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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 39320 | . . 3 β’ (πΎ β π· β πΆ = {π₯ β£ (π₯ β π΄ β§ ( β₯ β( β₯ βπ₯)) = π₯)}) |
5 | 4 | eleq2d 2813 | . 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 6899 | . . . . 5 β’ π΄ β V |
7 | 6 | ssex 5314 | . . . 4 β’ (π β π΄ β π β V) |
8 | 7 | adantr 480 | . . 3 β’ ((π β π΄ β§ ( β₯ β( β₯ βπ)) = π) β π β V) |
9 | sseq1 4002 | . . . 4 β’ (π₯ = π β (π₯ β π΄ β π β π΄)) | |
10 | 2fveq3 6890 | . . . . 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 2742 | . . . 4 β’ (π₯ = π β (( β₯ β( β₯ βπ₯)) = π₯ β ( β₯ β( β₯ βπ)) = π)) |
13 | 9, 12 | anbi12d 630 | . . 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 3671 | . 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 395 = wceq 1533 β wcel 2098 {cab 2703 Vcvv 3468 β wss 3943 βcfv 6537 Atomscatm 38646 β₯πcpolN 39286 PSubClcpscN 39318 |
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-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-rab 3427 df-v 3470 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-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-iota 6489 df-fun 6539 df-fv 6545 df-psubclN 39319 |
This theorem is referenced by: psubcliN 39322 psubcli2N 39323 0psubclN 39327 1psubclN 39328 atpsubclN 39329 pmapsubclN 39330 ispsubcl2N 39331 osumclN 39351 pexmidN 39353 pexmidlem6N 39359 |
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