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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 3487 | . 2 β’ (πΎ β π β πΎ β V) | |
2 | pclfval.c | . . 3 β’ π = (PClβπΎ) | |
3 | fveq2 6885 | . . . . . . 7 β’ (π = πΎ β (Atomsβπ) = (AtomsβπΎ)) | |
4 | pclfval.a | . . . . . . 7 β’ π΄ = (AtomsβπΎ) | |
5 | 3, 4 | eqtr4di 2784 | . . . . . 6 β’ (π = πΎ β (Atomsβπ) = π΄) |
6 | 5 | pweqd 4614 | . . . . 5 β’ (π = πΎ β π« (Atomsβπ) = π« π΄) |
7 | fveq2 6885 | . . . . . . . 8 β’ (π = πΎ β (PSubSpβπ) = (PSubSpβπΎ)) | |
8 | pclfval.s | . . . . . . . 8 β’ π = (PSubSpβπΎ) | |
9 | 7, 8 | eqtr4di 2784 | . . . . . . 7 β’ (π = πΎ β (PSubSpβπ) = π) |
10 | 9 | rabeqdv 3441 | . . . . . 6 β’ (π = πΎ β {π¦ β (PSubSpβπ) β£ π₯ β π¦} = {π¦ β π β£ π₯ β π¦}) |
11 | 10 | inteqd 4948 | . . . . 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 5232 | . . . 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 39272 | . . . 4 β’ PCl = (π β V β¦ (π₯ β π« (Atomsβπ) β¦ β© {π¦ β (PSubSpβπ) β£ π₯ β π¦})) | |
14 | 4 | fvexi 6899 | . . . . . 6 β’ π΄ β V |
15 | 14 | pwex 5371 | . . . . 5 β’ π« π΄ β V |
16 | 15 | mptex 7220 | . . . 4 β’ (π₯ β π« π΄ β¦ β© {π¦ β π β£ π₯ β π¦}) β V |
17 | 12, 13, 16 | fvmpt 6992 | . . 3 β’ (πΎ β V β (PClβπΎ) = (π₯ β π« π΄ β¦ β© {π¦ β π β£ π₯ β π¦})) |
18 | 2, 17 | eqtrid 2778 | . 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 1533 β wcel 2098 {crab 3426 Vcvv 3468 β wss 3943 π« cpw 4597 β© cint 4943 β¦ cmpt 5224 βcfv 6537 Atomscatm 38646 PSubSpcpsubsp 38880 PClcpclN 39271 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-pclN 39272 |
This theorem is referenced by: pclvalN 39274 |
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