (((πβπ΄)β2) + ((πβπ΅)β2))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 = wceq 1533 β wcel 2098 βcfv 6553 (class class class)co 7426 βcc 11144 0cc0 11146 + caddc 11149 2c2 12305 βcexp 14066 Basecbs 17187 +gcplusg 17240 Β·πcip 17245 Grpcgrp 18897 normcnm 24505 NrmGrpcngp 24506 βPreHilccph 25114 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 ax-mulf 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-rp 13015 df-fz 13525 df-seq 14007 df-exp 14067 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-grp 18900 df-minusg 18901 df-subg 19085 df-ghm 19175 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-rhm 20418 df-subrg 20515 df-drng 20633 df-staf 20732 df-srng 20733 df-lmod 20752 df-lmhm 20914 df-lvec 20995 df-sra 21065 df-rgmod 21066 df-cnfld 21287 df-phl 21565 df-ngp 24512 df-nlm 24515 df-clm 25010 df-cph 25116 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |