Theorem minplyval 33963

Description: Expand the value of the minimal polynomial (𝑀‘𝐴) for a given element 𝐴. It is defined as the unique monic polynomial of minimal degree which annihilates 𝐴. By ply1annig1p 33962, that polynomial generates the ideal of the annihilators of 𝐴. (Contributed by Thierry Arnoux, 9-Feb-2025.)

Hypotheses

Ref	Expression
ply1annig1p.o	⊢ 𝑂 = (𝐸 evalSub1 𝐹)
ply1annig1p.p	⊢ 𝑃 = (Poly1‘(𝐸 ↾s 𝐹))
ply1annig1p.b	⊢ 𝐵 = (Base‘𝐸)
ply1annig1p.e	⊢ (𝜑 → 𝐸 ∈ Field)
ply1annig1p.f	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
ply1annig1p.a	⊢ (𝜑 → 𝐴 ∈ 𝐵)
ply1annig1p.0	⊢ 0 = (0g‘𝐸)
ply1annig1p.q	⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }
ply1annig1p.k	⊢ 𝐾 = (RSpan‘𝑃)
ply1annig1p.g	⊢ 𝐺 = (idlGen1p‘(𝐸 ↾s 𝐹))
minplyval.1	⊢ 𝑀 = (𝐸 minPoly 𝐹)

Assertion

Ref	Expression
minplyval	⊢ (𝜑 → (𝑀‘𝐴) = (𝐺‘𝑄))

Distinct variable groups:   0 ,𝑞   𝐴,𝑞   𝑂,𝑞   𝑃,𝑞   𝜑,𝑞   𝐸,𝑞   𝐹,𝑞

Allowed substitution hints:   𝐵(𝑞)   𝑄(𝑞)   𝐺(𝑞)   𝐾(𝑞)   𝑀(𝑞)

Proof of Theorem minplyval

Dummy variables 𝑒 𝑓 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		minplyval.1	. . 3 ⊢ 𝑀 = (𝐸 minPoly 𝐹)
2		ply1annig1p.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Field)
3	2	elexd 3476	. . . 4 ⊢ (𝜑 → 𝐸 ∈ V)
4		ply1annig1p.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
5	4	elexd 3476	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
6		ply1annig1p.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐸)
7	6	fvexi 6876	. . . . . 6 ⊢ 𝐵 ∈ V
8	7	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
9	8	mptexd 7203	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 })) ∈ V)
10		fveq2 6862	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → (Base‘𝑒) = (Base‘𝐸))
11	10, 6	eqtr4di 2814	. . . . . . 7 ⊢ (𝑒 = 𝐸 → (Base‘𝑒) = 𝐵)
12	11	adantr 484	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (Base‘𝑒) = 𝐵)
13		oveq12 7400	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 ↾s 𝑓) = (𝐸 ↾s 𝐹))
14	13	fveq2d 6866	. . . . . . . 8 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (idlGen1p‘(𝑒 ↾s 𝑓)) = (idlGen1p‘(𝐸 ↾s 𝐹)))
15		ply1annig1p.g	. . . . . . . 8 ⊢ 𝐺 = (idlGen1p‘(𝐸 ↾s 𝐹))
16	14, 15	eqtr4di 2814	. . . . . . 7 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (idlGen1p‘(𝑒 ↾s 𝑓)) = 𝐺)
17		oveq12 7400	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 evalSub1 𝑓) = (𝐸 evalSub1 𝐹))
18		ply1annig1p.o	. . . . . . . . . 10 ⊢ 𝑂 = (𝐸 evalSub1 𝐹)
19	17, 18	eqtr4di 2814	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 evalSub1 𝑓) = 𝑂)
20	19	dmeqd 5877	. . . . . . . 8 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → dom (𝑒 evalSub1 𝑓) = dom 𝑂)
21	19	fveq1d 6864	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒 evalSub1 𝑓)‘𝑞) = (𝑂‘𝑞))
22	21	fveq1d 6864	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = ((𝑂‘𝑞)‘𝑥))
23		fveq2 6862	. . . . . . . . . . 11 ⊢ (𝑒 = 𝐸 → (0g‘𝑒) = (0g‘𝐸))
24	23	adantr 484	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (0g‘𝑒) = (0g‘𝐸))
25		ply1annig1p.0	. . . . . . . . . 10 ⊢ 0 = (0g‘𝐸)
26	24, 25	eqtr4di 2814	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (0g‘𝑒) = 0 )
27	22, 26	eqeq12d 2777	. . . . . . . 8 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = (0g‘𝑒) ↔ ((𝑂‘𝑞)‘𝑥) = 0 ))
28	20, 27	rabeqbidv 3431	. . . . . . 7 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → {𝑞 ∈ dom (𝑒 evalSub1 𝑓) ∣ (((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = (0g‘𝑒)} = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 })
29	16, 28	fveq12d 6869	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((idlGen1p‘(𝑒 ↾s 𝑓))‘{𝑞 ∈ dom (𝑒 evalSub1 𝑓) ∣ (((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = (0g‘𝑒)}) = (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 }))
30	12, 29	mpteq12dv 5184	. . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑥 ∈ (Base‘𝑒) ↦ ((idlGen1p‘(𝑒 ↾s 𝑓))‘{𝑞 ∈ dom (𝑒 evalSub1 𝑓) ∣ (((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = (0g‘𝑒)})) = (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 })))
31		df-minply 33958	. . . . 5 ⊢ minPoly = (𝑒 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (Base‘𝑒) ↦ ((idlGen1p‘(𝑒 ↾s 𝑓))‘{𝑞 ∈ dom (𝑒 evalSub1 𝑓) ∣ (((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = (0g‘𝑒)})))
32	30, 31	ovmpoga 7545	. . . 4 ⊢ ((𝐸 ∈ V ∧ 𝐹 ∈ V ∧ (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 })) ∈ V) → (𝐸 minPoly 𝐹) = (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 })))
33	3, 5, 9, 32	syl3anc 1389	. . 3 ⊢ (𝜑 → (𝐸 minPoly 𝐹) = (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 })))
34	1, 33	eqtrid 2808	. 2 ⊢ (𝜑 → 𝑀 = (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 })))
35		fveqeq2 6871	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑂‘𝑞)‘𝑥) = 0 ↔ ((𝑂‘𝑞)‘𝐴) = 0 ))
36	35	rabbidv 3420	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 } = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 })
37		ply1annig1p.q	. . . . 5 ⊢ 𝑄 = {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝐴) = 0 }
38	36, 37	eqtr4di 2814	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 } = 𝑄)
39	38	fveq2d 6866	. . 3 ⊢ (𝑥 = 𝐴 → (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 }) = (𝐺‘𝑄))
40	39	adantl 485	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 }) = (𝐺‘𝑄))
41		ply1annig1p.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
42		fvexd 6877	. 2 ⊢ (𝜑 → (𝐺‘𝑄) ∈ V)
43	34, 40, 41, 42	fvmptd 6978	1 ⊢ (𝜑 → (𝑀‘𝐴) = (𝐺‘𝑄))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  {crab 3413  Vcvv 3453   ↦ cmpt 5178  dom cdm 5643  ‘cfv 6516  (class class class)co 7391  Basecbs 17236   ↾_s cress 17257  0_gc0g 17459  Fieldcfield 20767  SubDRingcsdrg 20823  RSpancrsp 21265  Poly₁cpl1 22227   evalSub₁ ces1 22364  idlGen_1pcig1p 26178   minPoly cminply 33957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-minply 33958

This theorem is referenced by:  minplycl  33964  minplymindeg  33966  minplyann  33967  minplyirredlem  33968  minplyirred  33969  irngnminplynz  33970  minplym1p  33971  minplynzm1p  33972  irredminply  33974  algextdeglem4  33978  algextdeglem5  33979