Theorem minplyval 34387

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 34386, 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 3473	. . . 4 ⊢ (𝜑 → 𝐸 ∈ V)
4		ply1annig1p.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
5	4	elexd 3473	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
6		ply1annig1p.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐸)
7	6	fvexi 6904	. . . . . 6 ⊢ 𝐵 ∈ V
8	7	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
9	8	mptexd 7234	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 })) ∈ V)
10		fveq2 6890	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → (Base‘𝑒) = (Base‘𝐸))
11	10, 6	eqtr4di 2813	. . . . . . 7 ⊢ (𝑒 = 𝐸 → (Base‘𝑒) = 𝐵)
12	11	adantr 486	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (Base‘𝑒) = 𝐵)
13		oveq12 7433	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 ↾s 𝑓) = (𝐸 ↾s 𝐹))
14	13	fveq2d 6894	. . . . . . . 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 2813	. . . . . . 7 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (idlGen1p‘(𝑒 ↾s 𝑓)) = 𝐺)
17		oveq12 7433	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 evalSub1 𝑓) = (𝐸 evalSub1 𝐹))
18		ply1annig1p.o	. . . . . . . . . 10 ⊢ 𝑂 = (𝐸 evalSub1 𝐹)
19	17, 18	eqtr4di 2813	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 evalSub1 𝑓) = 𝑂)
20	19	dmeqd 5888	. . . . . . . 8 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → dom (𝑒 evalSub1 𝑓) = dom 𝑂)
21	19	fveq1d 6892	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒 evalSub1 𝑓)‘𝑞) = (𝑂‘𝑞))
22	21	fveq1d 6892	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = ((𝑂‘𝑞)‘𝑥))
23		fveq2 6890	. . . . . . . . . . 11 ⊢ (𝑒 = 𝐸 → (0g‘𝑒) = (0g‘𝐸))
24	23	adantr 486	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (0g‘𝑒) = (0g‘𝐸))
25		ply1annig1p.0	. . . . . . . . . 10 ⊢ 0 = (0g‘𝐸)
26	24, 25	eqtr4di 2813	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (0g‘𝑒) = 0 )
27	22, 26	eqeq12d 2776	. . . . . . . 8 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = (0g‘𝑒) ↔ ((𝑂‘𝑞)‘𝑥) = 0 ))
28	20, 27	rabeqbidv 3429	. . . . . . 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 6897	. . . . . 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 5191	. . . . 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 34382	. . . . 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 7578	. . . 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 1398	. . 3 ⊢ (𝜑 → (𝐸 minPoly 𝐹) = (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 })))
34	1, 33	eqtrid 2807	. 2 ⊢ (𝜑 → 𝑀 = (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 })))
35		fveqeq2 6899	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑂‘𝑞)‘𝑥) = 0 ↔ ((𝑂‘𝑞)‘𝐴) = 0 ))
36	35	rabbidv 3419	. . . . 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 2813	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 } = 𝑄)
39	38	fveq2d 6894	. . 3 ⊢ (𝑥 = 𝐴 → (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 }) = (𝐺‘𝑄))
40	39	adantl 487	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 }) = (𝐺‘𝑄))
41		ply1annig1p.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
42		fvexd 6905	. 2 ⊢ (𝜑 → (𝐺‘𝑄) ∈ V)
43	34, 40, 41, 42	fvmptd 7006	1 ⊢ (𝜑 → (𝑀‘𝐴) = (𝐺‘𝑄))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {crab 3412  Vcvv 3450   ↦ cmpt 5185  dom cdm 5651  ‘cfv 6543  (class class class)co 7424  Basecbs 17396   ↾_s cress 17417  0_gc0g 17619  Fieldcfield 20990  SubDRingcsdrg 21052  RSpancrsp 21494  Poly₁cpl1 22504   evalSub₁ ces1 22640  idlGen_1pcig1p 26457   minPoly cminply 34381

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-minply 34382

This theorem is used by:  minplycl  34388  minplymindeg  34390  minplyann  34391  minplyirredlem  34392  minplyirred  34393  irngnminplynz  34394  minplym1p  34395  minplynzm1p  34396  irredminply  34398  algextdeglem4  34402  algextdeglem5  34403