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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 2728 | . . 3 β’ (+gβπ) = (+gβπ) | |
| 2 | eqid 2728 | . . 3 β’ (LSSumβπ) = (LSSumβπ) | |
| 3 | eqid 2728 | . . 3 β’ (0gβπ) = (0gβπ) | |
| 4 | eqid 2728 | . . 3 β’ (Cntzβπ) = (Cntzβπ) | |
| 5 | phllmod 21555 | . . . . . 6 β’ (π β PreHil β π β LMod) | |
| 6 | 5 | adantr 480 | . . . . 5 β’ ((π β PreHil β§ π β dom πΎ) β π β LMod) |
| 7 | eqid 2728 | . . . . . 6 β’ (LSubSpβπ) = (LSubSpβπ) | |
| 8 | 7 | lsssssubg 20835 | . . . . 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 2728 | . . . . . 6 β’ (ocvβπ) = (ocvβπ) | |
| 12 | pjf.k | . . . . . 6 β’ πΎ = (projβπ) | |
| 13 | 10, 7, 11, 2, 12 | pjdm2 21638 | . . . . 5 β’ (π β PreHil β (π β dom πΎ β (π β (LSubSpβπ) β§ (π(LSSumβπ)((ocvβπ)βπ)) = π))) |
| 14 | 13 | simprbda 498 | . . . 4 β’ ((π β PreHil β§ π β dom πΎ) β π β (LSubSpβπ)) |
| 15 | 9, 14 | sseldd 3979 | . . 3 β’ ((π β PreHil β§ π β dom πΎ) β π β (SubGrpβπ)) |
| 16 | 10, 7 | lssss 20813 | . . . . . 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 21597 | . . . . 5 β’ ((π β PreHil β§ π β π) β ((ocvβπ)βπ) β (LSubSpβπ)) |
| 19 | 17, 18 | syldan 590 | . . . 4 β’ ((π β PreHil β§ π β dom πΎ) β ((ocvβπ)βπ) β (LSubSpβπ)) |
| 20 | 9, 19 | sseldd 3979 | . . 3 β’ ((π β PreHil β§ π β dom πΎ) β ((ocvβπ)βπ) β (SubGrpβπ)) |
| 21 | 11, 7, 3 | ocvin 21599 | . . . 4 β’ ((π β PreHil β§ π β (LSubSpβπ)) β (π β© ((ocvβπ)βπ)) = {(0gβπ)}) |
| 22 | 14, 21 | syldan 590 | . . 3 β’ ((π β PreHil β§ π β dom πΎ) β (π β© ((ocvβπ)βπ)) = {(0gβπ)}) |
| 23 | lmodabl 20785 | . . . . 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 19805 | . . 3 β’ ((π β PreHil β§ π β dom πΎ) β π β ((Cntzβπ)β((ocvβπ)βπ))) |
| 26 | eqid 2728 | . . 3 β’ (proj1βπ) = (proj1βπ) | |
| 27 | 1, 2, 3, 4, 15, 20, 22, 25, 26 | pj1f 19645 | . 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 21637 | . . . . 5 β’ (π β dom πΎ β (πΎβπ) = (π(proj1βπ)((ocvβπ)βπ))) |
| 29 | 28 | adantl 481 | . . . 4 β’ ((π β PreHil β§ π β dom πΎ) β (πΎβπ) = (π(proj1βπ)((ocvβπ)βπ))) |
| 30 | 29 | eqcomd 2734 | . . 3 β’ ((π β PreHil β§ π β dom πΎ) β (π(proj1βπ)((ocvβπ)βπ)) = (πΎβπ)) |
| 31 | 13 | simplbda 499 | . . 3 β’ ((π β PreHil β§ π β dom πΎ) β (π(LSSumβπ)((ocvβπ)βπ)) = π) |
| 32 | 30, 31 | feq12d 6704 | . 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 395 = wceq 1534 β wcel 2099 β© cin 3944 β wss 3945 {csn 4624 dom cdm 5672 βΆwf 6538 βcfv 6542 (class class class)co 7414 Basecbs 17173 +gcplusg 17226 0gc0g 17414 SubGrpcsubg 19068 Cntzccntz 19259 LSSumclsm 19582 proj1cpj1 19583 Abelcabl 19729 LModclmod 20736 LSubSpclss 20808 PreHilcphl 21549 ocvcocv 21585 projcpj 21627 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-sca 17242 df-vsca 17243 df-ip 17244 df-0g 17416 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-grp 18886 df-minusg 18887 df-sbg 18888 df-subg 19071 df-ghm 19161 df-cntz 19261 df-lsm 19584 df-pj1 19585 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-lmod 20738 df-lss 20809 df-lmhm 20900 df-lvec 20981 df-sra 21051 df-rgmod 21052 df-phl 21551 df-ocv 21588 df-pj 21630 |
| This theorem is referenced by: pjfo 21642 |
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