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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pclfvalN | Structured version Visualization version GIF version |
Description: The projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pclfval.a | β’ π΄ = (AtomsβπΎ) |
pclfval.s | β’ π = (PSubSpβπΎ) |
pclfval.c | β’ π = (PClβπΎ) |
Ref | Expression |
---|---|
pclfvalN | β’ (πΎ β π β π = (π₯ β π« π΄ β¦ β© {π¦ β π β£ π₯ β π¦})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3464 | . 2 β’ (πΎ β π β πΎ β V) | |
2 | pclfval.c | . . 3 β’ π = (PClβπΎ) | |
3 | fveq2 6843 | . . . . . . 7 β’ (π = πΎ β (Atomsβπ) = (AtomsβπΎ)) | |
4 | pclfval.a | . . . . . . 7 β’ π΄ = (AtomsβπΎ) | |
5 | 3, 4 | eqtr4di 2795 | . . . . . 6 β’ (π = πΎ β (Atomsβπ) = π΄) |
6 | 5 | pweqd 4578 | . . . . 5 β’ (π = πΎ β π« (Atomsβπ) = π« π΄) |
7 | fveq2 6843 | . . . . . . . 8 β’ (π = πΎ β (PSubSpβπ) = (PSubSpβπΎ)) | |
8 | pclfval.s | . . . . . . . 8 β’ π = (PSubSpβπΎ) | |
9 | 7, 8 | eqtr4di 2795 | . . . . . . 7 β’ (π = πΎ β (PSubSpβπ) = π) |
10 | 9 | rabeqdv 3423 | . . . . . 6 β’ (π = πΎ β {π¦ β (PSubSpβπ) β£ π₯ β π¦} = {π¦ β π β£ π₯ β π¦}) |
11 | 10 | inteqd 4913 | . . . . 5 β’ (π = πΎ β β© {π¦ β (PSubSpβπ) β£ π₯ β π¦} = β© {π¦ β π β£ π₯ β π¦}) |
12 | 6, 11 | mpteq12dv 5197 | . . . 4 β’ (π = πΎ β (π₯ β π« (Atomsβπ) β¦ β© {π¦ β (PSubSpβπ) β£ π₯ β π¦}) = (π₯ β π« π΄ β¦ β© {π¦ β π β£ π₯ β π¦})) |
13 | df-pclN 38354 | . . . 4 β’ PCl = (π β V β¦ (π₯ β π« (Atomsβπ) β¦ β© {π¦ β (PSubSpβπ) β£ π₯ β π¦})) | |
14 | 4 | fvexi 6857 | . . . . . 6 β’ π΄ β V |
15 | 14 | pwex 5336 | . . . . 5 β’ π« π΄ β V |
16 | 15 | mptex 7174 | . . . 4 β’ (π₯ β π« π΄ β¦ β© {π¦ β π β£ π₯ β π¦}) β V |
17 | 12, 13, 16 | fvmpt 6949 | . . 3 β’ (πΎ β V β (PClβπΎ) = (π₯ β π« π΄ β¦ β© {π¦ β π β£ π₯ β π¦})) |
18 | 2, 17 | eqtrid 2789 | . 2 β’ (πΎ β V β π = (π₯ β π« π΄ β¦ β© {π¦ β π β£ π₯ β π¦})) |
19 | 1, 18 | syl 17 | 1 β’ (πΎ β π β π = (π₯ β π« π΄ β¦ β© {π¦ β π β£ π₯ β π¦})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 {crab 3408 Vcvv 3446 β wss 3911 π« cpw 4561 β© cint 4908 β¦ cmpt 5189 βcfv 6497 Atomscatm 37728 PSubSpcpsubsp 37962 PClcpclN 38353 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-pclN 38354 |
This theorem is referenced by: pclvalN 38356 |
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