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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ispsubclN | Structured version Visualization version GIF version |
Description: The predicate "is a closed projective subspace". (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
psubclset.a | β’ π΄ = (AtomsβπΎ) |
psubclset.p | β’ β₯ = (β₯πβπΎ) |
psubclset.c | β’ πΆ = (PSubClβπΎ) |
Ref | Expression |
---|---|
ispsubclN | β’ (πΎ β π· β (π β πΆ β (π β π΄ β§ ( β₯ β( β₯ βπ)) = π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psubclset.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
2 | psubclset.p | . . . 4 β’ β₯ = (β₯πβπΎ) | |
3 | psubclset.c | . . . 4 β’ πΆ = (PSubClβπΎ) | |
4 | 1, 2, 3 | psubclsetN 38402 | . . 3 β’ (πΎ β π· β πΆ = {π₯ β£ (π₯ β π΄ β§ ( β₯ β( β₯ βπ₯)) = π₯)}) |
5 | 4 | eleq2d 2824 | . 2 β’ (πΎ β π· β (π β πΆ β π β {π₯ β£ (π₯ β π΄ β§ ( β₯ β( β₯ βπ₯)) = π₯)})) |
6 | 1 | fvexi 6857 | . . . . 5 β’ π΄ β V |
7 | 6 | ssex 5279 | . . . 4 β’ (π β π΄ β π β V) |
8 | 7 | adantr 482 | . . 3 β’ ((π β π΄ β§ ( β₯ β( β₯ βπ)) = π) β π β V) |
9 | sseq1 3970 | . . . 4 β’ (π₯ = π β (π₯ β π΄ β π β π΄)) | |
10 | 2fveq3 6848 | . . . . 5 β’ (π₯ = π β ( β₯ β( β₯ βπ₯)) = ( β₯ β( β₯ βπ))) | |
11 | id 22 | . . . . 5 β’ (π₯ = π β π₯ = π) | |
12 | 10, 11 | eqeq12d 2753 | . . . 4 β’ (π₯ = π β (( β₯ β( β₯ βπ₯)) = π₯ β ( β₯ β( β₯ βπ)) = π)) |
13 | 9, 12 | anbi12d 632 | . . 3 β’ (π₯ = π β ((π₯ β π΄ β§ ( β₯ β( β₯ βπ₯)) = π₯) β (π β π΄ β§ ( β₯ β( β₯ βπ)) = π))) |
14 | 8, 13 | elab3 3639 | . 2 β’ (π β {π₯ β£ (π₯ β π΄ β§ ( β₯ β( β₯ βπ₯)) = π₯)} β (π β π΄ β§ ( β₯ β( β₯ βπ)) = π)) |
15 | 5, 14 | bitrdi 287 | 1 β’ (πΎ β π· β (π β πΆ β (π β π΄ β§ ( β₯ β( β₯ βπ)) = π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 {cab 2714 Vcvv 3446 β wss 3911 βcfv 6497 Atomscatm 37728 β₯πcpolN 38368 PSubClcpscN 38400 |
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-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-rab 3409 df-v 3448 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-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-iota 6449 df-fun 6499 df-fv 6505 df-psubclN 38401 |
This theorem is referenced by: psubcliN 38404 psubcli2N 38405 0psubclN 38409 1psubclN 38410 atpsubclN 38411 pmapsubclN 38412 ispsubcl2N 38413 osumclN 38433 pexmidN 38435 pexmidlem6N 38441 |
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