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Theorem rtelextdg2lem 33767
Description: Lemma for rtelextdg2 33768: If an element 𝑋 is a solution of a quadratic equation, then the degree of its field extension is at most 2. (Contributed by Thierry Arnoux, 22-Jun-2025.)
Hypotheses
Ref Expression
rtelextdg2.1 𝐾 = (𝐸s 𝐹)
rtelextdg2.2 𝐿 = (𝐸s (𝐸 fldGen (𝐹 ∪ {𝑋})))
rtelextdg2.3 0 = (0g𝐸)
rtelextdg2.4 𝑃 = (Poly1𝐾)
rtelextdg2.5 𝑉 = (Base‘𝐸)
rtelextdg2.6 · = (.r𝐸)
rtelextdg2.7 + = (+g𝐸)
rtelextdg2.8 = (.g‘(mulGrp‘𝐸))
rtelextdg2.9 (𝜑𝐸 ∈ Field)
rtelextdg2.10 (𝜑𝐹 ∈ (SubDRing‘𝐸))
rtelextdg2.11 (𝜑𝑋𝑉)
rtelextdg2.12 (𝜑𝐴𝐹)
rtelextdg2.13 (𝜑𝐵𝐹)
rtelextdg2.14 (𝜑 → ((2 𝑋) + ((𝐴 · 𝑋) + 𝐵)) = 0 )
rtelextdg2lem.1 𝑌 = (var1𝐾)
rtelextdg2lem.2 = (+g𝑃)
rtelextdg2lem.3 = (.r𝑃)
rtelextdg2lem.4 = (.g‘(mulGrp‘𝑃))
rtelextdg2lem.5 𝑈 = (algSc‘𝑃)
rtelextdg2lem.6 𝐺 = ((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵)))
Assertion
Ref Expression
rtelextdg2lem (𝜑 → (𝐿[:]𝐾) ≤ 2)

Proof of Theorem rtelextdg2lem
Dummy variables 𝑖 𝑝 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rtelextdg2.1 . . . . 5 𝐾 = (𝐸s 𝐹)
2 rtelextdg2.2 . . . . 5 𝐿 = (𝐸s (𝐸 fldGen (𝐹 ∪ {𝑋})))
3 eqid 2737 . . . . 5 (deg1𝐸) = (deg1𝐸)
4 eqid 2737 . . . . 5 (𝐸 minPoly 𝐹) = (𝐸 minPoly 𝐹)
5 rtelextdg2.9 . . . . 5 (𝜑𝐸 ∈ Field)
6 rtelextdg2.10 . . . . 5 (𝜑𝐹 ∈ (SubDRing‘𝐸))
7 rtelextdg2.11 . . . . . 6 (𝜑𝑋𝑉)
8 fveq2 6906 . . . . . . . . 9 (𝑝 = 𝐺 → ((𝐸 evalSub1 𝐹)‘𝑝) = ((𝐸 evalSub1 𝐹)‘𝐺))
98fveq1d 6908 . . . . . . . 8 (𝑝 = 𝐺 → (((𝐸 evalSub1 𝐹)‘𝑝)‘𝑋) = (((𝐸 evalSub1 𝐹)‘𝐺)‘𝑋))
109eqeq1d 2739 . . . . . . 7 (𝑝 = 𝐺 → ((((𝐸 evalSub1 𝐹)‘𝑝)‘𝑋) = 0 ↔ (((𝐸 evalSub1 𝐹)‘𝐺)‘𝑋) = 0 ))
11 rtelextdg2lem.6 . . . . . . . . 9 𝐺 = ((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵)))
12 eqid 2737 . . . . . . . . . 10 (Base‘𝑃) = (Base‘𝑃)
13 rtelextdg2lem.2 . . . . . . . . . 10 = (+g𝑃)
14 fldsdrgfld 20799 . . . . . . . . . . . . . . . 16 ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸s 𝐹) ∈ Field)
155, 6, 14syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑 → (𝐸s 𝐹) ∈ Field)
1615fldcrngd 20742 . . . . . . . . . . . . . 14 (𝜑 → (𝐸s 𝐹) ∈ CRing)
171, 16eqeltrid 2845 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ CRing)
1817crngringd 20243 . . . . . . . . . . . 12 (𝜑𝐾 ∈ Ring)
19 rtelextdg2.4 . . . . . . . . . . . . 13 𝑃 = (Poly1𝐾)
2019ply1ring 22249 . . . . . . . . . . . 12 (𝐾 ∈ Ring → 𝑃 ∈ Ring)
2118, 20syl 17 . . . . . . . . . . 11 (𝜑𝑃 ∈ Ring)
2221ringgrpd 20239 . . . . . . . . . 10 (𝜑𝑃 ∈ Grp)
23 eqid 2737 . . . . . . . . . . . 12 (mulGrp‘𝑃) = (mulGrp‘𝑃)
2423, 12mgpbas 20142 . . . . . . . . . . 11 (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
25 rtelextdg2lem.4 . . . . . . . . . . 11 = (.g‘(mulGrp‘𝑃))
2623ringmgp 20236 . . . . . . . . . . . 12 (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
2721, 26syl 17 . . . . . . . . . . 11 (𝜑 → (mulGrp‘𝑃) ∈ Mnd)
28 2nn0 12543 . . . . . . . . . . . 12 2 ∈ ℕ0
2928a1i 11 . . . . . . . . . . 11 (𝜑 → 2 ∈ ℕ0)
30 rtelextdg2lem.1 . . . . . . . . . . . . 13 𝑌 = (var1𝐾)
3130, 19, 12vr1cl 22219 . . . . . . . . . . . 12 (𝐾 ∈ Ring → 𝑌 ∈ (Base‘𝑃))
3218, 31syl 17 . . . . . . . . . . 11 (𝜑𝑌 ∈ (Base‘𝑃))
3324, 25, 27, 29, 32mulgnn0cld 19113 . . . . . . . . . 10 (𝜑 → (2 𝑌) ∈ (Base‘𝑃))
34 rtelextdg2lem.3 . . . . . . . . . . . 12 = (.r𝑃)
35 rtelextdg2lem.5 . . . . . . . . . . . . 13 𝑈 = (algSc‘𝑃)
365fldcrngd 20742 . . . . . . . . . . . . 13 (𝜑𝐸 ∈ CRing)
37 sdrgsubrg 20792 . . . . . . . . . . . . . 14 (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
386, 37syl 17 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ (SubRing‘𝐸))
39 rtelextdg2.12 . . . . . . . . . . . . 13 (𝜑𝐴𝐹)
4019, 1, 35, 12, 36, 38, 39ressasclcl 33596 . . . . . . . . . . . 12 (𝜑 → (𝑈𝐴) ∈ (Base‘𝑃))
4112, 34, 21, 40, 32ringcld 20257 . . . . . . . . . . 11 (𝜑 → ((𝑈𝐴) 𝑌) ∈ (Base‘𝑃))
42 rtelextdg2.13 . . . . . . . . . . . 12 (𝜑𝐵𝐹)
4319, 1, 35, 12, 36, 38, 42ressasclcl 33596 . . . . . . . . . . 11 (𝜑 → (𝑈𝐵) ∈ (Base‘𝑃))
4412, 13, 22, 41, 43grpcld 18965 . . . . . . . . . 10 (𝜑 → (((𝑈𝐴) 𝑌) (𝑈𝐵)) ∈ (Base‘𝑃))
4512, 13, 22, 33, 44grpcld 18965 . . . . . . . . 9 (𝜑 → ((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))) ∈ (Base‘𝑃))
4611, 45eqeltrid 2845 . . . . . . . 8 (𝜑𝐺 ∈ (Base‘𝑃))
4711fveq2i 6909 . . . . . . . . . . . 12 (coe1𝐺) = (coe1‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))
4847fveq1i 6907 . . . . . . . . . . 11 ((coe1𝐺)‘2) = ((coe1‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))‘2)
49 eqid 2737 . . . . . . . . . . . . . 14 (+g𝐾) = (+g𝐾)
5019, 12, 13, 49coe1addfv 22268 . . . . . . . . . . . . 13 (((𝐾 ∈ Ring ∧ (2 𝑌) ∈ (Base‘𝑃) ∧ (((𝑈𝐴) 𝑌) (𝑈𝐵)) ∈ (Base‘𝑃)) ∧ 2 ∈ ℕ0) → ((coe1‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))‘2) = (((coe1‘(2 𝑌))‘2)(+g𝐾)((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘2)))
5118, 33, 44, 29, 50syl31anc 1375 . . . . . . . . . . . 12 (𝜑 → ((coe1‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))‘2) = (((coe1‘(2 𝑌))‘2)(+g𝐾)((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘2)))
52 eqid 2737 . . . . . . . . . . . . . . 15 (0g𝐾) = (0g𝐾)
53 eqid 2737 . . . . . . . . . . . . . . 15 (1r𝐾) = (1r𝐾)
5419, 30, 25, 18, 29, 52, 53coe1mon 33610 . . . . . . . . . . . . . 14 (𝜑 → (coe1‘(2 𝑌)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 2, (1r𝐾), (0g𝐾))))
55 simpr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑖 = 2) → 𝑖 = 2)
5655iftrued 4533 . . . . . . . . . . . . . 14 ((𝜑𝑖 = 2) → if(𝑖 = 2, (1r𝐾), (0g𝐾)) = (1r𝐾))
57 fvexd 6921 . . . . . . . . . . . . . 14 (𝜑 → (1r𝐾) ∈ V)
5854, 56, 29, 57fvmptd 7023 . . . . . . . . . . . . 13 (𝜑 → ((coe1‘(2 𝑌))‘2) = (1r𝐾))
5919, 12, 13, 49coe1addfv 22268 . . . . . . . . . . . . . . 15 (((𝐾 ∈ Ring ∧ ((𝑈𝐴) 𝑌) ∈ (Base‘𝑃) ∧ (𝑈𝐵) ∈ (Base‘𝑃)) ∧ 2 ∈ ℕ0) → ((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘2) = (((coe1‘((𝑈𝐴) 𝑌))‘2)(+g𝐾)((coe1‘(𝑈𝐵))‘2)))
6018, 41, 43, 29, 59syl31anc 1375 . . . . . . . . . . . . . 14 (𝜑 → ((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘2) = (((coe1‘((𝑈𝐴) 𝑌))‘2)(+g𝐾)((coe1‘(𝑈𝐵))‘2)))
61 rtelextdg2.5 . . . . . . . . . . . . . . . . . . . 20 𝑉 = (Base‘𝐸)
6261sdrgss 20794 . . . . . . . . . . . . . . . . . . 19 (𝐹 ∈ (SubDRing‘𝐸) → 𝐹𝑉)
631, 61ressbas2 17283 . . . . . . . . . . . . . . . . . . 19 (𝐹𝑉𝐹 = (Base‘𝐾))
646, 62, 633syl 18 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹 = (Base‘𝐾))
6539, 64eleqtrd 2843 . . . . . . . . . . . . . . . . 17 (𝜑𝐴 ∈ (Base‘𝐾))
66 eqid 2737 . . . . . . . . . . . . . . . . . 18 (Base‘𝐾) = (Base‘𝐾)
67 eqid 2737 . . . . . . . . . . . . . . . . . 18 (.r𝐾) = (.r𝐾)
6819, 12, 66, 35, 34, 67coe1sclmulfv 22286 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Ring ∧ (𝐴 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝑃)) ∧ 2 ∈ ℕ0) → ((coe1‘((𝑈𝐴) 𝑌))‘2) = (𝐴(.r𝐾)((coe1𝑌)‘2)))
6918, 65, 32, 29, 68syl121anc 1377 . . . . . . . . . . . . . . . 16 (𝜑 → ((coe1‘((𝑈𝐴) 𝑌))‘2) = (𝐴(.r𝐾)((coe1𝑌)‘2)))
7019, 30, 18, 52, 53coe1vr1 33613 . . . . . . . . . . . . . . . . . 18 (𝜑 → (coe1𝑌) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 1, (1r𝐾), (0g𝐾))))
71 1ne2 12474 . . . . . . . . . . . . . . . . . . . . . 22 1 ≠ 2
7271nesymi 2998 . . . . . . . . . . . . . . . . . . . . 21 ¬ 2 = 1
73 eqeq1 2741 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 2 → (𝑖 = 1 ↔ 2 = 1))
7472, 73mtbiri 327 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 2 → ¬ 𝑖 = 1)
7574adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 = 2) → ¬ 𝑖 = 1)
7675iffalsed 4536 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 = 2) → if(𝑖 = 1, (1r𝐾), (0g𝐾)) = (0g𝐾))
77 fvexd 6921 . . . . . . . . . . . . . . . . . 18 (𝜑 → (0g𝐾) ∈ V)
7870, 76, 29, 77fvmptd 7023 . . . . . . . . . . . . . . . . 17 (𝜑 → ((coe1𝑌)‘2) = (0g𝐾))
7978oveq2d 7447 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐴(.r𝐾)((coe1𝑌)‘2)) = (𝐴(.r𝐾)(0g𝐾)))
8066, 67, 52, 18, 65ringrzd 20293 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐴(.r𝐾)(0g𝐾)) = (0g𝐾))
8169, 79, 803eqtrd 2781 . . . . . . . . . . . . . . 15 (𝜑 → ((coe1‘((𝑈𝐴) 𝑌))‘2) = (0g𝐾))
8242, 64eleqtrd 2843 . . . . . . . . . . . . . . . . 17 (𝜑𝐵 ∈ (Base‘𝐾))
8319, 35, 66, 52coe1scl 22290 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Ring ∧ 𝐵 ∈ (Base‘𝐾)) → (coe1‘(𝑈𝐵)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, 𝐵, (0g𝐾))))
8418, 82, 83syl2anc 584 . . . . . . . . . . . . . . . 16 (𝜑 → (coe1‘(𝑈𝐵)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 0, 𝐵, (0g𝐾))))
85 0ne2 12473 . . . . . . . . . . . . . . . . . . . 20 0 ≠ 2
8685neii 2942 . . . . . . . . . . . . . . . . . . 19 ¬ 0 = 2
87 eqeq1 2741 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 0 → (𝑖 = 2 ↔ 0 = 2))
8886, 87mtbiri 327 . . . . . . . . . . . . . . . . . 18 (𝑖 = 0 → ¬ 𝑖 = 2)
8988, 55nsyl3 138 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 = 2) → ¬ 𝑖 = 0)
9089iffalsed 4536 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 = 2) → if(𝑖 = 0, 𝐵, (0g𝐾)) = (0g𝐾))
9184, 90, 29, 77fvmptd 7023 . . . . . . . . . . . . . . 15 (𝜑 → ((coe1‘(𝑈𝐵))‘2) = (0g𝐾))
9281, 91oveq12d 7449 . . . . . . . . . . . . . 14 (𝜑 → (((coe1‘((𝑈𝐴) 𝑌))‘2)(+g𝐾)((coe1‘(𝑈𝐵))‘2)) = ((0g𝐾)(+g𝐾)(0g𝐾)))
9318ringgrpd 20239 . . . . . . . . . . . . . . 15 (𝜑𝐾 ∈ Grp)
9466, 52grpidcl 18983 . . . . . . . . . . . . . . . 16 (𝐾 ∈ Grp → (0g𝐾) ∈ (Base‘𝐾))
9593, 94syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (0g𝐾) ∈ (Base‘𝐾))
9666, 49, 52, 93, 95grpridd 18988 . . . . . . . . . . . . . 14 (𝜑 → ((0g𝐾)(+g𝐾)(0g𝐾)) = (0g𝐾))
9760, 92, 963eqtrd 2781 . . . . . . . . . . . . 13 (𝜑 → ((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘2) = (0g𝐾))
9858, 97oveq12d 7449 . . . . . . . . . . . 12 (𝜑 → (((coe1‘(2 𝑌))‘2)(+g𝐾)((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘2)) = ((1r𝐾)(+g𝐾)(0g𝐾)))
9966, 53ringidcl 20262 . . . . . . . . . . . . . . 15 (𝐾 ∈ Ring → (1r𝐾) ∈ (Base‘𝐾))
10018, 99syl 17 . . . . . . . . . . . . . 14 (𝜑 → (1r𝐾) ∈ (Base‘𝐾))
10166, 49, 52, 93, 100grpridd 18988 . . . . . . . . . . . . 13 (𝜑 → ((1r𝐾)(+g𝐾)(0g𝐾)) = (1r𝐾))
10236crngringd 20243 . . . . . . . . . . . . . 14 (𝜑𝐸 ∈ Ring)
103 eqid 2737 . . . . . . . . . . . . . . . 16 (1r𝐸) = (1r𝐸)
104103subrg1cl 20580 . . . . . . . . . . . . . . 15 (𝐹 ∈ (SubRing‘𝐸) → (1r𝐸) ∈ 𝐹)
10538, 104syl 17 . . . . . . . . . . . . . 14 (𝜑 → (1r𝐸) ∈ 𝐹)
1066, 62syl 17 . . . . . . . . . . . . . 14 (𝜑𝐹𝑉)
1071, 61, 103ress1r 33238 . . . . . . . . . . . . . 14 ((𝐸 ∈ Ring ∧ (1r𝐸) ∈ 𝐹𝐹𝑉) → (1r𝐸) = (1r𝐾))
108102, 105, 106, 107syl3anc 1373 . . . . . . . . . . . . 13 (𝜑 → (1r𝐸) = (1r𝐾))
109101, 108eqtr4d 2780 . . . . . . . . . . . 12 (𝜑 → ((1r𝐾)(+g𝐾)(0g𝐾)) = (1r𝐸))
11051, 98, 1093eqtrd 2781 . . . . . . . . . . 11 (𝜑 → ((coe1‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))‘2) = (1r𝐸))
11148, 110eqtrid 2789 . . . . . . . . . 10 (𝜑 → ((coe1𝐺)‘2) = (1r𝐸))
1125flddrngd 20741 . . . . . . . . . . 11 (𝜑𝐸 ∈ DivRing)
113 drngnzr 20748 . . . . . . . . . . 11 (𝐸 ∈ DivRing → 𝐸 ∈ NzRing)
114 rtelextdg2.3 . . . . . . . . . . . 12 0 = (0g𝐸)
115103, 114nzrnz 20515 . . . . . . . . . . 11 (𝐸 ∈ NzRing → (1r𝐸) ≠ 0 )
116112, 113, 1153syl 18 . . . . . . . . . 10 (𝜑 → (1r𝐸) ≠ 0 )
117111, 116eqnetrd 3008 . . . . . . . . 9 (𝜑 → ((coe1𝐺)‘2) ≠ 0 )
118 fveq2 6906 . . . . . . . . . . 11 (𝐺 = (0g𝑃) → (coe1𝐺) = (coe1‘(0g𝑃)))
119118fveq1d 6908 . . . . . . . . . 10 (𝐺 = (0g𝑃) → ((coe1𝐺)‘2) = ((coe1‘(0g𝑃))‘2))
120 eqid 2737 . . . . . . . . . . . 12 (0g𝑃) = (0g𝑃)
12119, 120, 52, 18, 29coe1zfv 33612 . . . . . . . . . . 11 (𝜑 → ((coe1‘(0g𝑃))‘2) = (0g𝐾))
122102ringgrpd 20239 . . . . . . . . . . . . 13 (𝜑𝐸 ∈ Grp)
123122grpmndd 18964 . . . . . . . . . . . 12 (𝜑𝐸 ∈ Mnd)
124 subrgsubg 20577 . . . . . . . . . . . . . 14 (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
12538, 124syl 17 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ (SubGrp‘𝐸))
126114subg0cl 19152 . . . . . . . . . . . . 13 (𝐹 ∈ (SubGrp‘𝐸) → 0𝐹)
127125, 126syl 17 . . . . . . . . . . . 12 (𝜑0𝐹)
1281, 61, 114ress0g 18775 . . . . . . . . . . . 12 ((𝐸 ∈ Mnd ∧ 0𝐹𝐹𝑉) → 0 = (0g𝐾))
129123, 127, 106, 128syl3anc 1373 . . . . . . . . . . 11 (𝜑0 = (0g𝐾))
130121, 129eqtr4d 2780 . . . . . . . . . 10 (𝜑 → ((coe1‘(0g𝑃))‘2) = 0 )
131119, 130sylan9eqr 2799 . . . . . . . . 9 ((𝜑𝐺 = (0g𝑃)) → ((coe1𝐺)‘2) = 0 )
132117, 131mteqand 3033 . . . . . . . 8 (𝜑𝐺 ≠ (0g𝑃))
13311fveq2i 6909 . . . . . . . . . . 11 ((deg1𝐾)‘𝐺) = ((deg1𝐾)‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))
134 eqid 2737 . . . . . . . . . . . . 13 (deg1𝐾) = (deg1𝐾)
135 2re 12340 . . . . . . . . . . . . . . . . 17 2 ∈ ℝ
136135rexri 11319 . . . . . . . . . . . . . . . 16 2 ∈ ℝ*
137136a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → 2 ∈ ℝ*)
138134, 19, 12deg1xrcl 26121 . . . . . . . . . . . . . . . . 17 (((𝑈𝐴) 𝑌) ∈ (Base‘𝑃) → ((deg1𝐾)‘((𝑈𝐴) 𝑌)) ∈ ℝ*)
13941, 138syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((deg1𝐾)‘((𝑈𝐴) 𝑌)) ∈ ℝ*)
140 1xr 11320 . . . . . . . . . . . . . . . . 17 1 ∈ ℝ*
141140a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → 1 ∈ ℝ*)
142134, 19, 66, 12, 34, 35deg1mul3le 26156 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ Ring ∧ 𝐴 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝑃)) → ((deg1𝐾)‘((𝑈𝐴) 𝑌)) ≤ ((deg1𝐾)‘𝑌))
14318, 65, 32, 142syl3anc 1373 . . . . . . . . . . . . . . . . 17 (𝜑 → ((deg1𝐾)‘((𝑈𝐴) 𝑌)) ≤ ((deg1𝐾)‘𝑌))
1441, 15eqeltrid 2845 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐾 ∈ Field)
145144flddrngd 20741 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐾 ∈ DivRing)
146 drngnzr 20748 . . . . . . . . . . . . . . . . . . 19 (𝐾 ∈ DivRing → 𝐾 ∈ NzRing)
147145, 146syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝐾 ∈ NzRing)
148134, 19, 30, 147deg1vr 33614 . . . . . . . . . . . . . . . . 17 (𝜑 → ((deg1𝐾)‘𝑌) = 1)
149143, 148breqtrd 5169 . . . . . . . . . . . . . . . 16 (𝜑 → ((deg1𝐾)‘((𝑈𝐴) 𝑌)) ≤ 1)
150 1lt2 12437 . . . . . . . . . . . . . . . . 17 1 < 2
151150a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → 1 < 2)
152139, 141, 137, 149, 151xrlelttrd 13202 . . . . . . . . . . . . . . 15 (𝜑 → ((deg1𝐾)‘((𝑈𝐴) 𝑌)) < 2)
153134, 19, 12deg1xrcl 26121 . . . . . . . . . . . . . . . . 17 ((𝑈𝐵) ∈ (Base‘𝑃) → ((deg1𝐾)‘(𝑈𝐵)) ∈ ℝ*)
15443, 153syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((deg1𝐾)‘(𝑈𝐵)) ∈ ℝ*)
155 0xr 11308 . . . . . . . . . . . . . . . . 17 0 ∈ ℝ*
156155a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → 0 ∈ ℝ*)
157134, 19, 66, 35deg1sclle 26151 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Ring ∧ 𝐵 ∈ (Base‘𝐾)) → ((deg1𝐾)‘(𝑈𝐵)) ≤ 0)
15818, 82, 157syl2anc 584 . . . . . . . . . . . . . . . 16 (𝜑 → ((deg1𝐾)‘(𝑈𝐵)) ≤ 0)
159 2pos 12369 . . . . . . . . . . . . . . . . 17 0 < 2
160159a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → 0 < 2)
161154, 156, 137, 158, 160xrlelttrd 13202 . . . . . . . . . . . . . . 15 (𝜑 → ((deg1𝐾)‘(𝑈𝐵)) < 2)
16219, 134, 18, 12, 13, 41, 43, 137, 152, 161deg1addlt 33620 . . . . . . . . . . . . . 14 (𝜑 → ((deg1𝐾)‘(((𝑈𝐴) 𝑌) (𝑈𝐵))) < 2)
163134, 19, 30, 23, 25deg1pw 26160 . . . . . . . . . . . . . . 15 ((𝐾 ∈ NzRing ∧ 2 ∈ ℕ0) → ((deg1𝐾)‘(2 𝑌)) = 2)
164147, 29, 163syl2anc 584 . . . . . . . . . . . . . 14 (𝜑 → ((deg1𝐾)‘(2 𝑌)) = 2)
165162, 164breqtrrd 5171 . . . . . . . . . . . . 13 (𝜑 → ((deg1𝐾)‘(((𝑈𝐴) 𝑌) (𝑈𝐵))) < ((deg1𝐾)‘(2 𝑌)))
16619, 134, 18, 12, 13, 33, 44, 165deg1add 26142 . . . . . . . . . . . 12 (𝜑 → ((deg1𝐾)‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵)))) = ((deg1𝐾)‘(2 𝑌)))
167166, 164eqtrd 2777 . . . . . . . . . . 11 (𝜑 → ((deg1𝐾)‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵)))) = 2)
168133, 167eqtrid 2789 . . . . . . . . . 10 (𝜑 → ((deg1𝐾)‘𝐺) = 2)
169168fveq2d 6910 . . . . . . . . 9 (𝜑 → ((coe1𝐺)‘((deg1𝐾)‘𝐺)) = ((coe1𝐺)‘2))
170169, 111, 1083eqtrd 2781 . . . . . . . 8 (𝜑 → ((coe1𝐺)‘((deg1𝐾)‘𝐺)) = (1r𝐾))
171 eqid 2737 . . . . . . . . 9 (Monic1p𝐾) = (Monic1p𝐾)
17219, 12, 120, 134, 171, 53ismon1p 26182 . . . . . . . 8 (𝐺 ∈ (Monic1p𝐾) ↔ (𝐺 ∈ (Base‘𝑃) ∧ 𝐺 ≠ (0g𝑃) ∧ ((coe1𝐺)‘((deg1𝐾)‘𝐺)) = (1r𝐾)))
17346, 132, 170, 172syl3anbrc 1344 . . . . . . 7 (𝜑𝐺 ∈ (Monic1p𝐾))
174 eqid 2737 . . . . . . . . . . . 12 (𝐸 evalSub1 𝐹) = (𝐸 evalSub1 𝐹)
175 eqid 2737 . . . . . . . . . . . 12 (eval1𝐸) = (eval1𝐸)
176174, 61, 19, 1, 12, 175, 36, 38ressply1evl 22374 . . . . . . . . . . 11 (𝜑 → (𝐸 evalSub1 𝐹) = ((eval1𝐸) ↾ (Base‘𝑃)))
177176fveq1d 6908 . . . . . . . . . 10 (𝜑 → ((𝐸 evalSub1 𝐹)‘𝐺) = (((eval1𝐸) ↾ (Base‘𝑃))‘𝐺))
17846fvresd 6926 . . . . . . . . . 10 (𝜑 → (((eval1𝐸) ↾ (Base‘𝑃))‘𝐺) = ((eval1𝐸)‘𝐺))
179177, 178eqtrd 2777 . . . . . . . . 9 (𝜑 → ((𝐸 evalSub1 𝐹)‘𝐺) = ((eval1𝐸)‘𝐺))
180179fveq1d 6908 . . . . . . . 8 (𝜑 → (((𝐸 evalSub1 𝐹)‘𝐺)‘𝑋) = (((eval1𝐸)‘𝐺)‘𝑋))
181 eqid 2737 . . . . . . . . 9 (Poly1𝐸) = (Poly1𝐸)
182 eqid 2737 . . . . . . . . 9 (Base‘(Poly1𝐸)) = (Base‘(Poly1𝐸))
183 rtelextdg2.6 . . . . . . . . 9 · = (.r𝐸)
184 rtelextdg2.7 . . . . . . . . 9 + = (+g𝐸)
185 rtelextdg2.8 . . . . . . . . 9 = (.g‘(mulGrp‘𝐸))
186 eqid 2737 . . . . . . . . 9 (coe1𝐺) = (coe1𝐺)
187 eqid 2737 . . . . . . . . 9 ((coe1𝐺)‘2) = ((coe1𝐺)‘2)
188 eqid 2737 . . . . . . . . 9 ((coe1𝐺)‘1) = ((coe1𝐺)‘1)
189 eqid 2737 . . . . . . . . 9 ((coe1𝐺)‘0) = ((coe1𝐺)‘0)
190 eqid 2737 . . . . . . . . . . . 12 (PwSer1𝐾) = (PwSer1𝐾)
191 eqid 2737 . . . . . . . . . . . 12 (Base‘(PwSer1𝐾)) = (Base‘(PwSer1𝐾))
192181, 1, 19, 12, 38, 190, 191, 182ressply1bas2 22229 . . . . . . . . . . 11 (𝜑 → (Base‘𝑃) = ((Base‘(PwSer1𝐾)) ∩ (Base‘(Poly1𝐸))))
19346, 192eleqtrd 2843 . . . . . . . . . 10 (𝜑𝐺 ∈ ((Base‘(PwSer1𝐾)) ∩ (Base‘(Poly1𝐸))))
194193elin2d 4205 . . . . . . . . 9 (𝜑𝐺 ∈ (Base‘(Poly1𝐸)))
1951, 3, 19, 12, 46, 38ressdeg1 33591 . . . . . . . . . 10 (𝜑 → ((deg1𝐸)‘𝐺) = ((deg1𝐾)‘𝐺))
196195, 168eqtrd 2777 . . . . . . . . 9 (𝜑 → ((deg1𝐸)‘𝐺) = 2)
197181, 175, 61, 182, 183, 184, 185, 186, 3, 187, 188, 189, 36, 194, 196, 7evl1deg2 33602 . . . . . . . 8 (𝜑 → (((eval1𝐸)‘𝐺)‘𝑋) = ((((coe1𝐺)‘2) · (2 𝑋)) + ((((coe1𝐺)‘1) · 𝑋) + ((coe1𝐺)‘0))))
198111oveq1d 7446 . . . . . . . . . . 11 (𝜑 → (((coe1𝐺)‘2) · (2 𝑋)) = ((1r𝐸) · (2 𝑋)))
199 eqid 2737 . . . . . . . . . . . . . 14 (mulGrp‘𝐸) = (mulGrp‘𝐸)
200199, 61mgpbas 20142 . . . . . . . . . . . . 13 𝑉 = (Base‘(mulGrp‘𝐸))
201199ringmgp 20236 . . . . . . . . . . . . . 14 (𝐸 ∈ Ring → (mulGrp‘𝐸) ∈ Mnd)
202102, 201syl 17 . . . . . . . . . . . . 13 (𝜑 → (mulGrp‘𝐸) ∈ Mnd)
203200, 185, 202, 29, 7mulgnn0cld 19113 . . . . . . . . . . . 12 (𝜑 → (2 𝑋) ∈ 𝑉)
20461, 183, 103, 102, 203ringlidmd 20269 . . . . . . . . . . 11 (𝜑 → ((1r𝐸) · (2 𝑋)) = (2 𝑋))
205198, 204eqtrd 2777 . . . . . . . . . 10 (𝜑 → (((coe1𝐺)‘2) · (2 𝑋)) = (2 𝑋))
20647fveq1i 6907 . . . . . . . . . . . . 13 ((coe1𝐺)‘1) = ((coe1‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))‘1)
207 1nn0 12542 . . . . . . . . . . . . . . . 16 1 ∈ ℕ0
208207a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → 1 ∈ ℕ0)
20919, 12, 13, 49coe1addfv 22268 . . . . . . . . . . . . . . 15 (((𝐾 ∈ Ring ∧ (2 𝑌) ∈ (Base‘𝑃) ∧ (((𝑈𝐴) 𝑌) (𝑈𝐵)) ∈ (Base‘𝑃)) ∧ 1 ∈ ℕ0) → ((coe1‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))‘1) = (((coe1‘(2 𝑌))‘1)(+g𝐾)((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘1)))
21018, 33, 44, 208, 209syl31anc 1375 . . . . . . . . . . . . . 14 (𝜑 → ((coe1‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))‘1) = (((coe1‘(2 𝑌))‘1)(+g𝐾)((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘1)))
21171neii 2942 . . . . . . . . . . . . . . . . . 18 ¬ 1 = 2
212 eqeq1 2741 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 1 → (𝑖 = 2 ↔ 1 = 2))
213212notbid 318 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 1 → (¬ 𝑖 = 2 ↔ ¬ 1 = 2))
214213adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 = 1) → (¬ 𝑖 = 2 ↔ ¬ 1 = 2))
215211, 214mpbiri 258 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 = 1) → ¬ 𝑖 = 2)
216215iffalsed 4536 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 = 1) → if(𝑖 = 2, (1r𝐾), (0g𝐾)) = (0g𝐾))
21754, 216, 208, 77fvmptd 7023 . . . . . . . . . . . . . . 15 (𝜑 → ((coe1‘(2 𝑌))‘1) = (0g𝐾))
21819, 12, 13, 49coe1addfv 22268 . . . . . . . . . . . . . . . . 17 (((𝐾 ∈ Ring ∧ ((𝑈𝐴) 𝑌) ∈ (Base‘𝑃) ∧ (𝑈𝐵) ∈ (Base‘𝑃)) ∧ 1 ∈ ℕ0) → ((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘1) = (((coe1‘((𝑈𝐴) 𝑌))‘1)(+g𝐾)((coe1‘(𝑈𝐵))‘1)))
21918, 41, 43, 208, 218syl31anc 1375 . . . . . . . . . . . . . . . 16 (𝜑 → ((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘1) = (((coe1‘((𝑈𝐴) 𝑌))‘1)(+g𝐾)((coe1‘(𝑈𝐵))‘1)))
22019, 12, 66, 35, 34, 67coe1sclmulfv 22286 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ Ring ∧ (𝐴 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝑃)) ∧ 1 ∈ ℕ0) → ((coe1‘((𝑈𝐴) 𝑌))‘1) = (𝐴(.r𝐾)((coe1𝑌)‘1)))
22118, 65, 32, 208, 220syl121anc 1377 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((coe1‘((𝑈𝐴) 𝑌))‘1) = (𝐴(.r𝐾)((coe1𝑌)‘1)))
222 simpr 484 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 = 1) → 𝑖 = 1)
223222iftrued 4533 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 = 1) → if(𝑖 = 1, (1r𝐾), (0g𝐾)) = (1r𝐾))
22470, 223, 208, 57fvmptd 7023 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((coe1𝑌)‘1) = (1r𝐾))
225224oveq2d 7447 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴(.r𝐾)((coe1𝑌)‘1)) = (𝐴(.r𝐾)(1r𝐾)))
22666, 67, 53, 18, 65ringridmd 20270 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴(.r𝐾)(1r𝐾)) = 𝐴)
227221, 225, 2263eqtrd 2781 . . . . . . . . . . . . . . . . 17 (𝜑 → ((coe1‘((𝑈𝐴) 𝑌))‘1) = 𝐴)
228 0ne1 12337 . . . . . . . . . . . . . . . . . . . . . 22 0 ≠ 1
229228nesymi 2998 . . . . . . . . . . . . . . . . . . . . 21 ¬ 1 = 0
230 eqeq1 2741 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 1 → (𝑖 = 0 ↔ 1 = 0))
231229, 230mtbiri 327 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 1 → ¬ 𝑖 = 0)
232231adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 = 1) → ¬ 𝑖 = 0)
233232iffalsed 4536 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 = 1) → if(𝑖 = 0, 𝐵, (0g𝐾)) = (0g𝐾))
23484, 233, 208, 77fvmptd 7023 . . . . . . . . . . . . . . . . 17 (𝜑 → ((coe1‘(𝑈𝐵))‘1) = (0g𝐾))
235227, 234oveq12d 7449 . . . . . . . . . . . . . . . 16 (𝜑 → (((coe1‘((𝑈𝐴) 𝑌))‘1)(+g𝐾)((coe1‘(𝑈𝐵))‘1)) = (𝐴(+g𝐾)(0g𝐾)))
23666, 49, 52, 93, 65grpridd 18988 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐴(+g𝐾)(0g𝐾)) = 𝐴)
237219, 235, 2363eqtrd 2781 . . . . . . . . . . . . . . 15 (𝜑 → ((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘1) = 𝐴)
238217, 237oveq12d 7449 . . . . . . . . . . . . . 14 (𝜑 → (((coe1‘(2 𝑌))‘1)(+g𝐾)((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘1)) = ((0g𝐾)(+g𝐾)𝐴))
23966, 49, 52, 93, 65grplidd 18987 . . . . . . . . . . . . . 14 (𝜑 → ((0g𝐾)(+g𝐾)𝐴) = 𝐴)
240210, 238, 2393eqtrd 2781 . . . . . . . . . . . . 13 (𝜑 → ((coe1‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))‘1) = 𝐴)
241206, 240eqtrid 2789 . . . . . . . . . . . 12 (𝜑 → ((coe1𝐺)‘1) = 𝐴)
242241oveq1d 7446 . . . . . . . . . . 11 (𝜑 → (((coe1𝐺)‘1) · 𝑋) = (𝐴 · 𝑋))
24347fveq1i 6907 . . . . . . . . . . . 12 ((coe1𝐺)‘0) = ((coe1‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))‘0)
244 0nn0 12541 . . . . . . . . . . . . . . 15 0 ∈ ℕ0
245244a1i 11 . . . . . . . . . . . . . 14 (𝜑 → 0 ∈ ℕ0)
24619, 12, 13, 49coe1addfv 22268 . . . . . . . . . . . . . 14 (((𝐾 ∈ Ring ∧ (2 𝑌) ∈ (Base‘𝑃) ∧ (((𝑈𝐴) 𝑌) (𝑈𝐵)) ∈ (Base‘𝑃)) ∧ 0 ∈ ℕ0) → ((coe1‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))‘0) = (((coe1‘(2 𝑌))‘0)(+g𝐾)((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘0)))
24718, 33, 44, 245, 246syl31anc 1375 . . . . . . . . . . . . 13 (𝜑 → ((coe1‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))‘0) = (((coe1‘(2 𝑌))‘0)(+g𝐾)((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘0)))
24888adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑖 = 0) → ¬ 𝑖 = 2)
249248iffalsed 4536 . . . . . . . . . . . . . . 15 ((𝜑𝑖 = 0) → if(𝑖 = 2, (1r𝐾), (0g𝐾)) = (0g𝐾))
25054, 249, 245, 77fvmptd 7023 . . . . . . . . . . . . . 14 (𝜑 → ((coe1‘(2 𝑌))‘0) = (0g𝐾))
25119, 12, 13, 49coe1addfv 22268 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ Ring ∧ ((𝑈𝐴) 𝑌) ∈ (Base‘𝑃) ∧ (𝑈𝐵) ∈ (Base‘𝑃)) ∧ 0 ∈ ℕ0) → ((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘0) = (((coe1‘((𝑈𝐴) 𝑌))‘0)(+g𝐾)((coe1‘(𝑈𝐵))‘0)))
25218, 41, 43, 245, 251syl31anc 1375 . . . . . . . . . . . . . . 15 (𝜑 → ((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘0) = (((coe1‘((𝑈𝐴) 𝑌))‘0)(+g𝐾)((coe1‘(𝑈𝐵))‘0)))
25319, 12, 66, 35, 34, 67coe1sclmulfv 22286 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ Ring ∧ (𝐴 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝑃)) ∧ 0 ∈ ℕ0) → ((coe1‘((𝑈𝐴) 𝑌))‘0) = (𝐴(.r𝐾)((coe1𝑌)‘0)))
25418, 65, 32, 245, 253syl121anc 1377 . . . . . . . . . . . . . . . . 17 (𝜑 → ((coe1‘((𝑈𝐴) 𝑌))‘0) = (𝐴(.r𝐾)((coe1𝑌)‘0)))
255228neii 2942 . . . . . . . . . . . . . . . . . . . . . 22 ¬ 0 = 1
256 eqeq1 2741 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = 0 → (𝑖 = 1 ↔ 0 = 1))
257255, 256mtbiri 327 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 0 → ¬ 𝑖 = 1)
258257adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 = 0) → ¬ 𝑖 = 1)
259258iffalsed 4536 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 = 0) → if(𝑖 = 1, (1r𝐾), (0g𝐾)) = (0g𝐾))
26070, 259, 245, 77fvmptd 7023 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((coe1𝑌)‘0) = (0g𝐾))
261260oveq2d 7447 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐴(.r𝐾)((coe1𝑌)‘0)) = (𝐴(.r𝐾)(0g𝐾)))
262254, 261, 803eqtrd 2781 . . . . . . . . . . . . . . . 16 (𝜑 → ((coe1‘((𝑈𝐴) 𝑌))‘0) = (0g𝐾))
26319, 35, 66ply1sclid 22291 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ Ring ∧ 𝐵 ∈ (Base‘𝐾)) → 𝐵 = ((coe1‘(𝑈𝐵))‘0))
26418, 82, 263syl2anc 584 . . . . . . . . . . . . . . . . 17 (𝜑𝐵 = ((coe1‘(𝑈𝐵))‘0))
265264eqcomd 2743 . . . . . . . . . . . . . . . 16 (𝜑 → ((coe1‘(𝑈𝐵))‘0) = 𝐵)
266262, 265oveq12d 7449 . . . . . . . . . . . . . . 15 (𝜑 → (((coe1‘((𝑈𝐴) 𝑌))‘0)(+g𝐾)((coe1‘(𝑈𝐵))‘0)) = ((0g𝐾)(+g𝐾)𝐵))
26766, 49, 52, 93, 82grplidd 18987 . . . . . . . . . . . . . . 15 (𝜑 → ((0g𝐾)(+g𝐾)𝐵) = 𝐵)
268252, 266, 2673eqtrd 2781 . . . . . . . . . . . . . 14 (𝜑 → ((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘0) = 𝐵)
269250, 268oveq12d 7449 . . . . . . . . . . . . 13 (𝜑 → (((coe1‘(2 𝑌))‘0)(+g𝐾)((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘0)) = ((0g𝐾)(+g𝐾)𝐵))
270247, 269, 2673eqtrd 2781 . . . . . . . . . . . 12 (𝜑 → ((coe1‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))‘0) = 𝐵)
271243, 270eqtrid 2789 . . . . . . . . . . 11 (𝜑 → ((coe1𝐺)‘0) = 𝐵)
272242, 271oveq12d 7449 . . . . . . . . . 10 (𝜑 → ((((coe1𝐺)‘1) · 𝑋) + ((coe1𝐺)‘0)) = ((𝐴 · 𝑋) + 𝐵))
273205, 272oveq12d 7449 . . . . . . . . 9 (𝜑 → ((((coe1𝐺)‘2) · (2 𝑋)) + ((((coe1𝐺)‘1) · 𝑋) + ((coe1𝐺)‘0))) = ((2 𝑋) + ((𝐴 · 𝑋) + 𝐵)))
274 rtelextdg2.14 . . . . . . . . 9 (𝜑 → ((2 𝑋) + ((𝐴 · 𝑋) + 𝐵)) = 0 )
275273, 274eqtrd 2777 . . . . . . . 8 (𝜑 → ((((coe1𝐺)‘2) · (2 𝑋)) + ((((coe1𝐺)‘1) · 𝑋) + ((coe1𝐺)‘0))) = 0 )
276180, 197, 2753eqtrd 2781 . . . . . . 7 (𝜑 → (((𝐸 evalSub1 𝐹)‘𝐺)‘𝑋) = 0 )
27710, 173, 276rspcedvdw 3625 . . . . . 6 (𝜑 → ∃𝑝 ∈ (Monic1p𝐾)(((𝐸 evalSub1 𝐹)‘𝑝)‘𝑋) = 0 )
278174, 1, 61, 114, 36, 38elirng 33736 . . . . . 6 (𝜑 → (𝑋 ∈ (𝐸 IntgRing 𝐹) ↔ (𝑋𝑉 ∧ ∃𝑝 ∈ (Monic1p𝐾)(((𝐸 evalSub1 𝐹)‘𝑝)‘𝑋) = 0 )))
2797, 277, 278mpbir2and 713 . . . . 5 (𝜑𝑋 ∈ (𝐸 IntgRing 𝐹))
2801, 2, 3, 4, 5, 6, 279algextdeg 33766 . . . 4 (𝜑 → (𝐿[:]𝐾) = ((deg1𝐸)‘((𝐸 minPoly 𝐹)‘𝑋)))
2811fveq2i 6909 . . . . . . 7 (Poly1𝐾) = (Poly1‘(𝐸s 𝐹))
28219, 281eqtri 2765 . . . . . 6 𝑃 = (Poly1‘(𝐸s 𝐹))
283 eqid 2737 . . . . . 6 {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝑋) = 0 } = {𝑞 ∈ dom (𝐸 evalSub1 𝐹) ∣ (((𝐸 evalSub1 𝐹)‘𝑞)‘𝑋) = 0 }
284 eqid 2737 . . . . . 6 (RSpan‘𝑃) = (RSpan‘𝑃)
285 eqid 2737 . . . . . 6 (idlGen1p‘(𝐸s 𝐹)) = (idlGen1p‘(𝐸s 𝐹))
286174, 282, 61, 5, 6, 7, 114, 283, 284, 285, 4minplycl 33749 . . . . 5 (𝜑 → ((𝐸 minPoly 𝐹)‘𝑋) ∈ (Base‘𝑃))
2871, 3, 19, 12, 286, 38ressdeg1 33591 . . . 4 (𝜑 → ((deg1𝐸)‘((𝐸 minPoly 𝐹)‘𝑋)) = ((deg1𝐾)‘((𝐸 minPoly 𝐹)‘𝑋)))
288280, 287eqtrd 2777 . . 3 (𝜑 → (𝐿[:]𝐾) = ((deg1𝐾)‘((𝐸 minPoly 𝐹)‘𝑋)))
2891fveq2i 6909 . . . 4 (deg1𝐾) = (deg1‘(𝐸s 𝐹))
290174, 282, 61, 5, 6, 7, 114, 4, 289, 120, 12, 276, 46, 132minplymindeg 33751 . . 3 (𝜑 → ((deg1𝐾)‘((𝐸 minPoly 𝐹)‘𝑋)) ≤ ((deg1𝐾)‘𝐺))
291288, 290eqbrtrd 5165 . 2 (𝜑 → (𝐿[:]𝐾) ≤ ((deg1𝐾)‘𝐺))
292291, 168breqtrd 5169 1 (𝜑 → (𝐿[:]𝐾) ≤ 2)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wrex 3070  {crab 3436  Vcvv 3480  cun 3949  cin 3950  wss 3951  ifcif 4525  {csn 4626   class class class wbr 5143  cmpt 5225  dom cdm 5685  cres 5687  cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156  *cxr 11294   < clt 11295  cle 11296  2c2 12321  0cn0 12526  Basecbs 17247  s cress 17274  +gcplusg 17297  .rcmulr 17298  0gc0g 17484  Mndcmnd 18747  Grpcgrp 18951  .gcmg 19085  SubGrpcsubg 19138  mulGrpcmgp 20137  1rcur 20178  Ringcrg 20230  CRingccrg 20231  NzRingcnzr 20512  SubRingcsubrg 20569  DivRingcdr 20729  Fieldcfield 20730  SubDRingcsdrg 20787  RSpancrsp 21217  algSccascl 21872  PwSer1cps1 22176  var1cv1 22177  Poly1cpl1 22178  coe1cco1 22179   evalSub1 ces1 22317  eval1ce1 22318  deg1cdg1 26093  Monic1pcmn1 26165  idlGen1pcig1p 26169   fldGen cfldgen 33312  [:]cextdg 33692   IntgRing cirng 33733   minPoly cminply 33742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681  ax-ac2 10503  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-rpss 7743  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-ec 8747  df-qs 8751  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-inf 9483  df-oi 9550  df-r1 9804  df-rank 9805  df-dju 9941  df-card 9979  df-acn 9982  df-ac 10156  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-xnn0 12600  df-z 12614  df-dec 12734  df-uz 12879  df-ico 13393  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ocomp 17318  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-imas 17553  df-qus 17554  df-mre 17629  df-mrc 17630  df-mri 17631  df-acs 17632  df-proset 18340  df-drs 18341  df-poset 18359  df-ipo 18573  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-nsg 19142  df-eqg 19143  df-ghm 19231  df-gim 19277  df-cntz 19335  df-oppg 19364  df-lsm 19654  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-srg 20184  df-ring 20232  df-cring 20233  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-irred 20359  df-invr 20388  df-dvr 20401  df-rhm 20472  df-nzr 20513  df-subrng 20546  df-subrg 20570  df-rlreg 20694  df-domn 20695  df-idom 20696  df-drng 20731  df-field 20732  df-sdrg 20788  df-lmod 20860  df-lss 20930  df-lsp 20970  df-lmhm 21021  df-lmim 21022  df-lmic 21023  df-lbs 21074  df-lvec 21102  df-sra 21172  df-rgmod 21173  df-lidl 21218  df-rsp 21219  df-2idl 21260  df-lpidl 21332  df-lpir 21333  df-pid 21347  df-cnfld 21365  df-dsmm 21752  df-frlm 21767  df-uvc 21803  df-lindf 21826  df-linds 21827  df-assa 21873  df-asp 21874  df-ascl 21875  df-psr 21929  df-mvr 21930  df-mpl 21931  df-opsr 21933  df-evls 22098  df-evl 22099  df-psr1 22181  df-vr1 22182  df-ply1 22183  df-coe1 22184  df-evls1 22319  df-evl1 22320  df-mdeg 26094  df-deg1 26095  df-mon1 26170  df-uc1p 26171  df-q1p 26172  df-r1p 26173  df-ig1p 26174  df-fldgen 33313  df-mxidl 33488  df-dim 33650  df-fldext 33693  df-extdg 33694  df-irng 33734  df-minply 33743
This theorem is referenced by:  rtelextdg2  33768
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