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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 21567 | . . . . 5 β’ (π β PreHil β πΉ β *-Ring) |
3 | 2 | 3ad2ant1 1130 | . . . 4 β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β πΉ β *-Ring) |
4 | eqid 2725 | . . . . 5 β’ (*rfβπΉ) = (*rfβπΉ) | |
5 | eqid 2725 | . . . . 5 β’ (BaseβπΉ) = (BaseβπΉ) | |
6 | 4, 5 | srngf1o 20738 | . . . 4 β’ (πΉ β *-Ring β (*rfβπΉ):(BaseβπΉ)β1-1-ontoβ(BaseβπΉ)) |
7 | f1of1 6833 | . . . 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 21569 | . . 3 β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β (π΄ , π΅) β (BaseβπΉ)) |
12 | phllmod 21566 | . . . . 5 β’ (π β PreHil β π β LMod) | |
13 | 12 | 3ad2ant1 1130 | . . . 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 20775 | . . . 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 7268 | . . 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 2725 | . . . . . 6 β’ (*πβπΉ) = (*πβπΉ) | |
20 | 5, 19, 4 | stafval 20732 | . . . . 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 21570 | . . . 4 β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β ((*πβπΉ)β(π΄ , π΅)) = (π΅ , π΄)) |
23 | 21, 22 | eqtrd 2765 | . . 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 20732 | . . . . 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 20744 | . . . . 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 2765 | . . 3 β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β ((*rfβπΉ)βπ) = π) |
29 | 23, 28 | eqeq12d 2741 | . 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 1084 = wceq 1533 β wcel 2098 β1-1βwf1 6540 β1-1-ontoβwf1o 6542 βcfv 6543 (class class class)co 7416 Basecbs 17179 *πcstv 17234 Scalarcsca 17235 Β·πcip 17237 0gc0g 17420 *rfcstf 20727 *-Ringcsr 20728 LModclmod 20747 PreHilcphl 21560 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7992 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-ip 17250 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-grp 18897 df-ghm 19172 df-mgp 20079 df-ur 20126 df-ring 20179 df-oppr 20277 df-rhm 20415 df-staf 20729 df-srng 20730 df-lmod 20749 df-lmhm 20911 df-lvec 20992 df-sra 21062 df-rgmod 21063 df-phl 21562 |
This theorem is referenced by: ocvocv 21607 lsmcss 21628 cphorthcom 25147 |
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