Theorem minplyval 33285

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 33284, 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 3489	. . . 4 ⊢ (𝜑 → 𝐸 ∈ V)
4		ply1annig1p.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
5	4	elexd 3489	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
6		ply1annig1p.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐸)
7	6	fvexi 6899	. . . . . 6 ⊢ 𝐵 ∈ V
8	7	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
9	8	mptexd 7221	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 })) ∈ V)
10		fveq2 6885	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → (Base‘𝑒) = (Base‘𝐸))
11	10, 6	eqtr4di 2784	. . . . . . 7 ⊢ (𝑒 = 𝐸 → (Base‘𝑒) = 𝐵)
12	11	adantr 480	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (Base‘𝑒) = 𝐵)
13		oveq12 7414	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 ↾s 𝑓) = (𝐸 ↾s 𝐹))
14	13	fveq2d 6889	. . . . . . . 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 2784	. . . . . . 7 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (idlGen1p‘(𝑒 ↾s 𝑓)) = 𝐺)
17		oveq12 7414	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 evalSub1 𝑓) = (𝐸 evalSub1 𝐹))
18		ply1annig1p.o	. . . . . . . . . 10 ⊢ 𝑂 = (𝐸 evalSub1 𝐹)
19	17, 18	eqtr4di 2784	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 evalSub1 𝑓) = 𝑂)
20	19	dmeqd 5899	. . . . . . . 8 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → dom (𝑒 evalSub1 𝑓) = dom 𝑂)
21	19	fveq1d 6887	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒 evalSub1 𝑓)‘𝑞) = (𝑂‘𝑞))
22	21	fveq1d 6887	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = ((𝑂‘𝑞)‘𝑥))
23		fveq2 6885	. . . . . . . . . . 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 2784	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (0g‘𝑒) = 0 )
27	22, 26	eqeq12d 2742	. . . . . . . 8 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = (0g‘𝑒) ↔ ((𝑂‘𝑞)‘𝑥) = 0 ))
28	20, 27	rabeqbidv 3443	. . . . . . 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 6892	. . . . . 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 5232	. . . . 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 33276	. . . . 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 7558	. . . 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 1368	. . 3 ⊢ (𝜑 → (𝐸 minPoly 𝐹) = (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 })))
34	1, 33	eqtrid 2778	. 2 ⊢ (𝜑 → 𝑀 = (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 })))
35		fveqeq2 6894	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑂‘𝑞)‘𝑥) = 0 ↔ ((𝑂‘𝑞)‘𝐴) = 0 ))
36	35	rabbidv 3434	. . . . 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 2784	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 } = 𝑄)
39	38	fveq2d 6889	. . 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 6900	. 2 ⊢ (𝜑 → (𝐺‘𝑄) ∈ V)
43	34, 40, 41, 42	fvmptd 6999	1 ⊢ (𝜑 → (𝑀‘𝐴) = (𝐺‘𝑄))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {crab 3426  Vcvv 3468   ↦ cmpt 5224  dom cdm 5669  ‘cfv 6537  (class class class)co 7405  Basecbs 17153   ↾_s cress 17182  0_gc0g 17394  Fieldcfield 20588  SubDRingcsdrg 20637  RSpancrsp 21066  Poly₁cpl1 22051   evalSub₁ ces1 22187  idlGen_1pcig1p 26020   minPoly cminply 33275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-minply 33276

This theorem is referenced by:  minplycl  33286  minplyann  33288  minplyirredlem  33289  minplyirred  33290  irngnminplynz  33291  minplym1p  33292  irredminply  33293  algextdeglem4  33297  algextdeglem5  33298