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Mathbox for Norm Megill |
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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 4389 | . . 3 β’ β β π΄ | |
2 | eqid 2724 | . . . 4 β’ (ocβπΎ) = (ocβπΎ) | |
3 | polssat.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
4 | eqid 2724 | . . . 4 β’ (pmapβπΎ) = (pmapβπΎ) | |
5 | polssat.p | . . . 4 β’ β₯ = (β₯πβπΎ) | |
6 | 2, 3, 4, 5 | polvalN 39270 | . . 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 688 | . 2 β’ (πΎ β π΅ β ( β₯ ββ ) = (π΄ β© β© π β β ((pmapβπΎ)β((ocβπΎ)βπ)))) |
8 | 0iin 5058 | . . . 4 β’ β© π β β ((pmapβπΎ)β((ocβπΎ)βπ)) = V | |
9 | 8 | ineq2i 4202 | . . 3 β’ (π΄ β© β© π β β ((pmapβπΎ)β((ocβπΎ)βπ))) = (π΄ β© V) |
10 | inv1 4387 | . . 3 β’ (π΄ β© V) = π΄ | |
11 | 9, 10 | eqtri 2752 | . 2 β’ (π΄ β© β© π β β ((pmapβπΎ)β((ocβπΎ)βπ))) = π΄ |
12 | 7, 11 | eqtrdi 2780 | 1 β’ (πΎ β π΅ β ( β₯ ββ ) = π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3466 β© cin 3940 β wss 3941 β c0 4315 β© ciin 4989 βcfv 6534 occoc 17206 Atomscatm 38627 pmapcpmap 38862 β₯πcpolN 39267 |
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-iun 4990 df-iin 4991 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-polarityN 39268 |
This theorem is referenced by: 2pol0N 39276 1psubclN 39309 osumcllem9N 39329 pexmidN 39334 pexmidlem6N 39340 pexmidALTN 39343 |
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