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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 2724 | . . 3 β’ (+gβπ) = (+gβπ) | |
2 | eqid 2724 | . . 3 β’ (LSSumβπ) = (LSSumβπ) | |
3 | eqid 2724 | . . 3 β’ (0gβπ) = (0gβπ) | |
4 | eqid 2724 | . . 3 β’ (Cntzβπ) = (Cntzβπ) | |
5 | phllmod 21493 | . . . . . 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 2724 | . . . . . 6 β’ (LSubSpβπ) = (LSubSpβπ) | |
8 | 7 | lsssssubg 20797 | . . . . 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 2724 | . . . . . 6 β’ (ocvβπ) = (ocvβπ) | |
12 | pjf.k | . . . . . 6 β’ πΎ = (projβπ) | |
13 | 10, 7, 11, 2, 12 | pjdm2 21576 | . . . . 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 3976 | . . 3 β’ ((π β PreHil β§ π β dom πΎ) β π β (SubGrpβπ)) |
16 | 10, 7 | lssss 20775 | . . . . . 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 21535 | . . . . 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 3976 | . . 3 β’ ((π β PreHil β§ π β dom πΎ) β ((ocvβπ)βπ) β (SubGrpβπ)) |
21 | 11, 7, 3 | ocvin 21537 | . . . 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 20747 | . . . . 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 19769 | . . 3 β’ ((π β PreHil β§ π β dom πΎ) β π β ((Cntzβπ)β((ocvβπ)βπ))) |
26 | eqid 2724 | . . 3 β’ (proj1βπ) = (proj1βπ) | |
27 | 1, 2, 3, 4, 15, 20, 22, 25, 26 | pj1f 19609 | . 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 21575 | . . . . 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 2730 | . . 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 6696 | . 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 1533 β wcel 2098 β© cin 3940 β wss 3941 {csn 4621 dom cdm 5667 βΆwf 6530 βcfv 6534 (class class class)co 7402 Basecbs 17145 +gcplusg 17198 0gc0g 17386 SubGrpcsubg 19039 Cntzccntz 19223 LSSumclsm 19546 proj1cpj1 19547 Abelcabl 19693 LModclmod 20698 LSubSpclss 20770 PreHilcphl 21487 ocvcocv 21523 projcpj 21565 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-sca 17214 df-vsca 17215 df-ip 17216 df-0g 17388 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-sbg 18860 df-subg 19042 df-ghm 19131 df-cntz 19225 df-lsm 19548 df-pj1 19549 df-cmn 19694 df-abl 19695 df-mgp 20032 df-rng 20050 df-ur 20079 df-ring 20132 df-lmod 20700 df-lss 20771 df-lmhm 20862 df-lvec 20943 df-sra 21013 df-rgmod 21014 df-phl 21489 df-ocv 21526 df-pj 21568 |
This theorem is referenced by: pjfo 21580 |
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