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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 6902 | . . . . 5 β’ (π = π΄ β (degAAβπ) = (degAAβπ΄)) | |
2 | 1 | eqeq2d 2739 | . . . 4 β’ (π = π΄ β ((degβπ) = (degAAβπ) β (degβπ) = (degAAβπ΄))) |
3 | fveqeq2 6911 | . . . 4 β’ (π = π΄ β ((πβπ) = 0 β (πβπ΄) = 0)) | |
4 | 2fveq3 6907 | . . . . 5 β’ (π = π΄ β ((coeffβπ)β(degAAβπ)) = ((coeffβπ)β(degAAβπ΄))) | |
5 | 4 | eqeq1d 2730 | . . . 4 β’ (π = π΄ β (((coeffβπ)β(degAAβπ)) = 1 β ((coeffβπ)β(degAAβπ΄)) = 1)) |
6 | 2, 3, 5 | 3anbi123d 1432 | . . 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 7384 | . 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 42598 | . 2 β’ minPolyAA = (π β πΈ β¦ (β©π β (Polyββ)((degβπ) = (degAAβπ) β§ (πβπ) = 0 β§ ((coeffβπ)β(degAAβπ)) = 1))) | |
9 | riotaex 7386 | . 2 β’ (β©π β (Polyββ)((degβπ) = (degAAβπ΄) β§ (πβπ΄) = 0 β§ ((coeffβπ)β(degAAβπ΄)) = 1)) β V | |
10 | 7, 8, 9 | fvmpt 7010 | 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 1084 = wceq 1533 β wcel 2098 βcfv 6553 β©crio 7381 0cc0 11146 1c1 11147 βcq 12970 Polycply 26138 coeffccoe 26140 degcdgr 26141 πΈcaa 26269 degAAcdgraa 42595 minPolyAAcmpaa 42596 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-riota 7382 df-mpaa 42598 |
This theorem is referenced by: mpaalem 42607 |
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