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Theorem minplyeulem 32176
Description: An algebraic number 𝑋 over 𝐹 is a root of some monic polynomial 𝑝 with coefficients in 𝐹. (Contributed by Thierry Arnoux, 26-Jan-2025.)
Hypotheses
Ref Expression
algnbval.o 𝑂 = (𝐸 evalSub1 𝐹)
algnbval.z 𝑍 = (0g‘(Poly1𝐸))
algnbval.1 0 = (0g𝐸)
algnbval.2 (𝜑𝐸 ∈ Field)
algnbval.3 (𝜑𝐹 ∈ (SubDRing‘𝐸))
minplyeulem.x (𝜑𝑋 ∈ (𝐸 AlgNb 𝐹))
Assertion
Ref Expression
minplyeulem (𝜑 → ∃𝑝 ∈ (Monic1p𝐸)((𝑂𝑝)‘𝑋) = 0 )
Distinct variable groups:   𝐸,𝑝   𝐹,𝑝   𝑂,𝑝   𝑍,𝑝   0 ,𝑝   𝑋,𝑝   𝜑,𝑝

Proof of Theorem minplyeulem
Dummy variable 𝑞 is distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . . . 6 (Poly1𝐸) = (Poly1𝐸)
2 eqid 2737 . . . . . 6 (𝐸s 𝐹) = (𝐸s 𝐹)
3 eqid 2737 . . . . . 6 (Poly1‘(𝐸s 𝐹)) = (Poly1‘(𝐸s 𝐹))
4 eqid 2737 . . . . . 6 (Base‘(Poly1‘(𝐸s 𝐹))) = (Base‘(Poly1‘(𝐸s 𝐹)))
5 algnbval.3 . . . . . . . 8 (𝜑𝐹 ∈ (SubDRing‘𝐸))
65ad2antrr 724 . . . . . . 7 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → 𝐹 ∈ (SubDRing‘𝐸))
7 issdrg 20213 . . . . . . . 8 (𝐹 ∈ (SubDRing‘𝐸) ↔ (𝐸 ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸) ∧ (𝐸s 𝐹) ∈ DivRing))
87simp2bi 1146 . . . . . . 7 (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
96, 8syl 17 . . . . . 6 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → 𝐹 ∈ (SubRing‘𝐸))
10 eqid 2737 . . . . . 6 (Monic1p𝐸) = (Monic1p𝐸)
11 eqid 2737 . . . . . 6 (Monic1p‘(𝐸s 𝐹)) = (Monic1p‘(𝐸s 𝐹))
121, 2, 3, 4, 9, 10, 11ressply1mon1p 32094 . . . . 5 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → (Monic1p‘(𝐸s 𝐹)) = ((Base‘(Poly1‘(𝐸s 𝐹))) ∩ (Monic1p𝐸)))
13 inss2 4187 . . . . 5 ((Base‘(Poly1‘(𝐸s 𝐹))) ∩ (Monic1p𝐸)) ⊆ (Monic1p𝐸)
1412, 13eqsstrdi 3996 . . . 4 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → (Monic1p‘(𝐸s 𝐹)) ⊆ (Monic1p𝐸))
157simp3bi 1147 . . . . . . . 8 (𝐹 ∈ (SubDRing‘𝐸) → (𝐸s 𝐹) ∈ DivRing)
165, 15syl 17 . . . . . . 7 (𝜑 → (𝐸s 𝐹) ∈ DivRing)
1716ad2antrr 724 . . . . . 6 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → (𝐸s 𝐹) ∈ DivRing)
1817drngringd 20145 . . . . 5 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → (𝐸s 𝐹) ∈ Ring)
19 simplr 767 . . . . . . . 8 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → 𝑞 ∈ (dom 𝑂 ∖ {𝑍}))
2019eldifad 3920 . . . . . . 7 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → 𝑞 ∈ dom 𝑂)
21 algnbval.2 . . . . . . . . . . . 12 (𝜑𝐸 ∈ Field)
2221fldcrngd 20149 . . . . . . . . . . 11 (𝜑𝐸 ∈ CRing)
235, 8syl 17 . . . . . . . . . . 11 (𝜑𝐹 ∈ (SubRing‘𝐸))
24 algnbval.o . . . . . . . . . . . 12 𝑂 = (𝐸 evalSub1 𝐹)
25 eqid 2737 . . . . . . . . . . . 12 (Base‘𝐸) = (Base‘𝐸)
26 eqid 2737 . . . . . . . . . . . 12 (𝐸s (Base‘𝐸)) = (𝐸s (Base‘𝐸))
2724, 25, 26, 2, 3evls1rhm 21639 . . . . . . . . . . 11 ((𝐸 ∈ CRing ∧ 𝐹 ∈ (SubRing‘𝐸)) → 𝑂 ∈ ((Poly1‘(𝐸s 𝐹)) RingHom (𝐸s (Base‘𝐸))))
2822, 23, 27syl2anc 584 . . . . . . . . . 10 (𝜑𝑂 ∈ ((Poly1‘(𝐸s 𝐹)) RingHom (𝐸s (Base‘𝐸))))
29 eqid 2737 . . . . . . . . . . 11 (Base‘(𝐸s (Base‘𝐸))) = (Base‘(𝐸s (Base‘𝐸)))
304, 29rhmf 20110 . . . . . . . . . 10 (𝑂 ∈ ((Poly1‘(𝐸s 𝐹)) RingHom (𝐸s (Base‘𝐸))) → 𝑂:(Base‘(Poly1‘(𝐸s 𝐹)))⟶(Base‘(𝐸s (Base‘𝐸))))
3128, 30syl 17 . . . . . . . . 9 (𝜑𝑂:(Base‘(Poly1‘(𝐸s 𝐹)))⟶(Base‘(𝐸s (Base‘𝐸))))
3231fdmd 6676 . . . . . . . 8 (𝜑 → dom 𝑂 = (Base‘(Poly1‘(𝐸s 𝐹))))
3332ad2antrr 724 . . . . . . 7 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → dom 𝑂 = (Base‘(Poly1‘(𝐸s 𝐹))))
3420, 33eleqtrd 2840 . . . . . 6 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → 𝑞 ∈ (Base‘(Poly1‘(𝐸s 𝐹))))
35 eldifsni 4748 . . . . . . . 8 (𝑞 ∈ (dom 𝑂 ∖ {𝑍}) → 𝑞𝑍)
3635ad2antlr 725 . . . . . . 7 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → 𝑞𝑍)
37 algnbval.z . . . . . . . . 9 𝑍 = (0g‘(Poly1𝐸))
381, 2, 3, 4, 23, 37ressply10g 32093 . . . . . . . 8 (𝜑𝑍 = (0g‘(Poly1‘(𝐸s 𝐹))))
3938ad2antrr 724 . . . . . . 7 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → 𝑍 = (0g‘(Poly1‘(𝐸s 𝐹))))
4036, 39neeqtrd 3011 . . . . . 6 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → 𝑞 ≠ (0g‘(Poly1‘(𝐸s 𝐹))))
41 eqid 2737 . . . . . . 7 (0g‘(Poly1‘(𝐸s 𝐹))) = (0g‘(Poly1‘(𝐸s 𝐹)))
42 eqid 2737 . . . . . . 7 (Unic1p‘(𝐸s 𝐹)) = (Unic1p‘(𝐸s 𝐹))
433, 4, 41, 42drnguc1p 25486 . . . . . 6 (((𝐸s 𝐹) ∈ DivRing ∧ 𝑞 ∈ (Base‘(Poly1‘(𝐸s 𝐹))) ∧ 𝑞 ≠ (0g‘(Poly1‘(𝐸s 𝐹)))) → 𝑞 ∈ (Unic1p‘(𝐸s 𝐹)))
4417, 34, 40, 43syl3anc 1371 . . . . 5 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → 𝑞 ∈ (Unic1p‘(𝐸s 𝐹)))
45 eqid 2737 . . . . . 6 (.r‘(Poly1‘(𝐸s 𝐹))) = (.r‘(Poly1‘(𝐸s 𝐹)))
46 eqid 2737 . . . . . 6 (algSc‘(Poly1‘(𝐸s 𝐹))) = (algSc‘(Poly1‘(𝐸s 𝐹)))
47 eqid 2737 . . . . . 6 ( deg1 ‘(𝐸s 𝐹)) = ( deg1 ‘(𝐸s 𝐹))
48 eqid 2737 . . . . . 6 (invr‘(𝐸s 𝐹)) = (invr‘(𝐸s 𝐹))
4942, 11, 3, 45, 46, 47, 48uc1pmon1p 25467 . . . . 5 (((𝐸s 𝐹) ∈ Ring ∧ 𝑞 ∈ (Unic1p‘(𝐸s 𝐹))) → (((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞))))(.r‘(Poly1‘(𝐸s 𝐹)))𝑞) ∈ (Monic1p‘(𝐸s 𝐹)))
5018, 44, 49syl2anc 584 . . . 4 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → (((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞))))(.r‘(Poly1‘(𝐸s 𝐹)))𝑞) ∈ (Monic1p‘(𝐸s 𝐹)))
5114, 50sseldd 3943 . . 3 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → (((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞))))(.r‘(Poly1‘(𝐸s 𝐹)))𝑞) ∈ (Monic1p𝐸))
52 simpr 485 . . . . . 6 ((((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) ∧ 𝑝 = (((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞))))(.r‘(Poly1‘(𝐸s 𝐹)))𝑞)) → 𝑝 = (((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞))))(.r‘(Poly1‘(𝐸s 𝐹)))𝑞))
5352fveq2d 6843 . . . . 5 ((((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) ∧ 𝑝 = (((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞))))(.r‘(Poly1‘(𝐸s 𝐹)))𝑞)) → (𝑂𝑝) = (𝑂‘(((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞))))(.r‘(Poly1‘(𝐸s 𝐹)))𝑞)))
5453fveq1d 6841 . . . 4 ((((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) ∧ 𝑝 = (((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞))))(.r‘(Poly1‘(𝐸s 𝐹)))𝑞)) → ((𝑂𝑝)‘𝑋) = ((𝑂‘(((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞))))(.r‘(Poly1‘(𝐸s 𝐹)))𝑞))‘𝑋))
5554eqeq1d 2739 . . 3 ((((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) ∧ 𝑝 = (((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞))))(.r‘(Poly1‘(𝐸s 𝐹)))𝑞)) → (((𝑂𝑝)‘𝑋) = 0 ↔ ((𝑂‘(((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞))))(.r‘(Poly1‘(𝐸s 𝐹)))𝑞))‘𝑋) = 0 ))
56 eqid 2737 . . . . 5 (.r𝐸) = (.r𝐸)
5722ad2antrr 724 . . . . 5 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → 𝐸 ∈ CRing)
58 eqid 2737 . . . . . . 7 (Scalar‘(Poly1‘(𝐸s 𝐹))) = (Scalar‘(Poly1‘(𝐸s 𝐹)))
59 fldsdrgfld 20217 . . . . . . . . . . . 12 ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸s 𝐹) ∈ Field)
6021, 5, 59syl2anc 584 . . . . . . . . . . 11 (𝜑 → (𝐸s 𝐹) ∈ Field)
6160fldcrngd 20149 . . . . . . . . . 10 (𝜑 → (𝐸s 𝐹) ∈ CRing)
623ply1assa 21521 . . . . . . . . . 10 ((𝐸s 𝐹) ∈ CRing → (Poly1‘(𝐸s 𝐹)) ∈ AssAlg)
6361, 62syl 17 . . . . . . . . 9 (𝜑 → (Poly1‘(𝐸s 𝐹)) ∈ AssAlg)
6463ad2antrr 724 . . . . . . . 8 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → (Poly1‘(𝐸s 𝐹)) ∈ AssAlg)
65 assaring 21219 . . . . . . . 8 ((Poly1‘(𝐸s 𝐹)) ∈ AssAlg → (Poly1‘(𝐸s 𝐹)) ∈ Ring)
6664, 65syl 17 . . . . . . 7 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → (Poly1‘(𝐸s 𝐹)) ∈ Ring)
6761crngringd 19930 . . . . . . . . 9 (𝜑 → (𝐸s 𝐹) ∈ Ring)
683ply1lmod 21574 . . . . . . . . 9 ((𝐸s 𝐹) ∈ Ring → (Poly1‘(𝐸s 𝐹)) ∈ LMod)
6967, 68syl 17 . . . . . . . 8 (𝜑 → (Poly1‘(𝐸s 𝐹)) ∈ LMod)
7069ad2antrr 724 . . . . . . 7 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → (Poly1‘(𝐸s 𝐹)) ∈ LMod)
71 eqid 2737 . . . . . . 7 (Base‘(Scalar‘(Poly1‘(𝐸s 𝐹)))) = (Base‘(Scalar‘(Poly1‘(𝐸s 𝐹))))
7246, 58, 66, 70, 71, 4asclf 21237 . . . . . 6 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → (algSc‘(Poly1‘(𝐸s 𝐹))):(Base‘(Scalar‘(Poly1‘(𝐸s 𝐹))))⟶(Base‘(Poly1‘(𝐸s 𝐹))))
73 eqid 2737 . . . . . . . 8 (Base‘(𝐸s 𝐹)) = (Base‘(𝐸s 𝐹))
74 eqid 2737 . . . . . . . 8 (0g‘(𝐸s 𝐹)) = (0g‘(𝐸s 𝐹))
7547, 3, 41, 4deg1nn0cl 25404 . . . . . . . . . 10 (((𝐸s 𝐹) ∈ Ring ∧ 𝑞 ∈ (Base‘(Poly1‘(𝐸s 𝐹))) ∧ 𝑞 ≠ (0g‘(Poly1‘(𝐸s 𝐹)))) → (( deg1 ‘(𝐸s 𝐹))‘𝑞) ∈ ℕ0)
7618, 34, 40, 75syl3anc 1371 . . . . . . . . 9 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → (( deg1 ‘(𝐸s 𝐹))‘𝑞) ∈ ℕ0)
77 eqid 2737 . . . . . . . . . 10 (coe1𝑞) = (coe1𝑞)
7877, 4, 3, 73coe1fvalcl 21534 . . . . . . . . 9 ((𝑞 ∈ (Base‘(Poly1‘(𝐸s 𝐹))) ∧ (( deg1 ‘(𝐸s 𝐹))‘𝑞) ∈ ℕ0) → ((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞)) ∈ (Base‘(𝐸s 𝐹)))
7934, 76, 78syl2anc 584 . . . . . . . 8 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → ((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞)) ∈ (Base‘(𝐸s 𝐹)))
8047, 3, 41, 4, 74, 77deg1ldg 25408 . . . . . . . . 9 (((𝐸s 𝐹) ∈ Ring ∧ 𝑞 ∈ (Base‘(Poly1‘(𝐸s 𝐹))) ∧ 𝑞 ≠ (0g‘(Poly1‘(𝐸s 𝐹)))) → ((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞)) ≠ (0g‘(𝐸s 𝐹)))
8118, 34, 40, 80syl3anc 1371 . . . . . . . 8 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → ((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞)) ≠ (0g‘(𝐸s 𝐹)))
8273, 74, 48, 17, 79, 81drnginvrcld 20159 . . . . . . 7 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → ((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞))) ∈ (Base‘(𝐸s 𝐹)))
833ply1sca 21575 . . . . . . . . . 10 ((𝐸s 𝐹) ∈ Field → (𝐸s 𝐹) = (Scalar‘(Poly1‘(𝐸s 𝐹))))
8460, 83syl 17 . . . . . . . . 9 (𝜑 → (𝐸s 𝐹) = (Scalar‘(Poly1‘(𝐸s 𝐹))))
8584fveq2d 6843 . . . . . . . 8 (𝜑 → (Base‘(𝐸s 𝐹)) = (Base‘(Scalar‘(Poly1‘(𝐸s 𝐹)))))
8685ad2antrr 724 . . . . . . 7 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → (Base‘(𝐸s 𝐹)) = (Base‘(Scalar‘(Poly1‘(𝐸s 𝐹)))))
8782, 86eleqtrd 2840 . . . . . 6 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → ((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞))) ∈ (Base‘(Scalar‘(Poly1‘(𝐸s 𝐹)))))
8872, 87ffvelcdmd 7032 . . . . 5 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → ((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞)))) ∈ (Base‘(Poly1‘(𝐸s 𝐹))))
89 minplyeulem.x . . . . . . . 8 (𝜑𝑋 ∈ (𝐸 AlgNb 𝐹))
90 algnbval.1 . . . . . . . . 9 0 = (0g𝐸)
9124, 37, 90, 21, 5, 25isalgnb 32175 . . . . . . . 8 (𝜑 → (𝑋 ∈ (𝐸 AlgNb 𝐹) ↔ (𝑋 ∈ (Base‘𝐸) ∧ ∃𝑞 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑞)‘𝑋) = 0 )))
9289, 91mpbid 231 . . . . . . 7 (𝜑 → (𝑋 ∈ (Base‘𝐸) ∧ ∃𝑞 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑞)‘𝑋) = 0 ))
9392simpld 495 . . . . . 6 (𝜑𝑋 ∈ (Base‘𝐸))
9493ad2antrr 724 . . . . 5 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → 𝑋 ∈ (Base‘𝐸))
9524, 25, 3, 2, 4, 45, 56, 57, 9, 88, 34, 94evls1muld 32090 . . . 4 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → ((𝑂‘(((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞))))(.r‘(Poly1‘(𝐸s 𝐹)))𝑞))‘𝑋) = (((𝑂‘((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞)))))‘𝑋)(.r𝐸)((𝑂𝑞)‘𝑋)))
96 simpr 485 . . . . 5 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → ((𝑂𝑞)‘𝑋) = 0 )
9796oveq2d 7367 . . . 4 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → (((𝑂‘((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞)))))‘𝑋)(.r𝐸)((𝑂𝑞)‘𝑋)) = (((𝑂‘((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞)))))‘𝑋)(.r𝐸) 0 ))
9822crngringd 19930 . . . . . 6 (𝜑𝐸 ∈ Ring)
9998ad2antrr 724 . . . . 5 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → 𝐸 ∈ Ring)
10021ad2antrr 724 . . . . . . 7 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → 𝐸 ∈ Field)
101 fvexd 6854 . . . . . . 7 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → (Base‘𝐸) ∈ V)
10228ad2antrr 724 . . . . . . . . 9 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → 𝑂 ∈ ((Poly1‘(𝐸s 𝐹)) RingHom (𝐸s (Base‘𝐸))))
103102, 30syl 17 . . . . . . . 8 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → 𝑂:(Base‘(Poly1‘(𝐸s 𝐹)))⟶(Base‘(𝐸s (Base‘𝐸))))
104103, 88ffvelcdmd 7032 . . . . . . 7 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → (𝑂‘((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞))))) ∈ (Base‘(𝐸s (Base‘𝐸))))
10526, 25, 29, 100, 101, 104pwselbas 17330 . . . . . 6 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → (𝑂‘((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞))))):(Base‘𝐸)⟶(Base‘𝐸))
106105, 94ffvelcdmd 7032 . . . . 5 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → ((𝑂‘((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞)))))‘𝑋) ∈ (Base‘𝐸))
10725, 56, 90ringrz 19964 . . . . 5 ((𝐸 ∈ Ring ∧ ((𝑂‘((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞)))))‘𝑋) ∈ (Base‘𝐸)) → (((𝑂‘((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞)))))‘𝑋)(.r𝐸) 0 ) = 0 )
10899, 106, 107syl2anc 584 . . . 4 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → (((𝑂‘((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞)))))‘𝑋)(.r𝐸) 0 ) = 0 )
10995, 97, 1083eqtrd 2781 . . 3 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → ((𝑂‘(((algSc‘(Poly1‘(𝐸s 𝐹)))‘((invr‘(𝐸s 𝐹))‘((coe1𝑞)‘(( deg1 ‘(𝐸s 𝐹))‘𝑞))))(.r‘(Poly1‘(𝐸s 𝐹)))𝑞))‘𝑋) = 0 )
11051, 55, 109rspcedvd 3581 . 2 (((𝜑𝑞 ∈ (dom 𝑂 ∖ {𝑍})) ∧ ((𝑂𝑞)‘𝑋) = 0 ) → ∃𝑝 ∈ (Monic1p𝐸)((𝑂𝑝)‘𝑋) = 0 )
11192simprd 496 . 2 (𝜑 → ∃𝑞 ∈ (dom 𝑂 ∖ {𝑍})((𝑂𝑞)‘𝑋) = 0 )
112110, 111r19.29a 3157 1 (𝜑 → ∃𝑝 ∈ (Monic1p𝐸)((𝑂𝑝)‘𝑋) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2941  wrex 3071  Vcvv 3443  cdif 3905  cin 3907  {csn 4584  dom cdm 5631  wf 6489  cfv 6493  (class class class)co 7351  0cn0 12371  Basecbs 17042  s cress 17071  .rcmulr 17093  Scalarcsca 17095  0gc0g 17280  s cpws 17287  Ringcrg 19917  CRingccrg 19918  invrcinvr 20052   RingHom crh 20095  DivRingcdr 20137  Fieldcfield 20138  SubRingcsubrg 20170  SubDRingcsdrg 20211  LModclmod 20274  AssAlgcasa 21208  algSccascl 21210  Poly1cpl1 21499  coe1cco1 21500   evalSub1 ces1 21630   deg1 cdg1 25367  Monic1pcmn1 25441  Unic1pcuc1p 25442   AlgNb calgnb 32170
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 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-addf 11088  ax-mulf 11089
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-iin 4955  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-se 5587  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-of 7609  df-ofr 7610  df-om 7795  df-1st 7913  df-2nd 7914  df-supp 8085  df-tpos 8149  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-1o 8404  df-er 8606  df-map 8725  df-pm 8726  df-ixp 8794  df-en 8842  df-dom 8843  df-sdom 8844  df-fin 8845  df-fsupp 9264  df-sup 9336  df-oi 9404  df-card 9833  df-pnf 11149  df-mnf 11150  df-xr 11151  df-ltxr 11152  df-le 11153  df-sub 11345  df-neg 11346  df-nn 12112  df-2 12174  df-3 12175  df-4 12176  df-5 12177  df-6 12178  df-7 12179  df-8 12180  df-9 12181  df-n0 12372  df-z 12458  df-dec 12577  df-uz 12722  df-fz 13379  df-fzo 13522  df-seq 13861  df-hash 14184  df-struct 16978  df-sets 16995  df-slot 17013  df-ndx 17025  df-base 17043  df-ress 17072  df-plusg 17105  df-mulr 17106  df-starv 17107  df-sca 17108  df-vsca 17109  df-ip 17110  df-tset 17111  df-ple 17112  df-ds 17114  df-unif 17115  df-hom 17116  df-cco 17117  df-0g 17282  df-gsum 17283  df-prds 17288  df-pws 17290  df-mre 17425  df-mrc 17426  df-acs 17428  df-mgm 18456  df-sgrp 18505  df-mnd 18516  df-mhm 18560  df-submnd 18561  df-grp 18710  df-minusg 18711  df-sbg 18712  df-mulg 18831  df-subg 18883  df-ghm 18964  df-cntz 19055  df-cmn 19522  df-abl 19523  df-mgp 19855  df-ur 19872  df-srg 19876  df-ring 19919  df-cring 19920  df-oppr 20001  df-dvdsr 20022  df-unit 20023  df-invr 20053  df-rnghom 20098  df-drng 20139  df-field 20140  df-subrg 20172  df-sdrg 20212  df-lmod 20276  df-lss 20345  df-lsp 20385  df-rlreg 20705  df-cnfld 20749  df-assa 21211  df-asp 21212  df-ascl 21213  df-psr 21263  df-mvr 21264  df-mpl 21265  df-opsr 21267  df-evls 21433  df-evl 21434  df-psr1 21502  df-vr1 21503  df-ply1 21504  df-coe1 21505  df-evls1 21632  df-evl1 21633  df-mdeg 25368  df-deg1 25369  df-mon1 25446  df-uc1p 25447  df-algnb 32172
This theorem is referenced by: (None)
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