![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > pol0N | Structured version Visualization version GIF version |
Description: The polarity of the empty projective subspace is the whole space. (Contributed by NM, 29-Oct-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
polssat.a | β’ π΄ = (AtomsβπΎ) |
polssat.p | β’ β₯ = (β₯πβπΎ) |
Ref | Expression |
---|---|
pol0N | β’ (πΎ β π΅ β ( β₯ ββ ) = π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4392 | . . 3 β’ β β π΄ | |
2 | eqid 2728 | . . . 4 β’ (ocβπΎ) = (ocβπΎ) | |
3 | polssat.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
4 | eqid 2728 | . . . 4 β’ (pmapβπΎ) = (pmapβπΎ) | |
5 | polssat.p | . . . 4 β’ β₯ = (β₯πβπΎ) | |
6 | 2, 3, 4, 5 | polvalN 39372 | . . 3 β’ ((πΎ β π΅ β§ β β π΄) β ( β₯ ββ ) = (π΄ β© β© π β β ((pmapβπΎ)β((ocβπΎ)βπ)))) |
7 | 1, 6 | mpan2 690 | . 2 β’ (πΎ β π΅ β ( β₯ ββ ) = (π΄ β© β© π β β ((pmapβπΎ)β((ocβπΎ)βπ)))) |
8 | 0iin 5061 | . . . 4 β’ β© π β β ((pmapβπΎ)β((ocβπΎ)βπ)) = V | |
9 | 8 | ineq2i 4205 | . . 3 β’ (π΄ β© β© π β β ((pmapβπΎ)β((ocβπΎ)βπ))) = (π΄ β© V) |
10 | inv1 4390 | . . 3 β’ (π΄ β© V) = π΄ | |
11 | 9, 10 | eqtri 2756 | . 2 β’ (π΄ β© β© π β β ((pmapβπΎ)β((ocβπΎ)βπ))) = π΄ |
12 | 7, 11 | eqtrdi 2784 | 1 β’ (πΎ β π΅ β ( β₯ ββ ) = π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 Vcvv 3470 β© cin 3944 β wss 3945 β c0 4318 β© ciin 4992 βcfv 6542 occoc 17234 Atomscatm 38729 pmapcpmap 38964 β₯πcpolN 39369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-polarityN 39370 |
This theorem is referenced by: 2pol0N 39378 1psubclN 39411 osumcllem9N 39431 pexmidN 39436 pexmidlem6N 39442 pexmidALTN 39445 |
Copyright terms: Public domain | W3C validator |