![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > iporthcom | Structured version Visualization version GIF version |
Description: Orthogonality (meaning inner product is 0) is commutative. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
phlsrng.f | β’ πΉ = (Scalarβπ) |
phllmhm.h | β’ , = (Β·πβπ) |
phllmhm.v | β’ π = (Baseβπ) |
ip0l.z | β’ π = (0gβπΉ) |
Ref | Expression |
---|---|
iporthcom | β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β ((π΄ , π΅) = π β (π΅ , π΄) = π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phlsrng.f | . . . . . 6 β’ πΉ = (Scalarβπ) | |
2 | 1 | phlsrng 21175 | . . . . 5 β’ (π β PreHil β πΉ β *-Ring) |
3 | 2 | 3ad2ant1 1133 | . . . 4 β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β πΉ β *-Ring) |
4 | eqid 2732 | . . . . 5 β’ (*rfβπΉ) = (*rfβπΉ) | |
5 | eqid 2732 | . . . . 5 β’ (BaseβπΉ) = (BaseβπΉ) | |
6 | 4, 5 | srngf1o 20454 | . . . 4 β’ (πΉ β *-Ring β (*rfβπΉ):(BaseβπΉ)β1-1-ontoβ(BaseβπΉ)) |
7 | f1of1 6829 | . . . 4 β’ ((*rfβπΉ):(BaseβπΉ)β1-1-ontoβ(BaseβπΉ) β (*rfβπΉ):(BaseβπΉ)β1-1β(BaseβπΉ)) | |
8 | 3, 6, 7 | 3syl 18 | . . 3 β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β (*rfβπΉ):(BaseβπΉ)β1-1β(BaseβπΉ)) |
9 | phllmhm.h | . . . 4 β’ , = (Β·πβπ) | |
10 | phllmhm.v | . . . 4 β’ π = (Baseβπ) | |
11 | 1, 9, 10, 5 | ipcl 21177 | . . 3 β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β (π΄ , π΅) β (BaseβπΉ)) |
12 | phllmod 21174 | . . . . 5 β’ (π β PreHil β π β LMod) | |
13 | 12 | 3ad2ant1 1133 | . . . 4 β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β π β LMod) |
14 | ip0l.z | . . . . 5 β’ π = (0gβπΉ) | |
15 | 1, 5, 14 | lmod0cl 20490 | . . . 4 β’ (π β LMod β π β (BaseβπΉ)) |
16 | 13, 15 | syl 17 | . . 3 β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β π β (BaseβπΉ)) |
17 | f1fveq 7257 | . . 3 β’ (((*rfβπΉ):(BaseβπΉ)β1-1β(BaseβπΉ) β§ ((π΄ , π΅) β (BaseβπΉ) β§ π β (BaseβπΉ))) β (((*rfβπΉ)β(π΄ , π΅)) = ((*rfβπΉ)βπ) β (π΄ , π΅) = π)) | |
18 | 8, 11, 16, 17 | syl12anc 835 | . 2 β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β (((*rfβπΉ)β(π΄ , π΅)) = ((*rfβπΉ)βπ) β (π΄ , π΅) = π)) |
19 | eqid 2732 | . . . . . 6 β’ (*πβπΉ) = (*πβπΉ) | |
20 | 5, 19, 4 | stafval 20448 | . . . . 5 β’ ((π΄ , π΅) β (BaseβπΉ) β ((*rfβπΉ)β(π΄ , π΅)) = ((*πβπΉ)β(π΄ , π΅))) |
21 | 11, 20 | syl 17 | . . . 4 β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β ((*rfβπΉ)β(π΄ , π΅)) = ((*πβπΉ)β(π΄ , π΅))) |
22 | 1, 9, 10, 19 | ipcj 21178 | . . . 4 β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β ((*πβπΉ)β(π΄ , π΅)) = (π΅ , π΄)) |
23 | 21, 22 | eqtrd 2772 | . . 3 β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β ((*rfβπΉ)β(π΄ , π΅)) = (π΅ , π΄)) |
24 | 5, 19, 4 | stafval 20448 | . . . . 5 β’ (π β (BaseβπΉ) β ((*rfβπΉ)βπ) = ((*πβπΉ)βπ)) |
25 | 16, 24 | syl 17 | . . . 4 β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β ((*rfβπΉ)βπ) = ((*πβπΉ)βπ)) |
26 | 19, 14 | srng0 20460 | . . . . 5 β’ (πΉ β *-Ring β ((*πβπΉ)βπ) = π) |
27 | 3, 26 | syl 17 | . . . 4 β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β ((*πβπΉ)βπ) = π) |
28 | 25, 27 | eqtrd 2772 | . . 3 β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β ((*rfβπΉ)βπ) = π) |
29 | 23, 28 | eqeq12d 2748 | . 2 β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β (((*rfβπΉ)β(π΄ , π΅)) = ((*rfβπΉ)βπ) β (π΅ , π΄) = π)) |
30 | 18, 29 | bitr3d 280 | 1 β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β ((π΄ , π΅) = π β (π΅ , π΄) = π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ w3a 1087 = wceq 1541 β wcel 2106 β1-1βwf1 6537 β1-1-ontoβwf1o 6539 βcfv 6540 (class class class)co 7405 Basecbs 17140 *πcstv 17195 Scalarcsca 17196 Β·πcip 17198 0gc0g 17381 *rfcstf 20443 *-Ringcsr 20444 LModclmod 20463 PreHilcphl 21168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-grp 18818 df-ghm 19084 df-mgp 19982 df-ur 19999 df-ring 20051 df-oppr 20142 df-rnghom 20243 df-staf 20445 df-srng 20446 df-lmod 20465 df-lmhm 20625 df-lvec 20706 df-sra 20777 df-rgmod 20778 df-phl 21170 |
This theorem is referenced by: ocvocv 21215 lsmcss 21236 cphorthcom 24709 |
Copyright terms: Public domain | W3C validator |