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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 | ⊢ (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) = (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6672 | . . . . 5 ⊢ (𝑎 = 𝐴 → (degAA‘𝑎) = (degAA‘𝐴)) | |
2 | 1 | eqeq2d 2834 | . . . 4 ⊢ (𝑎 = 𝐴 → ((deg‘𝑝) = (degAA‘𝑎) ↔ (deg‘𝑝) = (degAA‘𝐴))) |
3 | fveqeq2 6681 | . . . 4 ⊢ (𝑎 = 𝐴 → ((𝑝‘𝑎) = 0 ↔ (𝑝‘𝐴) = 0)) | |
4 | 2fveq3 6677 | . . . . 5 ⊢ (𝑎 = 𝐴 → ((coeff‘𝑝)‘(degAA‘𝑎)) = ((coeff‘𝑝)‘(degAA‘𝐴))) | |
5 | 4 | eqeq1d 2825 | . . . 4 ⊢ (𝑎 = 𝐴 → (((coeff‘𝑝)‘(degAA‘𝑎)) = 1 ↔ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1)) |
6 | 2, 3, 5 | 3anbi123d 1432 | . . 3 ⊢ (𝑎 = 𝐴 → (((deg‘𝑝) = (degAA‘𝑎) ∧ (𝑝‘𝑎) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝑎)) = 1) ↔ ((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1))) |
7 | 6 | riotabidv 7118 | . 2 ⊢ (𝑎 = 𝐴 → (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝑎) ∧ (𝑝‘𝑎) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝑎)) = 1)) = (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1))) |
8 | df-mpaa 39750 | . 2 ⊢ minPolyAA = (𝑎 ∈ 𝔸 ↦ (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝑎) ∧ (𝑝‘𝑎) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝑎)) = 1))) | |
9 | riotaex 7120 | . 2 ⊢ (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1)) ∈ V | |
10 | 7, 8, 9 | fvmpt 6770 | 1 ⊢ (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) = (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (degAA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(degAA‘𝐴)) = 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 ℩crio 7115 0cc0 10539 1c1 10540 ℚcq 12351 Polycply 24776 coeffccoe 24778 degcdgr 24779 𝔸caa 24905 degAAcdgraa 39747 minPolyAAcmpaa 39748 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-riota 7116 df-mpaa 39750 |
This theorem is referenced by: mpaalem 39759 |
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