mpaalem - Mathbox for Stefan O'Rear

	Mathbox for Stefan O'Rear		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > mpaalem		Structured version Visualization version GIF version

Theorem mpaalem 42209

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 42208	. . 3 ⊢ (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) = (℩𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (deg_AA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(deg_AA‘𝐴)) = 1)))
2		mpaaeu 42207	. . . 4 ⊢ (𝐴 ∈ 𝔸 → ∃!𝑝 ∈ (Poly‘ℚ)((deg‘𝑝) = (deg_AA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(deg_AA‘𝐴)) = 1))
3		riotacl2 7385	. . . 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 2832	. 2 ⊢ (𝐴 ∈ 𝔸 → (minPolyAA‘𝐴) ∈ {𝑝 ∈ (Poly‘ℚ) ∣ ((deg‘𝑝) = (deg_AA‘𝐴) ∧ (𝑝‘𝐴) = 0 ∧ ((coeff‘𝑝)‘(deg_AA‘𝐴)) = 1)})
6		fveqeq2 6900	. . . 4 ⊢ (𝑝 = (minPolyAA‘𝐴) → ((deg‘𝑝) = (deg_AA‘𝐴) ↔ (deg‘(minPolyAA‘𝐴)) = (deg_AA‘𝐴)))
7		fveq1 6890	. . . . 5 ⊢ (𝑝 = (minPolyAA‘𝐴) → (𝑝‘𝐴) = ((minPolyAA‘𝐴)‘𝐴))
8	7	eqeq1d 2733	. . . 4 ⊢ (𝑝 = (minPolyAA‘𝐴) → ((𝑝‘𝐴) = 0 ↔ ((minPolyAA‘𝐴)‘𝐴) = 0))
9		fveq2 6891	. . . . . 6 ⊢ (𝑝 = (minPolyAA‘𝐴) → (coeff‘𝑝) = (coeff‘(minPolyAA‘𝐴)))
10	9	fveq1d 6893	. . . . 5 ⊢ (𝑝 = (minPolyAA‘𝐴) → ((coeff‘𝑝)‘(deg_AA‘𝐴)) = ((coeff‘(minPolyAA‘𝐴))‘(deg_AA‘𝐴)))
11	10	eqeq1d 2733	. . . 4 ⊢ (𝑝 = (minPolyAA‘𝐴) → (((coeff‘𝑝)‘(deg_AA‘𝐴)) = 1 ↔ ((coeff‘(minPolyAA‘𝐴))‘(deg_AA‘𝐴)) = 1))
12	6, 8, 11	3anbi123d 1435	. . 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 3683	. 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 1086 = wceq 1540 ∈ wcel 2105 ∃!wreu 3373 {crab 3431 ‘cfv 6543 ℩crio 7367 0cc0 11116 1c1 11117 ℚcq 12939 Polycply 25947 coeffccoe 25949 degcdgr 25950 𝔸caa 26077 deg_AAcdgraa 42197 minPolyAAcmpaa 42198

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-map 8828 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-q 12940 df-rp 12982 df-fz 13492 df-fzo 13635 df-fl 13764 df-mod 13842 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-rlim 15440 df-sum 15640 df-0p 25432 df-ply 25951 df-coe 25953 df-dgr 25954 df-aa 26078 df-dgraa 42199 df-mpaa 42200

This theorem is referenced by: mpaacl 42210 mpaadgr 42211 mpaaroot 42212 mpaamn 42213

W3C validator