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Mirrors > Home > MPE Home > Th. List > Mathboxes > prjsprellsp | Structured version Visualization version GIF version |
Description: Two vectors are equivalent iff their spans are equal. (Contributed by Steven Nguyen, 31-May-2023.) |
Ref | Expression |
---|---|
prjsprel.1 | β’ βΌ = {β¨π₯, π¦β© β£ ((π₯ β π΅ β§ π¦ β π΅) β§ βπ β πΎ π₯ = (π Β· π¦))} |
prjspertr.b | β’ π΅ = ((Baseβπ) β {(0gβπ)}) |
prjspertr.s | β’ π = (Scalarβπ) |
prjspertr.x | β’ Β· = ( Β·π βπ) |
prjspertr.k | β’ πΎ = (Baseβπ) |
prjsprellsp.n | β’ π = (LSpanβπ) |
Ref | Expression |
---|---|
prjsprellsp | β’ ((π β LVec β§ (π β π΅ β§ π β π΅)) β (π βΌ π β (πβ{π}) = (πβ{π}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ibar 530 | . . . 4 β’ ((π β π΅ β§ π β π΅) β (βπ β (πΎ β {(0gβπ)})π = (π Β· π) β ((π β π΅ β§ π β π΅) β§ βπ β (πΎ β {(0gβπ)})π = (π Β· π)))) | |
2 | 1 | bicomd 222 | . . 3 β’ ((π β π΅ β§ π β π΅) β (((π β π΅ β§ π β π΅) β§ βπ β (πΎ β {(0gβπ)})π = (π Β· π)) β βπ β (πΎ β {(0gβπ)})π = (π Β· π))) |
3 | 2 | adantl 483 | . 2 β’ ((π β LVec β§ (π β π΅ β§ π β π΅)) β (((π β π΅ β§ π β π΅) β§ βπ β (πΎ β {(0gβπ)})π = (π Β· π)) β βπ β (πΎ β {(0gβπ)})π = (π Β· π))) |
4 | prjsprel.1 | . . . 4 β’ βΌ = {β¨π₯, π¦β© β£ ((π₯ β π΅ β§ π¦ β π΅) β§ βπ β πΎ π₯ = (π Β· π¦))} | |
5 | prjspertr.b | . . . 4 β’ π΅ = ((Baseβπ) β {(0gβπ)}) | |
6 | prjspertr.s | . . . 4 β’ π = (Scalarβπ) | |
7 | prjspertr.x | . . . 4 β’ Β· = ( Β·π βπ) | |
8 | prjspertr.k | . . . 4 β’ πΎ = (Baseβπ) | |
9 | eqid 2737 | . . . 4 β’ (0gβπ) = (0gβπ) | |
10 | 4, 5, 6, 7, 8, 9 | prjspreln0 40976 | . . 3 β’ (π β LVec β (π βΌ π β ((π β π΅ β§ π β π΅) β§ βπ β (πΎ β {(0gβπ)})π = (π Β· π)))) |
11 | 10 | adantr 482 | . 2 β’ ((π β LVec β§ (π β π΅ β§ π β π΅)) β (π βΌ π β ((π β π΅ β§ π β π΅) β§ βπ β (πΎ β {(0gβπ)})π = (π Β· π)))) |
12 | eqid 2737 | . . 3 β’ (Baseβπ) = (Baseβπ) | |
13 | prjsprellsp.n | . . 3 β’ π = (LSpanβπ) | |
14 | simpl 484 | . . 3 β’ ((π β LVec β§ (π β π΅ β§ π β π΅)) β π β LVec) | |
15 | eldifi 4091 | . . . . 5 β’ (π β ((Baseβπ) β {(0gβπ)}) β π β (Baseβπ)) | |
16 | 15, 5 | eleq2s 2856 | . . . 4 β’ (π β π΅ β π β (Baseβπ)) |
17 | 16 | ad2antrl 727 | . . 3 β’ ((π β LVec β§ (π β π΅ β§ π β π΅)) β π β (Baseβπ)) |
18 | eldifi 4091 | . . . . 5 β’ (π β ((Baseβπ) β {(0gβπ)}) β π β (Baseβπ)) | |
19 | 18, 5 | eleq2s 2856 | . . . 4 β’ (π β π΅ β π β (Baseβπ)) |
20 | 19 | ad2antll 728 | . . 3 β’ ((π β LVec β§ (π β π΅ β§ π β π΅)) β π β (Baseβπ)) |
21 | 12, 6, 8, 9, 7, 13, 14, 17, 20 | lspsneq 20599 | . 2 β’ ((π β LVec β§ (π β π΅ β§ π β π΅)) β ((πβ{π}) = (πβ{π}) β βπ β (πΎ β {(0gβπ)})π = (π Β· π))) |
22 | 3, 11, 21 | 3bitr4d 311 | 1 β’ ((π β LVec β§ (π β π΅ β§ π β π΅)) β (π βΌ π β (πβ{π}) = (πβ{π}))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 βwrex 3074 β cdif 3912 {csn 4591 class class class wbr 5110 {copab 5172 βcfv 6501 (class class class)co 7362 Basecbs 17090 Scalarcsca 17143 Β·π cvsca 17144 0gc0g 17328 LSpanclspn 20448 LVecclvec 20579 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-tpos 8162 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-0g 17330 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-grp 18758 df-minusg 18759 df-sbg 18760 df-mgp 19904 df-ur 19921 df-ring 19973 df-oppr 20056 df-dvdsr 20077 df-unit 20078 df-invr 20108 df-drng 20201 df-lmod 20340 df-lss 20409 df-lsp 20449 df-lvec 20580 |
This theorem is referenced by: (None) |
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