![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > polvalN | Structured version Visualization version GIF version |
Description: Value of the projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
polfval.o | β’ β₯ = (ocβπΎ) |
polfval.a | β’ π΄ = (AtomsβπΎ) |
polfval.m | β’ π = (pmapβπΎ) |
polfval.p | β’ π = (β₯πβπΎ) |
Ref | Expression |
---|---|
polvalN | β’ ((πΎ β π΅ β§ π β π΄) β (πβπ) = (π΄ β© β© π β π (πβ( β₯ βπ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | polfval.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
2 | 1 | fvexi 6904 | . . 3 β’ π΄ β V |
3 | 2 | elpw2 5344 | . 2 β’ (π β π« π΄ β π β π΄) |
4 | polfval.o | . . . . 5 β’ β₯ = (ocβπΎ) | |
5 | polfval.m | . . . . 5 β’ π = (pmapβπΎ) | |
6 | polfval.p | . . . . 5 β’ π = (β₯πβπΎ) | |
7 | 4, 1, 5, 6 | polfvalN 39078 | . . . 4 β’ (πΎ β π΅ β π = (π β π« π΄ β¦ (π΄ β© β© π β π (πβ( β₯ βπ))))) |
8 | 7 | fveq1d 6892 | . . 3 β’ (πΎ β π΅ β (πβπ) = ((π β π« π΄ β¦ (π΄ β© β© π β π (πβ( β₯ βπ))))βπ)) |
9 | iineq1 5013 | . . . . 5 β’ (π = π β β© π β π (πβ( β₯ βπ)) = β© π β π (πβ( β₯ βπ))) | |
10 | 9 | ineq2d 4211 | . . . 4 β’ (π = π β (π΄ β© β© π β π (πβ( β₯ βπ))) = (π΄ β© β© π β π (πβ( β₯ βπ)))) |
11 | eqid 2730 | . . . 4 β’ (π β π« π΄ β¦ (π΄ β© β© π β π (πβ( β₯ βπ)))) = (π β π« π΄ β¦ (π΄ β© β© π β π (πβ( β₯ βπ)))) | |
12 | 2 | inex1 5316 | . . . 4 β’ (π΄ β© β© π β π (πβ( β₯ βπ))) β V |
13 | 10, 11, 12 | fvmpt 6997 | . . 3 β’ (π β π« π΄ β ((π β π« π΄ β¦ (π΄ β© β© π β π (πβ( β₯ βπ))))βπ) = (π΄ β© β© π β π (πβ( β₯ βπ)))) |
14 | 8, 13 | sylan9eq 2790 | . 2 β’ ((πΎ β π΅ β§ π β π« π΄) β (πβπ) = (π΄ β© β© π β π (πβ( β₯ βπ)))) |
15 | 3, 14 | sylan2br 593 | 1 β’ ((πΎ β π΅ β§ π β π΄) β (πβπ) = (π΄ β© β© π β π (πβ( β₯ βπ)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1539 β wcel 2104 β© cin 3946 β wss 3947 π« cpw 4601 β© ciin 4997 β¦ cmpt 5230 βcfv 6542 occoc 17209 Atomscatm 38436 pmapcpmap 38671 β₯πcpolN 39076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 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 39077 |
This theorem is referenced by: polval2N 39080 pol0N 39083 polcon3N 39091 polatN 39105 |
Copyright terms: Public domain | W3C validator |