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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 39465 | . . 3 β’ (πΎ β π· β πΆ = {π₯ β£ (π₯ β π΄ β§ ( β₯ β( β₯ βπ₯)) = π₯)}) |
5 | 4 | eleq2d 2811 | . 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 6906 | . . . . 5 β’ π΄ β V |
7 | 6 | ssex 5316 | . . . 4 β’ (π β π΄ β π β V) |
8 | 7 | adantr 479 | . . 3 β’ ((π β π΄ β§ ( β₯ β( β₯ βπ)) = π) β π β V) |
9 | sseq1 3998 | . . . 4 β’ (π₯ = π β (π₯ β π΄ β π β π΄)) | |
10 | 2fveq3 6897 | . . . . 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 2741 | . . . 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 3667 | . 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 286 | 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 394 = wceq 1533 β wcel 2098 {cab 2702 Vcvv 3463 β wss 3939 βcfv 6543 Atomscatm 38791 β₯πcpolN 39431 PSubClcpscN 39463 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6495 df-fun 6545 df-fv 6551 df-psubclN 39464 |
This theorem is referenced by: psubcliN 39467 psubcli2N 39468 0psubclN 39472 1psubclN 39473 atpsubclN 39474 pmapsubclN 39475 ispsubcl2N 39476 osumclN 39496 pexmidN 39498 pexmidlem6N 39504 |
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