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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elpclN | Structured version Visualization version GIF version |
Description: Membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pclfval.a | β’ π΄ = (AtomsβπΎ) |
pclfval.s | β’ π = (PSubSpβπΎ) |
pclfval.c | β’ π = (PClβπΎ) |
elpcl.q | β’ π β V |
Ref | Expression |
---|---|
elpclN | β’ ((πΎ β π β§ π β π΄) β (π β (πβπ) β βπ¦ β π (π β π¦ β π β π¦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pclfval.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
2 | pclfval.s | . . . 4 β’ π = (PSubSpβπΎ) | |
3 | pclfval.c | . . . 4 β’ π = (PClβπΎ) | |
4 | 1, 2, 3 | pclvalN 39265 | . . 3 β’ ((πΎ β π β§ π β π΄) β (πβπ) = β© {π¦ β π β£ π β π¦}) |
5 | 4 | eleq2d 2811 | . 2 β’ ((πΎ β π β§ π β π΄) β (π β (πβπ) β π β β© {π¦ β π β£ π β π¦})) |
6 | elpcl.q | . . 3 β’ π β V | |
7 | 6 | elintrab 4955 | . 2 β’ (π β β© {π¦ β π β£ π β π¦} β βπ¦ β π (π β π¦ β π β π¦)) |
8 | 5, 7 | bitrdi 287 | 1 β’ ((πΎ β π β§ π β π΄) β (π β (πβπ) β βπ¦ β π (π β π¦ β π β π¦))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 βwral 3053 {crab 3424 Vcvv 3466 β wss 3941 β© cint 4941 βcfv 6534 Atomscatm 38637 PSubSpcpsubsp 38871 PClcpclN 39262 |
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 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 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-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-id 5565 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-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-psubsp 38878 df-pclN 39263 |
This theorem is referenced by: pclfinclN 39325 |
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