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Mathbox for Norm Megill |
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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 6857 | . . 3 β’ π΄ β V |
3 | 2 | elpw2 5303 | . 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 38370 | . . . 4 β’ (πΎ β π΅ β π = (π β π« π΄ β¦ (π΄ β© β© π β π (πβ( β₯ βπ))))) |
8 | 7 | fveq1d 6845 | . . 3 β’ (πΎ β π΅ β (πβπ) = ((π β π« π΄ β¦ (π΄ β© β© π β π (πβ( β₯ βπ))))βπ)) |
9 | iineq1 4972 | . . . . 5 β’ (π = π β β© π β π (πβ( β₯ βπ)) = β© π β π (πβ( β₯ βπ))) | |
10 | 9 | ineq2d 4173 | . . . 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 2737 | . . . 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 5275 | . . . 4 β’ (π΄ β© β© π β π (πβ( β₯ βπ))) β V |
13 | 10, 11, 12 | fvmpt 6949 | . . 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 2797 | . 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 596 | 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 397 = wceq 1542 β wcel 2107 β© cin 3910 β wss 3911 π« cpw 4561 β© ciin 4956 β¦ cmpt 5189 βcfv 6497 occoc 17142 Atomscatm 37728 pmapcpmap 37963 β₯πcpolN 38368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-polarityN 38369 |
This theorem is referenced by: polval2N 38372 pol0N 38375 polcon3N 38383 polatN 38397 |
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