mpaalem - Mathbox for Stefan O'Rear

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Theorem mpaalem 43610

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 43609	. . 3 ⊢ (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) = (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (deg_AA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(deg_AA‘𝐴)) = 1)))
2		mpaaeu 43608	. . . 4 ⊢ (𝐴 ∈ 𝔸 → ∃!𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (deg_AA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(deg_AA‘𝐴)) = 1))
3		riotacl2 7332	. . . 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 2841	. 2 ⊢ (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) ∈ {𝑝 ∈ (Poly‘ℚ) ∣ ((deg‘𝑝) = (deg_AA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(deg_AA‘𝐴)) = 1)})
6		fveqeq2 6839	. . . 4 ⊢ (𝑝 = (minPolyAA‘𝐴) → ((deg‘𝑝) = (deg_AA‘𝐴) ↔ (deg‘(minPolyAA‘𝐴)) = (deg_AA‘𝐴)))
7		fveq1 6829	. . . . 5 ⊢ (𝑝 = (minPolyAA‘𝐴) → (𝑝‘𝐴) = ((minPolyAA‘𝐴)‘𝐴))
8	7	eqeq1d 2743	. . . 4 ⊢ (𝑝 = (minPolyAA‘𝐴) → ((𝑝‘𝐴) = 0 ↔ ((minPolyAA‘𝐴)‘𝐴) = 0))
9		fveq2 6830	. . . . . 6 ⊢ (𝑝 = (minPolyAA‘𝐴) → (coeff‘𝑝) = (coeff‘(minPolyAA‘𝐴)))
10	9	fveq1d 6832	. . . . 5 ⊢ (𝑝 = (minPolyAA‘𝐴) → ((coeff‘𝑝)‘(deg_AA‘𝐴)) = ((coeff‘(minPolyAA‘𝐴))‘(deg_AA‘𝐴)))
11	10	eqeq1d 2743	. . . 4 ⊢ (𝑝 = (minPolyAA‘𝐴) → (((coeff‘𝑝)‘(deg_AA‘𝐴)) = 1 ↔ ((coeff‘(minPolyAA‘𝐴))‘(deg_AA‘𝐴)) = 1))
12	6, 8, 11	3anbi123d 1445	. . 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 3630	. 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 220	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 397 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 ∃!wreu 3344 {crab 3393 ‘cfv 6488 ℩crio 7315 0cc0 11034 1c1 11035 ℚcq 12893 Polycply 26170 coeffccoe 26172 degcdgr 26173 𝔸caa 26301 deg_AAcdgraa 43598 minPolyAAcmpaa 43599

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7681 ax-inf2 9557 ax-cnex 11090 ax-resscn 11091 ax-1cn 11092 ax-icn 11093 ax-addcl 11094 ax-addrcl 11095 ax-mulcl 11096 ax-mulrcl 11097 ax-mulcom 11098 ax-addass 11099 ax-mulass 11100 ax-distr 11101 ax-i2m1 11102 ax-1ne0 11103 ax-1rid 11104 ax-rnegex 11105 ax-rrecex 11106 ax-cnre 11107 ax-pre-lttri 11108 ax-pre-lttrn 11109 ax-pre-ltadd 11110 ax-pre-mulgt0 11111 ax-pre-sup 11112

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3725 df-csb 3833 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-pss 3904 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-isom 6497 df-riota 7316 df-ov 7362 df-oprab 7363 df-mpo 7364 df-of 7623 df-om 7810 df-1st 7933 df-2nd 7934 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9858 df-pnf 11177 df-mnf 11178 df-xr 11179 df-ltxr 11180 df-le 11181 df-sub 11375 df-neg 11376 df-div 11804 df-nn 12170 df-2 12239 df-3 12240 df-n0 12433 df-z 12520 df-uz 12784 df-q 12894 df-rp 12938 df-fz 13457 df-fzo 13604 df-fl 13746 df-mod 13824 df-seq 13959 df-exp 14019 df-hash 14288 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-clim 15445 df-rlim 15446 df-sum 15644 df-0p 25658 df-ply 26174 df-coe 26176 df-dgr 26177 df-aa 26302 df-dgraa 43600 df-mpaa 43601

This theorem is referenced by: mpaacl 43611 mpaadgr 43612 mpaaroot 43613 mpaamn 43614

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