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Mirrors > Home > MPE Home > Th. List > pjff | Structured version Visualization version GIF version |
Description: A projection is a linear operator. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
pjf.k | β’ πΎ = (projβπ) |
Ref | Expression |
---|---|
pjff | β’ (π β PreHil β πΎ:dom πΎβΆ(π LMHom π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . . 4 β’ (LSubSpβπ) = (LSubSpβπ) | |
2 | eqid 2737 | . . . 4 β’ (LSSumβπ) = (LSSumβπ) | |
3 | eqid 2737 | . . . 4 β’ (0gβπ) = (0gβπ) | |
4 | eqid 2737 | . . . 4 β’ (proj1βπ) = (proj1βπ) | |
5 | phllmod 21050 | . . . . 5 β’ (π β PreHil β π β LMod) | |
6 | 5 | adantr 482 | . . . 4 β’ ((π β PreHil β§ π₯ β dom πΎ) β π β LMod) |
7 | eqid 2737 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) | |
8 | eqid 2737 | . . . . . 6 β’ (ocvβπ) = (ocvβπ) | |
9 | pjf.k | . . . . . 6 β’ πΎ = (projβπ) | |
10 | 7, 1, 8, 2, 9 | pjdm2 21133 | . . . . 5 β’ (π β PreHil β (π₯ β dom πΎ β (π₯ β (LSubSpβπ) β§ (π₯(LSSumβπ)((ocvβπ)βπ₯)) = (Baseβπ)))) |
11 | 10 | simprbda 500 | . . . 4 β’ ((π β PreHil β§ π₯ β dom πΎ) β π₯ β (LSubSpβπ)) |
12 | 7, 1 | lssss 20413 | . . . . . 6 β’ (π₯ β (LSubSpβπ) β π₯ β (Baseβπ)) |
13 | 11, 12 | syl 17 | . . . . 5 β’ ((π β PreHil β§ π₯ β dom πΎ) β π₯ β (Baseβπ)) |
14 | 7, 8, 1 | ocvlss 21092 | . . . . 5 β’ ((π β PreHil β§ π₯ β (Baseβπ)) β ((ocvβπ)βπ₯) β (LSubSpβπ)) |
15 | 13, 14 | syldan 592 | . . . 4 β’ ((π β PreHil β§ π₯ β dom πΎ) β ((ocvβπ)βπ₯) β (LSubSpβπ)) |
16 | 8, 1, 3 | ocvin 21094 | . . . . 5 β’ ((π β PreHil β§ π₯ β (LSubSpβπ)) β (π₯ β© ((ocvβπ)βπ₯)) = {(0gβπ)}) |
17 | 11, 16 | syldan 592 | . . . 4 β’ ((π β PreHil β§ π₯ β dom πΎ) β (π₯ β© ((ocvβπ)βπ₯)) = {(0gβπ)}) |
18 | 1, 2, 3, 4, 6, 11, 15, 17 | pj1lmhm 20577 | . . 3 β’ ((π β PreHil β§ π₯ β dom πΎ) β (π₯(proj1βπ)((ocvβπ)βπ₯)) β ((π βΎs (π₯(LSSumβπ)((ocvβπ)βπ₯))) LMHom π)) |
19 | 10 | simplbda 501 | . . . . . 6 β’ ((π β PreHil β§ π₯ β dom πΎ) β (π₯(LSSumβπ)((ocvβπ)βπ₯)) = (Baseβπ)) |
20 | 19 | oveq2d 7378 | . . . . 5 β’ ((π β PreHil β§ π₯ β dom πΎ) β (π βΎs (π₯(LSSumβπ)((ocvβπ)βπ₯))) = (π βΎs (Baseβπ))) |
21 | 7 | ressid 17132 | . . . . . 6 β’ (π β PreHil β (π βΎs (Baseβπ)) = π) |
22 | 21 | adantr 482 | . . . . 5 β’ ((π β PreHil β§ π₯ β dom πΎ) β (π βΎs (Baseβπ)) = π) |
23 | 20, 22 | eqtrd 2777 | . . . 4 β’ ((π β PreHil β§ π₯ β dom πΎ) β (π βΎs (π₯(LSSumβπ)((ocvβπ)βπ₯))) = π) |
24 | 23 | oveq1d 7377 | . . 3 β’ ((π β PreHil β§ π₯ β dom πΎ) β ((π βΎs (π₯(LSSumβπ)((ocvβπ)βπ₯))) LMHom π) = (π LMHom π)) |
25 | 18, 24 | eleqtrd 2840 | . 2 β’ ((π β PreHil β§ π₯ β dom πΎ) β (π₯(proj1βπ)((ocvβπ)βπ₯)) β (π LMHom π)) |
26 | 8, 4, 9 | pjfval2 21131 | . 2 β’ πΎ = (π₯ β dom πΎ β¦ (π₯(proj1βπ)((ocvβπ)βπ₯))) |
27 | 25, 26 | fmptd 7067 | 1 β’ (π β PreHil β πΎ:dom πΎβΆ(π LMHom π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β© cin 3914 β wss 3915 {csn 4591 dom cdm 5638 βΆwf 6497 βcfv 6501 (class class class)co 7362 Basecbs 17090 βΎs cress 17119 0gc0g 17328 LSSumclsm 19423 proj1cpj1 19424 LModclmod 20338 LSubSpclss 20408 LMHom clmhm 20496 PreHilcphl 21044 ocvcocv 21080 projcpj 21122 |
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-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-map 8774 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-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-sca 17156 df-vsca 17157 df-ip 17158 df-0g 17330 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-submnd 18609 df-grp 18758 df-minusg 18759 df-sbg 18760 df-subg 18932 df-ghm 19013 df-cntz 19104 df-lsm 19425 df-pj1 19426 df-cmn 19571 df-abl 19572 df-mgp 19904 df-ur 19921 df-ring 19973 df-lmod 20340 df-lss 20409 df-lmhm 20499 df-lvec 20580 df-sra 20649 df-rgmod 20650 df-phl 21046 df-ocv 21083 df-pj 21125 |
This theorem is referenced by: (None) |
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