Theorem minplyval 32761

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 32760, 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 3494	. . . 4 ⊢ (𝜑 → 𝐸 ∈ V)
4		ply1annig1p.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
5	4	elexd 3494	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
6		ply1annig1p.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐸)
7	6	fvexi 6905	. . . . . 6 ⊢ 𝐵 ∈ V
8	7	a1i 11	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
9	8	mptexd 7225	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 })) ∈ V)
10		fveq2 6891	. . . . . . . 8 ⊢ (𝑒 = 𝐸 → (Base‘𝑒) = (Base‘𝐸))
11	10, 6	eqtr4di 2790	. . . . . . 7 ⊢ (𝑒 = 𝐸 → (Base‘𝑒) = 𝐵)
12	11	adantr 481	. . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (Base‘𝑒) = 𝐵)
13		oveq12 7417	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 ↾s 𝑓) = (𝐸 ↾s 𝐹))
14	13	fveq2d 6895	. . . . . . . 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 2790	. . . . . . 7 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (idlGen1p‘(𝑒 ↾s 𝑓)) = 𝐺)
17		oveq12 7417	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 evalSub1 𝑓) = (𝐸 evalSub1 𝐹))
18		ply1annig1p.o	. . . . . . . . . 10 ⊢ 𝑂 = (𝐸 evalSub1 𝐹)
19	17, 18	eqtr4di 2790	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒 evalSub1 𝑓) = 𝑂)
20	19	dmeqd 5905	. . . . . . . 8 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → dom (𝑒 evalSub1 𝑓) = dom 𝑂)
21	19	fveq1d 6893	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒 evalSub1 𝑓)‘𝑞) = (𝑂‘𝑞))
22	21	fveq1d 6893	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = ((𝑂‘𝑞)‘𝑥))
23		fveq2 6891	. . . . . . . . . . 11 ⊢ (𝑒 = 𝐸 → (0g‘𝑒) = (0g‘𝐸))
24	23	adantr 481	. . . . . . . . . 10 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (0g‘𝑒) = (0g‘𝐸))
25		ply1annig1p.0	. . . . . . . . . 10 ⊢ 0 = (0g‘𝐸)
26	24, 25	eqtr4di 2790	. . . . . . . . 9 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (0g‘𝑒) = 0 )
27	22, 26	eqeq12d 2748	. . . . . . . 8 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((((𝑒 evalSub1 𝑓)‘𝑞)‘𝑥) = (0g‘𝑒) ↔ ((𝑂‘𝑞)‘𝑥) = 0 ))
28	20, 27	rabeqbidv 3449	. . . . . . 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 6898	. . . . . 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 32752	. . . . 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 7561	. . . 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 1371	. . 3 ⊢ (𝜑 → (𝐸 minPoly 𝐹) = (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 })))
34	1, 33	eqtrid 2784	. 2 ⊢ (𝜑 → 𝑀 = (𝑥 ∈ 𝐵 ↦ (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 })))
35		fveqeq2 6900	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑂‘𝑞)‘𝑥) = 0 ↔ ((𝑂‘𝑞)‘𝐴) = 0 ))
36	35	rabbidv 3440	. . . . 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 2790	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 } = 𝑄)
39	38	fveq2d 6895	. . 3 ⊢ (𝑥 = 𝐴 → (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 }) = (𝐺‘𝑄))
40	39	adantl 482	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐺‘{𝑞 ∈ dom 𝑂 ∣ ((𝑂‘𝑞)‘𝑥) = 0 }) = (𝐺‘𝑄))
41		ply1annig1p.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
42		fvexd 6906	. 2 ⊢ (𝜑 → (𝐺‘𝑄) ∈ V)
43	34, 40, 41, 42	fvmptd 7005	1 ⊢ (𝜑 → (𝑀‘𝐴) = (𝐺‘𝑄))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3432  Vcvv 3474   ↦ cmpt 5231  dom cdm 5676  ‘cfv 6543  (class class class)co 7408  Basecbs 17143   ↾_s cress 17172  0_gc0g 17384  Fieldcfield 20357  SubDRingcsdrg 20401  RSpancrsp 20783  Poly₁cpl1 21700   evalSub₁ ces1 21831  idlGen_1pcig1p 25646   minPoly cminply 32751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  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-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-minply 32752

This theorem is referenced by:  minplycl  32762  minplyirredlem  32764  minplyirred  32765  irngnminplynz  32766  algextdeglem1  32767