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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 2732 | . . 3 β’ (+gβπ) = (+gβπ) | |
| 2 | eqid 2732 | . . 3 β’ (LSSumβπ) = (LSSumβπ) | |
| 3 | eqid 2732 | . . 3 β’ (0gβπ) = (0gβπ) | |
| 4 | eqid 2732 | . . 3 β’ (Cntzβπ) = (Cntzβπ) | |
| 5 | phllmod 21174 | . . . . . 6 β’ (π β PreHil β π β LMod) | |
| 6 | 5 | adantr 481 | . . . . 5 β’ ((π β PreHil β§ π β dom πΎ) β π β LMod) |
| 7 | eqid 2732 | . . . . . 6 β’ (LSubSpβπ) = (LSubSpβπ) | |
| 8 | 7 | lsssssubg 20561 | . . . . 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 2732 | . . . . . 6 β’ (ocvβπ) = (ocvβπ) | |
| 12 | pjf.k | . . . . . 6 β’ πΎ = (projβπ) | |
| 13 | 10, 7, 11, 2, 12 | pjdm2 21257 | . . . . 5 β’ (π β PreHil β (π β dom πΎ β (π β (LSubSpβπ) β§ (π(LSSumβπ)((ocvβπ)βπ)) = π))) |
| 14 | 13 | simprbda 499 | . . . 4 β’ ((π β PreHil β§ π β dom πΎ) β π β (LSubSpβπ)) |
| 15 | 9, 14 | sseldd 3982 | . . 3 β’ ((π β PreHil β§ π β dom πΎ) β π β (SubGrpβπ)) |
| 16 | 10, 7 | lssss 20539 | . . . . . 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 21216 | . . . . 5 β’ ((π β PreHil β§ π β π) β ((ocvβπ)βπ) β (LSubSpβπ)) |
| 19 | 17, 18 | syldan 591 | . . . 4 β’ ((π β PreHil β§ π β dom πΎ) β ((ocvβπ)βπ) β (LSubSpβπ)) |
| 20 | 9, 19 | sseldd 3982 | . . 3 β’ ((π β PreHil β§ π β dom πΎ) β ((ocvβπ)βπ) β (SubGrpβπ)) |
| 21 | 11, 7, 3 | ocvin 21218 | . . . 4 β’ ((π β PreHil β§ π β (LSubSpβπ)) β (π β© ((ocvβπ)βπ)) = {(0gβπ)}) |
| 22 | 14, 21 | syldan 591 | . . 3 β’ ((π β PreHil β§ π β dom πΎ) β (π β© ((ocvβπ)βπ)) = {(0gβπ)}) |
| 23 | lmodabl 20511 | . . . . 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 19719 | . . 3 β’ ((π β PreHil β§ π β dom πΎ) β π β ((Cntzβπ)β((ocvβπ)βπ))) |
| 26 | eqid 2732 | . . 3 β’ (proj1βπ) = (proj1βπ) | |
| 27 | 1, 2, 3, 4, 15, 20, 22, 25, 26 | pj1f 19559 | . 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 21256 | . . . . 5 β’ (π β dom πΎ β (πΎβπ) = (π(proj1βπ)((ocvβπ)βπ))) |
| 29 | 28 | adantl 482 | . . . 4 β’ ((π β PreHil β§ π β dom πΎ) β (πΎβπ) = (π(proj1βπ)((ocvβπ)βπ))) |
| 30 | 29 | eqcomd 2738 | . . 3 β’ ((π β PreHil β§ π β dom πΎ) β (π(proj1βπ)((ocvβπ)βπ)) = (πΎβπ)) |
| 31 | 13 | simplbda 500 | . . 3 β’ ((π β PreHil β§ π β dom πΎ) β (π(LSSumβπ)((ocvβπ)βπ)) = π) |
| 32 | 30, 31 | feq12d 6702 | . 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 396 = wceq 1541 β wcel 2106 β© cin 3946 β wss 3947 {csn 4627 dom cdm 5675 βΆwf 6536 βcfv 6540 (class class class)co 7405 Basecbs 17140 +gcplusg 17193 0gc0g 17381 SubGrpcsubg 18994 Cntzccntz 19173 LSSumclsm 19496 proj1cpj1 19497 Abelcabl 19643 LModclmod 20463 LSubSpclss 20534 PreHilcphl 21168 ocvcocv 21204 projcpj 21246 |
| 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-int 4950 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-1st 7971 df-2nd 7972 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-ress 17170 df-plusg 17206 df-sca 17209 df-vsca 17210 df-ip 17211 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-ghm 19084 df-cntz 19175 df-lsm 19498 df-pj1 19499 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-lmod 20465 df-lss 20535 df-lmhm 20625 df-lvec 20706 df-sra 20777 df-rgmod 20778 df-phl 21170 df-ocv 21207 df-pj 21249 |
| This theorem is referenced by: pjfo 21261 |
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