mpaalem - Mathbox for Stefan O'Rear

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Theorem mpaalem 42454

Description: Properties of the minimal polynomial of an algebraic number. (Contributed by Stefan O'Rear, 25-Nov-2014.)

**Assertion**
Ref	Expression
mpaalem	⊢ (𝐴 ∈ 𝔸 → ((minPolyAA‘𝐴) ∈ (Poly‘ℚ) ∧ ((deg‘(minPolyAA‘𝐴)) = (deg_AA‘𝐴) ∧ ((minPolyAA‘𝐴)‘𝐴) = 0 ∧ ((coeff‘(minPolyAA‘𝐴))‘(deg_AA‘𝐴)) = 1)))

Proof of Theorem mpaalem Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpaaval 42453	. . 3 ⊢ (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) = (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (deg_AA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(deg_AA‘𝐴)) = 1)))
2		mpaaeu 42452	. . . 4 ⊢ (𝐴 ∈ 𝔸 → ∃!𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (deg_AA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(deg_AA‘𝐴)) = 1))
3		riotacl2 7377	. . . 4 ⊢ (∃!𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (deg_AA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(deg_AA‘𝐴)) = 1) → (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (deg_AA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(deg_AA‘𝐴)) = 1)) ∈ {𝑝 ∈ (Poly‘ℚ) ∣ ((deg‘𝑝) = (deg_AA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(deg_AA‘𝐴)) = 1)})
4	2, 3	syl 17	. . 3 ⊢ (𝐴 ∈ 𝔸 → (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (deg_AA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(deg_AA‘𝐴)) = 1)) ∈ {𝑝 ∈ (Poly‘ℚ) ∣ ((deg‘𝑝) = (deg_AA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(deg_AA‘𝐴)) = 1)})
5	1, 4	eqeltrd 2827	. 2 ⊢ (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) ∈ {𝑝 ∈ (Poly‘ℚ) ∣ ((deg‘𝑝) = (deg_AA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(deg_AA‘𝐴)) = 1)})
6		fveqeq2 6893	. . . 4 ⊢ (𝑝 = (minPolyAA‘𝐴) → ((deg‘𝑝) = (deg_AA‘𝐴) ↔ (deg‘(minPolyAA‘𝐴)) = (deg_AA‘𝐴)))
7		fveq1 6883	. . . . 5 ⊢ (𝑝 = (minPolyAA‘𝐴) → (𝑝‘𝐴) = ((minPolyAA‘𝐴)‘𝐴))
8	7	eqeq1d 2728	. . . 4 ⊢ (𝑝 = (minPolyAA‘𝐴) → ((𝑝‘𝐴) = 0 ↔ ((minPolyAA‘𝐴)‘𝐴) = 0))
9		fveq2 6884	. . . . . 6 ⊢ (𝑝 = (minPolyAA‘𝐴) → (coeff‘𝑝) = (coeff‘(minPolyAA‘𝐴)))
10	9	fveq1d 6886	. . . . 5 ⊢ (𝑝 = (minPolyAA‘𝐴) → ((coeff‘𝑝)‘(deg_AA‘𝐴)) = ((coeff‘(minPolyAA‘𝐴))‘(deg_AA‘𝐴)))
11	10	eqeq1d 2728	. . . 4 ⊢ (𝑝 = (minPolyAA‘𝐴) → (((coeff‘𝑝)‘(deg_AA‘𝐴)) = 1 ↔ ((coeff‘(minPolyAA‘𝐴))‘(deg_AA‘𝐴)) = 1))
12	6, 8, 11	3anbi123d 1432	. . 3 ⊢ (𝑝 = (minPolyAA‘𝐴) → (((deg‘𝑝) = (deg_AA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(deg_AA‘𝐴)) = 1) ↔ ((deg‘(minPolyAA‘𝐴)) = (deg_AA‘𝐴) ∧ ((minPolyAA‘𝐴)‘𝐴) = 0 ∧ ((coeff‘(minPolyAA‘𝐴))‘(deg_AA‘𝐴)) = 1)))
13	12	elrab 3678	. 2 ⊢ ((minPolyAA‘𝐴) ∈ {𝑝 ∈ (Poly‘ℚ) ∣ ((deg‘𝑝) = (deg_AA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(deg_AA‘𝐴)) = 1)} ↔ ((minPolyAA‘𝐴) ∈ (Poly‘ℚ) ∧ ((deg‘(minPolyAA‘𝐴)) = (deg_AA‘𝐴) ∧ ((minPolyAA‘𝐴)‘𝐴) = 0 ∧ ((coeff‘(minPolyAA‘𝐴))‘(deg_AA‘𝐴)) = 1)))
14	5, 13	sylib 217	1 ⊢ (𝐴 ∈ 𝔸 → ((minPolyAA‘𝐴) ∈ (Poly‘ℚ) ∧ ((deg‘(minPolyAA‘𝐴)) = (deg_AA‘𝐴) ∧ ((minPolyAA‘𝐴)‘𝐴) = 0 ∧ ((coeff‘(minPolyAA‘𝐴))‘(deg_AA‘𝐴)) = 1)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∃!wreu 3368 {crab 3426 ‘cfv 6536 ℩crio 7359 0cc0 11109 1c1 11110 ℚcq 12933 Polycply 26068 coeffccoe 26070 degcdgr 26071 𝔸caa 26199 deg_AAcdgraa 42442 minPolyAAcmpaa 42443

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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-q 12934 df-rp 12978 df-fz 13488 df-fzo 13631 df-fl 13760 df-mod 13838 df-seq 13970 df-exp 14030 df-hash 14293 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-clim 15435 df-rlim 15436 df-sum 15636 df-0p 25549 df-ply 26072 df-coe 26074 df-dgr 26075 df-aa 26200 df-dgraa 42444 df-mpaa 42445

This theorem is referenced by: mpaacl 42455 mpaadgr 42456 mpaaroot 42457 mpaamn 42458

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