Theorem minplyval 33713

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 33712, 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 3502	. . . 4 ⊢ (𝜑 → 𝐸 ∈ V)
4		ply1annig1p.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
5	4	elexd 3502	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
6		ply1annig1p.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐸)
7	6	fvexi 6921	. . . . . 6 ⊢ 𝐵 ∈ V
8	7	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
9	8	mptexd 7244	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 })) ∈ V)
10		fveq2 6907	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → (Base‘𝑒) = (Base‘𝐸))
11	10, 6	eqtr4di 2793	. . . . . . 7 ⊢ (𝑒 = 𝐸 → (Base‘𝑒) = 𝐵)
12	11	adantr 480	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (Base‘𝑒) = 𝐵)
13		oveq12 7440	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 ↾s 𝑓) = (𝐸 ↾s 𝐹))
14	13	fveq2d 6911	. . . . . . . 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 2793	. . . . . . 7 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (idlGen1p‘(𝑒 ↾s 𝑓)) = 𝐺)
17		oveq12 7440	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 evalSub1 𝑓) = (𝐸 evalSub1 𝐹))
18		ply1annig1p.o	. . . . . . . . . 10 ⊢ 𝑂 = (𝐸 evalSub1 𝐹)
19	17, 18	eqtr4di 2793	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 evalSub1 𝑓) = 𝑂)
20	19	dmeqd 5919	. . . . . . . 8 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → dom (𝑒 evalSub1 𝑓) = dom 𝑂)
21	19	fveq1d 6909	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒 evalSub1 𝑓)‘𝑞) = (𝑂‘𝑞))
22	21	fveq1d 6909	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = ((𝑂‘𝑞)‘𝑥))
23		fveq2 6907	. . . . . . . . . . 11 ⊢ (𝑒 = 𝐸 → (0g‘𝑒) = (0g‘𝐸))
24	23	adantr 480	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (0g‘𝑒) = (0g‘𝐸))
25		ply1annig1p.0	. . . . . . . . . 10 ⊢ 0 = (0g‘𝐸)
26	24, 25	eqtr4di 2793	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (0g‘𝑒) = 0 )
27	22, 26	eqeq12d 2751	. . . . . . . 8 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = (0g‘𝑒) ↔ ((𝑂‘𝑞)‘𝑥) = 0 ))
28	20, 27	rabeqbidv 3452	. . . . . . 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 6914	. . . . . 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 5239	. . . . 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 33708	. . . . 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 7587	. . . 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 1370	. . 3 ⊢ (𝜑 → (𝐸 minPoly 𝐹) = (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 })))
34	1, 33	eqtrid 2787	. 2 ⊢ (𝜑 → 𝑀 = (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 })))
35		fveqeq2 6916	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑂‘𝑞)‘𝑥) = 0 ↔ ((𝑂‘𝑞)‘𝐴) = 0 ))
36	35	rabbidv 3441	. . . . 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 2793	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 } = 𝑄)
39	38	fveq2d 6911	. . 3 ⊢ (𝑥 = 𝐴 → (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 }) = (𝐺‘𝑄))
40	39	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 }) = (𝐺‘𝑄))
41		ply1annig1p.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
42		fvexd 6922	. 2 ⊢ (𝜑 → (𝐺‘𝑄) ∈ V)
43	34, 40, 41, 42	fvmptd 7023	1 ⊢ (𝜑 → (𝑀‘𝐴) = (𝐺‘𝑄))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {crab 3433  Vcvv 3478   ↦ cmpt 5231  dom cdm 5689  ‘cfv 6563  (class class class)co 7431  Basecbs 17245   ↾_s cress 17274  0_gc0g 17486  Fieldcfield 20747  SubDRingcsdrg 20804  RSpancrsp 21235  Poly₁cpl1 22194   evalSub₁ ces1 22333  idlGen_1pcig1p 26184   minPoly cminply 33707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-minply 33708

This theorem is referenced by:  minplycl  33714  minplymindeg  33716  minplyann  33717  minplyirredlem  33718  minplyirred  33719  irngnminplynz  33720  minplym1p  33721  irredminply  33722  algextdeglem4  33726  algextdeglem5  33727