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Mirrors > Home > MPE Home > Th. List > pjf2 | Structured version Visualization version GIF version |
Description: A projection is a function from the base set to the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
pjf.k | β’ πΎ = (projβπ) |
pjf.v | β’ π = (Baseβπ) |
Ref | Expression |
---|---|
pjf2 | β’ ((π β PreHil β§ π β dom πΎ) β (πΎβπ):πβΆπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . . 3 β’ (+gβπ) = (+gβπ) | |
2 | eqid 2737 | . . 3 β’ (LSSumβπ) = (LSSumβπ) | |
3 | eqid 2737 | . . 3 β’ (0gβπ) = (0gβπ) | |
4 | eqid 2737 | . . 3 β’ (Cntzβπ) = (Cntzβπ) | |
5 | phllmod 21050 | . . . . . 6 β’ (π β PreHil β π β LMod) | |
6 | 5 | adantr 482 | . . . . 5 β’ ((π β PreHil β§ π β dom πΎ) β π β LMod) |
7 | eqid 2737 | . . . . . 6 β’ (LSubSpβπ) = (LSubSpβπ) | |
8 | 7 | lsssssubg 20435 | . . . . 5 β’ (π β LMod β (LSubSpβπ) β (SubGrpβπ)) |
9 | 6, 8 | syl 17 | . . . 4 β’ ((π β PreHil β§ π β dom πΎ) β (LSubSpβπ) β (SubGrpβπ)) |
10 | pjf.v | . . . . . 6 β’ π = (Baseβπ) | |
11 | eqid 2737 | . . . . . 6 β’ (ocvβπ) = (ocvβπ) | |
12 | pjf.k | . . . . . 6 β’ πΎ = (projβπ) | |
13 | 10, 7, 11, 2, 12 | pjdm2 21133 | . . . . 5 β’ (π β PreHil β (π β dom πΎ β (π β (LSubSpβπ) β§ (π(LSSumβπ)((ocvβπ)βπ)) = π))) |
14 | 13 | simprbda 500 | . . . 4 β’ ((π β PreHil β§ π β dom πΎ) β π β (LSubSpβπ)) |
15 | 9, 14 | sseldd 3950 | . . 3 β’ ((π β PreHil β§ π β dom πΎ) β π β (SubGrpβπ)) |
16 | 10, 7 | lssss 20413 | . . . . . 6 β’ (π β (LSubSpβπ) β π β π) |
17 | 14, 16 | syl 17 | . . . . 5 β’ ((π β PreHil β§ π β dom πΎ) β π β π) |
18 | 10, 11, 7 | ocvlss 21092 | . . . . 5 β’ ((π β PreHil β§ π β π) β ((ocvβπ)βπ) β (LSubSpβπ)) |
19 | 17, 18 | syldan 592 | . . . 4 β’ ((π β PreHil β§ π β dom πΎ) β ((ocvβπ)βπ) β (LSubSpβπ)) |
20 | 9, 19 | sseldd 3950 | . . 3 β’ ((π β PreHil β§ π β dom πΎ) β ((ocvβπ)βπ) β (SubGrpβπ)) |
21 | 11, 7, 3 | ocvin 21094 | . . . 4 β’ ((π β PreHil β§ π β (LSubSpβπ)) β (π β© ((ocvβπ)βπ)) = {(0gβπ)}) |
22 | 14, 21 | syldan 592 | . . 3 β’ ((π β PreHil β§ π β dom πΎ) β (π β© ((ocvβπ)βπ)) = {(0gβπ)}) |
23 | lmodabl 20385 | . . . . 5 β’ (π β LMod β π β Abel) | |
24 | 6, 23 | syl 17 | . . . 4 β’ ((π β PreHil β§ π β dom πΎ) β π β Abel) |
25 | 4, 24, 15, 20 | ablcntzd 19642 | . . 3 β’ ((π β PreHil β§ π β dom πΎ) β π β ((Cntzβπ)β((ocvβπ)βπ))) |
26 | eqid 2737 | . . 3 β’ (proj1βπ) = (proj1βπ) | |
27 | 1, 2, 3, 4, 15, 20, 22, 25, 26 | pj1f 19486 | . 2 β’ ((π β PreHil β§ π β dom πΎ) β (π(proj1βπ)((ocvβπ)βπ)):(π(LSSumβπ)((ocvβπ)βπ))βΆπ) |
28 | 11, 26, 12 | pjval 21132 | . . . . 5 β’ (π β dom πΎ β (πΎβπ) = (π(proj1βπ)((ocvβπ)βπ))) |
29 | 28 | adantl 483 | . . . 4 β’ ((π β PreHil β§ π β dom πΎ) β (πΎβπ) = (π(proj1βπ)((ocvβπ)βπ))) |
30 | 29 | eqcomd 2743 | . . 3 β’ ((π β PreHil β§ π β dom πΎ) β (π(proj1βπ)((ocvβπ)βπ)) = (πΎβπ)) |
31 | 13 | simplbda 501 | . . 3 β’ ((π β PreHil β§ π β dom πΎ) β (π(LSSumβπ)((ocvβπ)βπ)) = π) |
32 | 30, 31 | feq12d 6661 | . 2 β’ ((π β PreHil β§ π β dom πΎ) β ((π(proj1βπ)((ocvβπ)βπ)):(π(LSSumβπ)((ocvβπ)βπ))βΆπ β (πΎβπ):πβΆπ)) |
33 | 27, 32 | mpbid 231 | 1 β’ ((π β PreHil β§ π β dom πΎ) β (πΎβπ):πβΆπ) |
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 +gcplusg 17140 0gc0g 17328 SubGrpcsubg 18929 Cntzccntz 19102 LSSumclsm 19423 proj1cpj1 19424 Abelcabl 19570 LModclmod 20338 LSubSpclss 20408 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-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: pjfo 21137 |
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