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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mpaaval | Structured version Visualization version GIF version |
Description: Value of the minimal polynomial of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.) |
Ref | Expression |
---|---|
mpaaval | β’ (π΄ β πΈ β (minPolyAAβπ΄) = (β©π β (Polyββ)((degβπ) = (degAAβπ΄) β§ (πβπ΄) = 0 β§ ((coeffβπ)β(degAAβπ΄)) = 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6843 | . . . . 5 β’ (π = π΄ β (degAAβπ) = (degAAβπ΄)) | |
2 | 1 | eqeq2d 2744 | . . . 4 β’ (π = π΄ β ((degβπ) = (degAAβπ) β (degβπ) = (degAAβπ΄))) |
3 | fveqeq2 6852 | . . . 4 β’ (π = π΄ β ((πβπ) = 0 β (πβπ΄) = 0)) | |
4 | 2fveq3 6848 | . . . . 5 β’ (π = π΄ β ((coeffβπ)β(degAAβπ)) = ((coeffβπ)β(degAAβπ΄))) | |
5 | 4 | eqeq1d 2735 | . . . 4 β’ (π = π΄ β (((coeffβπ)β(degAAβπ)) = 1 β ((coeffβπ)β(degAAβπ΄)) = 1)) |
6 | 2, 3, 5 | 3anbi123d 1437 | . . 3 β’ (π = π΄ β (((degβπ) = (degAAβπ) β§ (πβπ) = 0 β§ ((coeffβπ)β(degAAβπ)) = 1) β ((degβπ) = (degAAβπ΄) β§ (πβπ΄) = 0 β§ ((coeffβπ)β(degAAβπ΄)) = 1))) |
7 | 6 | riotabidv 7316 | . 2 β’ (π = π΄ β (β©π β (Polyββ)((degβπ) = (degAAβπ) β§ (πβπ) = 0 β§ ((coeffβπ)β(degAAβπ)) = 1)) = (β©π β (Polyββ)((degβπ) = (degAAβπ΄) β§ (πβπ΄) = 0 β§ ((coeffβπ)β(degAAβπ΄)) = 1))) |
8 | df-mpaa 41513 | . 2 β’ minPolyAA = (π β πΈ β¦ (β©π β (Polyββ)((degβπ) = (degAAβπ) β§ (πβπ) = 0 β§ ((coeffβπ)β(degAAβπ)) = 1))) | |
9 | riotaex 7318 | . 2 β’ (β©π β (Polyββ)((degβπ) = (degAAβπ΄) β§ (πβπ΄) = 0 β§ ((coeffβπ)β(degAAβπ΄)) = 1)) β V | |
10 | 7, 8, 9 | fvmpt 6949 | 1 β’ (π΄ β πΈ β (minPolyAAβπ΄) = (β©π β (Polyββ)((degβπ) = (degAAβπ΄) β§ (πβπ΄) = 0 β§ ((coeffβπ)β(degAAβπ΄)) = 1))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 βcfv 6497 β©crio 7313 0cc0 11056 1c1 11057 βcq 12878 Polycply 25561 coeffccoe 25563 degcdgr 25564 πΈcaa 25690 degAAcdgraa 41510 minPolyAAcmpaa 41511 |
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 2704 ax-sep 5257 ax-nul 5264 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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 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-iota 6449 df-fun 6499 df-fv 6505 df-riota 7314 df-mpaa 41513 |
This theorem is referenced by: mpaalem 41522 |
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