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Theorem rtelextdg2lem 33732
Description: Lemma for rtelextdg2 33733: 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 2735 . . . . 5 (deg1𝐸) = (deg1𝐸)
4 eqid 2735 . . . . 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 6907 . . . . . . . . 9 (𝑝 = 𝐺 → ((𝐸 evalSub1 𝐹)‘𝑝) = ((𝐸 evalSub1 𝐹)‘𝐺))
98fveq1d 6909 . . . . . . . 8 (𝑝 = 𝐺 → (((𝐸 evalSub1 𝐹)‘𝑝)‘𝑋) = (((𝐸 evalSub1 𝐹)‘𝐺)‘𝑋))
109eqeq1d 2737 . . . . . . 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 2735 . . . . . . . . . 10 (Base‘𝑃) = (Base‘𝑃)
13 rtelextdg2lem.2 . . . . . . . . . 10 = (+g𝑃)
14 fldsdrgfld 20816 . . . . . . . . . . . . . . . 16 ((𝐸 ∈ Field ∧ 𝐹 ∈ (SubDRing‘𝐸)) → (𝐸s 𝐹) ∈ Field)
155, 6, 14syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑 → (𝐸s 𝐹) ∈ Field)
1615fldcrngd 20759 . . . . . . . . . . . . . 14 (𝜑 → (𝐸s 𝐹) ∈ CRing)
171, 16eqeltrid 2843 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ CRing)
1817crngringd 20264 . . . . . . . . . . . 12 (𝜑𝐾 ∈ Ring)
19 rtelextdg2.4 . . . . . . . . . . . . 13 𝑃 = (Poly1𝐾)
2019ply1ring 22265 . . . . . . . . . . . 12 (𝐾 ∈ Ring → 𝑃 ∈ Ring)
2118, 20syl 17 . . . . . . . . . . 11 (𝜑𝑃 ∈ Ring)
2221ringgrpd 20260 . . . . . . . . . 10 (𝜑𝑃 ∈ Grp)
23 eqid 2735 . . . . . . . . . . . 12 (mulGrp‘𝑃) = (mulGrp‘𝑃)
2423, 12mgpbas 20158 . . . . . . . . . . 11 (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
25 rtelextdg2lem.4 . . . . . . . . . . 11 = (.g‘(mulGrp‘𝑃))
2623ringmgp 20257 . . . . . . . . . . . 12 (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
2721, 26syl 17 . . . . . . . . . . 11 (𝜑 → (mulGrp‘𝑃) ∈ Mnd)
28 2nn0 12541 . . . . . . . . . . . 12 2 ∈ ℕ0
2928a1i 11 . . . . . . . . . . 11 (𝜑 → 2 ∈ ℕ0)
30 rtelextdg2lem.1 . . . . . . . . . . . . 13 𝑌 = (var1𝐾)
3130, 19, 12vr1cl 22235 . . . . . . . . . . . 12 (𝐾 ∈ Ring → 𝑌 ∈ (Base‘𝑃))
3218, 31syl 17 . . . . . . . . . . 11 (𝜑𝑌 ∈ (Base‘𝑃))
3324, 25, 27, 29, 32mulgnn0cld 19126 . . . . . . . . . 10 (𝜑 → (2 𝑌) ∈ (Base‘𝑃))
34 rtelextdg2lem.3 . . . . . . . . . . . 12 = (.r𝑃)
35 rtelextdg2lem.5 . . . . . . . . . . . . 13 𝑈 = (algSc‘𝑃)
365fldcrngd 20759 . . . . . . . . . . . . 13 (𝜑𝐸 ∈ CRing)
37 sdrgsubrg 20809 . . . . . . . . . . . . . 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 33576 . . . . . . . . . . . 12 (𝜑 → (𝑈𝐴) ∈ (Base‘𝑃))
4112, 34, 21, 40, 32ringcld 20277 . . . . . . . . . . 11 (𝜑 → ((𝑈𝐴) 𝑌) ∈ (Base‘𝑃))
42 rtelextdg2.13 . . . . . . . . . . . 12 (𝜑𝐵𝐹)
4319, 1, 35, 12, 36, 38, 42ressasclcl 33576 . . . . . . . . . . 11 (𝜑 → (𝑈𝐵) ∈ (Base‘𝑃))
4412, 13, 22, 41, 43grpcld 18978 . . . . . . . . . 10 (𝜑 → (((𝑈𝐴) 𝑌) (𝑈𝐵)) ∈ (Base‘𝑃))
4512, 13, 22, 33, 44grpcld 18978 . . . . . . . . 9 (𝜑 → ((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))) ∈ (Base‘𝑃))
4611, 45eqeltrid 2843 . . . . . . . 8 (𝜑𝐺 ∈ (Base‘𝑃))
4711fveq2i 6910 . . . . . . . . . . . 12 (coe1𝐺) = (coe1‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))
4847fveq1i 6908 . . . . . . . . . . 11 ((coe1𝐺)‘2) = ((coe1‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))‘2)
49 eqid 2735 . . . . . . . . . . . . . 14 (+g𝐾) = (+g𝐾)
5019, 12, 13, 49coe1addfv 22284 . . . . . . . . . . . . 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 1372 . . . . . . . . . . . 12 (𝜑 → ((coe1‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))‘2) = (((coe1‘(2 𝑌))‘2)(+g𝐾)((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘2)))
52 eqid 2735 . . . . . . . . . . . . . . 15 (0g𝐾) = (0g𝐾)
53 eqid 2735 . . . . . . . . . . . . . . 15 (1r𝐾) = (1r𝐾)
5419, 30, 25, 18, 29, 52, 53coe1mon 33590 . . . . . . . . . . . . . 14 (𝜑 → (coe1‘(2 𝑌)) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 2, (1r𝐾), (0g𝐾))))
55 simpr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑖 = 2) → 𝑖 = 2)
5655iftrued 4539 . . . . . . . . . . . . . 14 ((𝜑𝑖 = 2) → if(𝑖 = 2, (1r𝐾), (0g𝐾)) = (1r𝐾))
57 fvexd 6922 . . . . . . . . . . . . . 14 (𝜑 → (1r𝐾) ∈ V)
5854, 56, 29, 57fvmptd 7023 . . . . . . . . . . . . 13 (𝜑 → ((coe1‘(2 𝑌))‘2) = (1r𝐾))
5919, 12, 13, 49coe1addfv 22284 . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . 14 (𝜑 → ((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘2) = (((coe1‘((𝑈𝐴) 𝑌))‘2)(+g𝐾)((coe1‘(𝑈𝐵))‘2)))
61 rtelextdg2.5 . . . . . . . . . . . . . . . . . . . 20 𝑉 = (Base‘𝐸)
6261sdrgss 20811 . . . . . . . . . . . . . . . . . . 19 (𝐹 ∈ (SubDRing‘𝐸) → 𝐹𝑉)
631, 61ressbas2 17283 . . . . . . . . . . . . . . . . . . 19 (𝐹𝑉𝐹 = (Base‘𝐾))
646, 62, 633syl 18 . . . . . . . . . . . . . . . . . 18 (𝜑𝐹 = (Base‘𝐾))
6539, 64eleqtrd 2841 . . . . . . . . . . . . . . . . 17 (𝜑𝐴 ∈ (Base‘𝐾))
66 eqid 2735 . . . . . . . . . . . . . . . . . 18 (Base‘𝐾) = (Base‘𝐾)
67 eqid 2735 . . . . . . . . . . . . . . . . . 18 (.r𝐾) = (.r𝐾)
6819, 12, 66, 35, 34, 67coe1sclmulfv 22302 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Ring ∧ (𝐴 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝑃)) ∧ 2 ∈ ℕ0) → ((coe1‘((𝑈𝐴) 𝑌))‘2) = (𝐴(.r𝐾)((coe1𝑌)‘2)))
6918, 65, 32, 29, 68syl121anc 1374 . . . . . . . . . . . . . . . 16 (𝜑 → ((coe1‘((𝑈𝐴) 𝑌))‘2) = (𝐴(.r𝐾)((coe1𝑌)‘2)))
7019, 30, 18, 52, 53coe1vr1 33593 . . . . . . . . . . . . . . . . . 18 (𝜑 → (coe1𝑌) = (𝑖 ∈ ℕ0 ↦ if(𝑖 = 1, (1r𝐾), (0g𝐾))))
71 1ne2 12472 . . . . . . . . . . . . . . . . . . . . . 22 1 ≠ 2
7271nesymi 2996 . . . . . . . . . . . . . . . . . . . . 21 ¬ 2 = 1
73 eqeq1 2739 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 2 → (𝑖 = 1 ↔ 2 = 1))
7472, 73mtbiri 327 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 2 → ¬ 𝑖 = 1)
7574adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 = 2) → ¬ 𝑖 = 1)
7675iffalsed 4542 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑖 = 2) → if(𝑖 = 1, (1r𝐾), (0g𝐾)) = (0g𝐾))
77 fvexd 6922 . . . . . . . . . . . . . . . . . 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 20310 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐴(.r𝐾)(0g𝐾)) = (0g𝐾))
8169, 79, 803eqtrd 2779 . . . . . . . . . . . . . . 15 (𝜑 → ((coe1‘((𝑈𝐴) 𝑌))‘2) = (0g𝐾))
8242, 64eleqtrd 2841 . . . . . . . . . . . . . . . . 17 (𝜑𝐵 ∈ (Base‘𝐾))
8319, 35, 66, 52coe1scl 22306 . . . . . . . . . . . . . . . . 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 12471 . . . . . . . . . . . . . . . . . . . 20 0 ≠ 2
8685neii 2940 . . . . . . . . . . . . . . . . . . 19 ¬ 0 = 2
87 eqeq1 2739 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 0 → (𝑖 = 2 ↔ 0 = 2))
8886, 87mtbiri 327 . . . . . . . . . . . . . . . . . 18 (𝑖 = 0 → ¬ 𝑖 = 2)
8988, 55nsyl3 138 . . . . . . . . . . . . . . . . 17 ((𝜑𝑖 = 2) → ¬ 𝑖 = 0)
9089iffalsed 4542 . . . . . . . . . . . . . . . 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 20260 . . . . . . . . . . . . . . 15 (𝜑𝐾 ∈ Grp)
9466, 52grpidcl 18996 . . . . . . . . . . . . . . . 16 (𝐾 ∈ Grp → (0g𝐾) ∈ (Base‘𝐾))
9593, 94syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (0g𝐾) ∈ (Base‘𝐾))
9666, 49, 52, 93, 95grpridd 19001 . . . . . . . . . . . . . 14 (𝜑 → ((0g𝐾)(+g𝐾)(0g𝐾)) = (0g𝐾))
9760, 92, 963eqtrd 2779 . . . . . . . . . . . . 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 20280 . . . . . . . . . . . . . . 15 (𝐾 ∈ Ring → (1r𝐾) ∈ (Base‘𝐾))
10018, 99syl 17 . . . . . . . . . . . . . 14 (𝜑 → (1r𝐾) ∈ (Base‘𝐾))
10166, 49, 52, 93, 100grpridd 19001 . . . . . . . . . . . . 13 (𝜑 → ((1r𝐾)(+g𝐾)(0g𝐾)) = (1r𝐾))
10236crngringd 20264 . . . . . . . . . . . . . 14 (𝜑𝐸 ∈ Ring)
103 eqid 2735 . . . . . . . . . . . . . . . 16 (1r𝐸) = (1r𝐸)
104103subrg1cl 20597 . . . . . . . . . . . . . . 15 (𝐹 ∈ (SubRing‘𝐸) → (1r𝐸) ∈ 𝐹)
10538, 104syl 17 . . . . . . . . . . . . . 14 (𝜑 → (1r𝐸) ∈ 𝐹)
1066, 62syl 17 . . . . . . . . . . . . . 14 (𝜑𝐹𝑉)
1071, 61, 103ress1r 33224 . . . . . . . . . . . . . 14 ((𝐸 ∈ Ring ∧ (1r𝐸) ∈ 𝐹𝐹𝑉) → (1r𝐸) = (1r𝐾))
108102, 105, 106, 107syl3anc 1370 . . . . . . . . . . . . 13 (𝜑 → (1r𝐸) = (1r𝐾))
109101, 108eqtr4d 2778 . . . . . . . . . . . 12 (𝜑 → ((1r𝐾)(+g𝐾)(0g𝐾)) = (1r𝐸))
11051, 98, 1093eqtrd 2779 . . . . . . . . . . 11 (𝜑 → ((coe1‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))‘2) = (1r𝐸))
11148, 110eqtrid 2787 . . . . . . . . . 10 (𝜑 → ((coe1𝐺)‘2) = (1r𝐸))
1125flddrngd 20758 . . . . . . . . . . 11 (𝜑𝐸 ∈ DivRing)
113 drngnzr 20765 . . . . . . . . . . 11 (𝐸 ∈ DivRing → 𝐸 ∈ NzRing)
114 rtelextdg2.3 . . . . . . . . . . . 12 0 = (0g𝐸)
115103, 114nzrnz 20532 . . . . . . . . . . 11 (𝐸 ∈ NzRing → (1r𝐸) ≠ 0 )
116112, 113, 1153syl 18 . . . . . . . . . 10 (𝜑 → (1r𝐸) ≠ 0 )
117111, 116eqnetrd 3006 . . . . . . . . 9 (𝜑 → ((coe1𝐺)‘2) ≠ 0 )
118 fveq2 6907 . . . . . . . . . . 11 (𝐺 = (0g𝑃) → (coe1𝐺) = (coe1‘(0g𝑃)))
119118fveq1d 6909 . . . . . . . . . 10 (𝐺 = (0g𝑃) → ((coe1𝐺)‘2) = ((coe1‘(0g𝑃))‘2))
120 eqid 2735 . . . . . . . . . . . 12 (0g𝑃) = (0g𝑃)
12119, 120, 52, 18, 29coe1zfv 33592 . . . . . . . . . . 11 (𝜑 → ((coe1‘(0g𝑃))‘2) = (0g𝐾))
122102ringgrpd 20260 . . . . . . . . . . . . 13 (𝜑𝐸 ∈ Grp)
123122grpmndd 18977 . . . . . . . . . . . 12 (𝜑𝐸 ∈ Mnd)
124 subrgsubg 20594 . . . . . . . . . . . . . 14 (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
12538, 124syl 17 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ (SubGrp‘𝐸))
126114subg0cl 19165 . . . . . . . . . . . . 13 (𝐹 ∈ (SubGrp‘𝐸) → 0𝐹)
127125, 126syl 17 . . . . . . . . . . . 12 (𝜑0𝐹)
1281, 61, 114ress0g 18788 . . . . . . . . . . . 12 ((𝐸 ∈ Mnd ∧ 0𝐹𝐹𝑉) → 0 = (0g𝐾))
129123, 127, 106, 128syl3anc 1370 . . . . . . . . . . 11 (𝜑0 = (0g𝐾))
130121, 129eqtr4d 2778 . . . . . . . . . 10 (𝜑 → ((coe1‘(0g𝑃))‘2) = 0 )
131119, 130sylan9eqr 2797 . . . . . . . . 9 ((𝜑𝐺 = (0g𝑃)) → ((coe1𝐺)‘2) = 0 )
132117, 131mteqand 3031 . . . . . . . 8 (𝜑𝐺 ≠ (0g𝑃))
13311fveq2i 6910 . . . . . . . . . . 11 ((deg1𝐾)‘𝐺) = ((deg1𝐾)‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))
134 eqid 2735 . . . . . . . . . . . . 13 (deg1𝐾) = (deg1𝐾)
135 2re 12338 . . . . . . . . . . . . . . . . 17 2 ∈ ℝ
136135rexri 11317 . . . . . . . . . . . . . . . 16 2 ∈ ℝ*
137136a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → 2 ∈ ℝ*)
138134, 19, 12deg1xrcl 26136 . . . . . . . . . . . . . . . . 17 (((𝑈𝐴) 𝑌) ∈ (Base‘𝑃) → ((deg1𝐾)‘((𝑈𝐴) 𝑌)) ∈ ℝ*)
13941, 138syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((deg1𝐾)‘((𝑈𝐴) 𝑌)) ∈ ℝ*)
140 1xr 11318 . . . . . . . . . . . . . . . . 17 1 ∈ ℝ*
141140a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → 1 ∈ ℝ*)
142134, 19, 66, 12, 34, 35deg1mul3le 26171 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ Ring ∧ 𝐴 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝑃)) → ((deg1𝐾)‘((𝑈𝐴) 𝑌)) ≤ ((deg1𝐾)‘𝑌))
14318, 65, 32, 142syl3anc 1370 . . . . . . . . . . . . . . . . 17 (𝜑 → ((deg1𝐾)‘((𝑈𝐴) 𝑌)) ≤ ((deg1𝐾)‘𝑌))
1441, 15eqeltrid 2843 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐾 ∈ Field)
145144flddrngd 20758 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐾 ∈ DivRing)
146 drngnzr 20765 . . . . . . . . . . . . . . . . . . 19 (𝐾 ∈ DivRing → 𝐾 ∈ NzRing)
147145, 146syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝐾 ∈ NzRing)
148134, 19, 30, 147deg1vr 33594 . . . . . . . . . . . . . . . . 17 (𝜑 → ((deg1𝐾)‘𝑌) = 1)
149143, 148breqtrd 5174 . . . . . . . . . . . . . . . 16 (𝜑 → ((deg1𝐾)‘((𝑈𝐴) 𝑌)) ≤ 1)
150 1lt2 12435 . . . . . . . . . . . . . . . . 17 1 < 2
151150a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → 1 < 2)
152139, 141, 137, 149, 151xrlelttrd 13199 . . . . . . . . . . . . . . 15 (𝜑 → ((deg1𝐾)‘((𝑈𝐴) 𝑌)) < 2)
153134, 19, 12deg1xrcl 26136 . . . . . . . . . . . . . . . . 17 ((𝑈𝐵) ∈ (Base‘𝑃) → ((deg1𝐾)‘(𝑈𝐵)) ∈ ℝ*)
15443, 153syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((deg1𝐾)‘(𝑈𝐵)) ∈ ℝ*)
155 0xr 11306 . . . . . . . . . . . . . . . . 17 0 ∈ ℝ*
156155a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → 0 ∈ ℝ*)
157134, 19, 66, 35deg1sclle 26166 . . . . . . . . . . . . . . . . 17 ((𝐾 ∈ Ring ∧ 𝐵 ∈ (Base‘𝐾)) → ((deg1𝐾)‘(𝑈𝐵)) ≤ 0)
15818, 82, 157syl2anc 584 . . . . . . . . . . . . . . . 16 (𝜑 → ((deg1𝐾)‘(𝑈𝐵)) ≤ 0)
159 2pos 12367 . . . . . . . . . . . . . . . . 17 0 < 2
160159a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → 0 < 2)
161154, 156, 137, 158, 160xrlelttrd 13199 . . . . . . . . . . . . . . 15 (𝜑 → ((deg1𝐾)‘(𝑈𝐵)) < 2)
16219, 134, 18, 12, 13, 41, 43, 137, 152, 161deg1addlt 33600 . . . . . . . . . . . . . 14 (𝜑 → ((deg1𝐾)‘(((𝑈𝐴) 𝑌) (𝑈𝐵))) < 2)
163134, 19, 30, 23, 25deg1pw 26175 . . . . . . . . . . . . . . 15 ((𝐾 ∈ NzRing ∧ 2 ∈ ℕ0) → ((deg1𝐾)‘(2 𝑌)) = 2)
164147, 29, 163syl2anc 584 . . . . . . . . . . . . . 14 (𝜑 → ((deg1𝐾)‘(2 𝑌)) = 2)
165162, 164breqtrrd 5176 . . . . . . . . . . . . 13 (𝜑 → ((deg1𝐾)‘(((𝑈𝐴) 𝑌) (𝑈𝐵))) < ((deg1𝐾)‘(2 𝑌)))
16619, 134, 18, 12, 13, 33, 44, 165deg1add 26157 . . . . . . . . . . . 12 (𝜑 → ((deg1𝐾)‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵)))) = ((deg1𝐾)‘(2 𝑌)))
167166, 164eqtrd 2775 . . . . . . . . . . 11 (𝜑 → ((deg1𝐾)‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵)))) = 2)
168133, 167eqtrid 2787 . . . . . . . . . 10 (𝜑 → ((deg1𝐾)‘𝐺) = 2)
169168fveq2d 6911 . . . . . . . . 9 (𝜑 → ((coe1𝐺)‘((deg1𝐾)‘𝐺)) = ((coe1𝐺)‘2))
170169, 111, 1083eqtrd 2779 . . . . . . . 8 (𝜑 → ((coe1𝐺)‘((deg1𝐾)‘𝐺)) = (1r𝐾))
171 eqid 2735 . . . . . . . . 9 (Monic1p𝐾) = (Monic1p𝐾)
17219, 12, 120, 134, 171, 53ismon1p 26197 . . . . . . . 8 (𝐺 ∈ (Monic1p𝐾) ↔ (𝐺 ∈ (Base‘𝑃) ∧ 𝐺 ≠ (0g𝑃) ∧ ((coe1𝐺)‘((deg1𝐾)‘𝐺)) = (1r𝐾)))
17346, 132, 170, 172syl3anbrc 1342 . . . . . . 7 (𝜑𝐺 ∈ (Monic1p𝐾))
174 eqid 2735 . . . . . . . . . . . 12 (𝐸 evalSub1 𝐹) = (𝐸 evalSub1 𝐹)
175 eqid 2735 . . . . . . . . . . . 12 (eval1𝐸) = (eval1𝐸)
176174, 61, 19, 1, 12, 175, 36, 38ressply1evl 22390 . . . . . . . . . . 11 (𝜑 → (𝐸 evalSub1 𝐹) = ((eval1𝐸) ↾ (Base‘𝑃)))
177176fveq1d 6909 . . . . . . . . . 10 (𝜑 → ((𝐸 evalSub1 𝐹)‘𝐺) = (((eval1𝐸) ↾ (Base‘𝑃))‘𝐺))
17846fvresd 6927 . . . . . . . . . 10 (𝜑 → (((eval1𝐸) ↾ (Base‘𝑃))‘𝐺) = ((eval1𝐸)‘𝐺))
179177, 178eqtrd 2775 . . . . . . . . 9 (𝜑 → ((𝐸 evalSub1 𝐹)‘𝐺) = ((eval1𝐸)‘𝐺))
180179fveq1d 6909 . . . . . . . 8 (𝜑 → (((𝐸 evalSub1 𝐹)‘𝐺)‘𝑋) = (((eval1𝐸)‘𝐺)‘𝑋))
181 eqid 2735 . . . . . . . . 9 (Poly1𝐸) = (Poly1𝐸)
182 eqid 2735 . . . . . . . . 9 (Base‘(Poly1𝐸)) = (Base‘(Poly1𝐸))
183 rtelextdg2.6 . . . . . . . . 9 · = (.r𝐸)
184 rtelextdg2.7 . . . . . . . . 9 + = (+g𝐸)
185 rtelextdg2.8 . . . . . . . . 9 = (.g‘(mulGrp‘𝐸))
186 eqid 2735 . . . . . . . . 9 (coe1𝐺) = (coe1𝐺)
187 eqid 2735 . . . . . . . . 9 ((coe1𝐺)‘2) = ((coe1𝐺)‘2)
188 eqid 2735 . . . . . . . . 9 ((coe1𝐺)‘1) = ((coe1𝐺)‘1)
189 eqid 2735 . . . . . . . . 9 ((coe1𝐺)‘0) = ((coe1𝐺)‘0)
190 eqid 2735 . . . . . . . . . . . 12 (PwSer1𝐾) = (PwSer1𝐾)
191 eqid 2735 . . . . . . . . . . . 12 (Base‘(PwSer1𝐾)) = (Base‘(PwSer1𝐾))
192181, 1, 19, 12, 38, 190, 191, 182ressply1bas2 22245 . . . . . . . . . . 11 (𝜑 → (Base‘𝑃) = ((Base‘(PwSer1𝐾)) ∩ (Base‘(Poly1𝐸))))
19346, 192eleqtrd 2841 . . . . . . . . . 10 (𝜑𝐺 ∈ ((Base‘(PwSer1𝐾)) ∩ (Base‘(Poly1𝐸))))
194193elin2d 4215 . . . . . . . . 9 (𝜑𝐺 ∈ (Base‘(Poly1𝐸)))
1951, 3, 19, 12, 46, 38ressdeg1 33571 . . . . . . . . . 10 (𝜑 → ((deg1𝐸)‘𝐺) = ((deg1𝐾)‘𝐺))
196195, 168eqtrd 2775 . . . . . . . . 9 (𝜑 → ((deg1𝐸)‘𝐺) = 2)
197181, 175, 61, 182, 183, 184, 185, 186, 3, 187, 188, 189, 36, 194, 196, 7evl1deg2 33582 . . . . . . . 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 2735 . . . . . . . . . . . . . 14 (mulGrp‘𝐸) = (mulGrp‘𝐸)
200199, 61mgpbas 20158 . . . . . . . . . . . . 13 𝑉 = (Base‘(mulGrp‘𝐸))
201199ringmgp 20257 . . . . . . . . . . . . . 14 (𝐸 ∈ Ring → (mulGrp‘𝐸) ∈ Mnd)
202102, 201syl 17 . . . . . . . . . . . . 13 (𝜑 → (mulGrp‘𝐸) ∈ Mnd)
203200, 185, 202, 29, 7mulgnn0cld 19126 . . . . . . . . . . . 12 (𝜑 → (2 𝑋) ∈ 𝑉)
20461, 183, 103, 102, 203ringlidmd 20286 . . . . . . . . . . 11 (𝜑 → ((1r𝐸) · (2 𝑋)) = (2 𝑋))
205198, 204eqtrd 2775 . . . . . . . . . 10 (𝜑 → (((coe1𝐺)‘2) · (2 𝑋)) = (2 𝑋))
20647fveq1i 6908 . . . . . . . . . . . . 13 ((coe1𝐺)‘1) = ((coe1‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))‘1)
207 1nn0 12540 . . . . . . . . . . . . . . . 16 1 ∈ ℕ0
208207a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → 1 ∈ ℕ0)
20919, 12, 13, 49coe1addfv 22284 . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . 14 (𝜑 → ((coe1‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))‘1) = (((coe1‘(2 𝑌))‘1)(+g𝐾)((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘1)))
21171neii 2940 . . . . . . . . . . . . . . . . . 18 ¬ 1 = 2
212 eqeq1 2739 . . . . . . . . . . . . . . . . . . . 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 4542 . . . . . . . . . . . . . . . 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 22284 . . . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . . . 16 (𝜑 → ((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘1) = (((coe1‘((𝑈𝐴) 𝑌))‘1)(+g𝐾)((coe1‘(𝑈𝐵))‘1)))
22019, 12, 66, 35, 34, 67coe1sclmulfv 22302 . . . . . . . . . . . . . . . . . . 19 ((𝐾 ∈ Ring ∧ (𝐴 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝑃)) ∧ 1 ∈ ℕ0) → ((coe1‘((𝑈𝐴) 𝑌))‘1) = (𝐴(.r𝐾)((coe1𝑌)‘1)))
22118, 65, 32, 208, 220syl121anc 1374 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((coe1‘((𝑈𝐴) 𝑌))‘1) = (𝐴(.r𝐾)((coe1𝑌)‘1)))
222 simpr 484 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑖 = 1) → 𝑖 = 1)
223222iftrued 4539 . . . . . . . . . . . . . . . . . . . 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 20287 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐴(.r𝐾)(1r𝐾)) = 𝐴)
227221, 225, 2263eqtrd 2779 . . . . . . . . . . . . . . . . 17 (𝜑 → ((coe1‘((𝑈𝐴) 𝑌))‘1) = 𝐴)
228 0ne1 12335 . . . . . . . . . . . . . . . . . . . . . 22 0 ≠ 1
229228nesymi 2996 . . . . . . . . . . . . . . . . . . . . 21 ¬ 1 = 0
230 eqeq1 2739 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 1 → (𝑖 = 0 ↔ 1 = 0))
231229, 230mtbiri 327 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 1 → ¬ 𝑖 = 0)
232231adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑖 = 1) → ¬ 𝑖 = 0)
233232iffalsed 4542 . . . . . . . . . . . . . . . . . 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 19001 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐴(+g𝐾)(0g𝐾)) = 𝐴)
237219, 235, 2363eqtrd 2779 . . . . . . . . . . . . . . 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 19000 . . . . . . . . . . . . . 14 (𝜑 → ((0g𝐾)(+g𝐾)𝐴) = 𝐴)
240210, 238, 2393eqtrd 2779 . . . . . . . . . . . . 13 (𝜑 → ((coe1‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))‘1) = 𝐴)
241206, 240eqtrid 2787 . . . . . . . . . . . 12 (𝜑 → ((coe1𝐺)‘1) = 𝐴)
242241oveq1d 7446 . . . . . . . . . . 11 (𝜑 → (((coe1𝐺)‘1) · 𝑋) = (𝐴 · 𝑋))
24347fveq1i 6908 . . . . . . . . . . . 12 ((coe1𝐺)‘0) = ((coe1‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))‘0)
244 0nn0 12539 . . . . . . . . . . . . . . 15 0 ∈ ℕ0
245244a1i 11 . . . . . . . . . . . . . 14 (𝜑 → 0 ∈ ℕ0)
24619, 12, 13, 49coe1addfv 22284 . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . 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 4542 . . . . . . . . . . . . . . 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 22284 . . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . . 15 (𝜑 → ((coe1‘(((𝑈𝐴) 𝑌) (𝑈𝐵)))‘0) = (((coe1‘((𝑈𝐴) 𝑌))‘0)(+g𝐾)((coe1‘(𝑈𝐵))‘0)))
25319, 12, 66, 35, 34, 67coe1sclmulfv 22302 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ Ring ∧ (𝐴 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝑃)) ∧ 0 ∈ ℕ0) → ((coe1‘((𝑈𝐴) 𝑌))‘0) = (𝐴(.r𝐾)((coe1𝑌)‘0)))
25418, 65, 32, 245, 253syl121anc 1374 . . . . . . . . . . . . . . . . 17 (𝜑 → ((coe1‘((𝑈𝐴) 𝑌))‘0) = (𝐴(.r𝐾)((coe1𝑌)‘0)))
255228neii 2940 . . . . . . . . . . . . . . . . . . . . . 22 ¬ 0 = 1
256 eqeq1 2739 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = 0 → (𝑖 = 1 ↔ 0 = 1))
257255, 256mtbiri 327 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 0 → ¬ 𝑖 = 1)
258257adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑖 = 0) → ¬ 𝑖 = 1)
259258iffalsed 4542 . . . . . . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . . . . . 16 (𝜑 → ((coe1‘((𝑈𝐴) 𝑌))‘0) = (0g𝐾))
26319, 35, 66ply1sclid 22307 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ Ring ∧ 𝐵 ∈ (Base‘𝐾)) → 𝐵 = ((coe1‘(𝑈𝐵))‘0))
26418, 82, 263syl2anc 584 . . . . . . . . . . . . . . . . 17 (𝜑𝐵 = ((coe1‘(𝑈𝐵))‘0))
265264eqcomd 2741 . . . . . . . . . . . . . . . 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 19000 . . . . . . . . . . . . . . 15 (𝜑 → ((0g𝐾)(+g𝐾)𝐵) = 𝐵)
268252, 266, 2673eqtrd 2779 . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . 12 (𝜑 → ((coe1‘((2 𝑌) (((𝑈𝐴) 𝑌) (𝑈𝐵))))‘0) = 𝐵)
271243, 270eqtrid 2787 . . . . . . . . . . 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 2775 . . . . . . . 8 (𝜑 → ((((coe1𝐺)‘2) · (2 𝑋)) + ((((coe1𝐺)‘1) · 𝑋) + ((coe1𝐺)‘0))) = 0 )
276180, 197, 2753eqtrd 2779 . . . . . . 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 33701 . . . . . 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 33731 . . . 4 (𝜑 → (𝐿[:]𝐾) = ((deg1𝐸)‘((𝐸 minPoly 𝐹)‘𝑋)))
2811fveq2i 6910 . . . . . . 7 (Poly1𝐾) = (Poly1‘(𝐸s 𝐹))
28219, 281eqtri 2763 . . . . . 6 𝑃 = (Poly1‘(𝐸s 𝐹))
283 eqid 2735 . . . . . 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 2735 . . . . . 6 (RSpan‘𝑃) = (RSpan‘𝑃)
285 eqid 2735 . . . . . 6 (idlGen1p‘(𝐸s 𝐹)) = (idlGen1p‘(𝐸s 𝐹))
286174, 282, 61, 5, 6, 7, 114, 283, 284, 285, 4minplycl 33714 . . . . 5 (𝜑 → ((𝐸 minPoly 𝐹)‘𝑋) ∈ (Base‘𝑃))
2871, 3, 19, 12, 286, 38ressdeg1 33571 . . . 4 (𝜑 → ((deg1𝐸)‘((𝐸 minPoly 𝐹)‘𝑋)) = ((deg1𝐾)‘((𝐸 minPoly 𝐹)‘𝑋)))
288280, 287eqtrd 2775 . . 3 (𝜑 → (𝐿[:]𝐾) = ((deg1𝐾)‘((𝐸 minPoly 𝐹)‘𝑋)))
2891fveq2i 6910 . . . 4 (deg1𝐾) = (deg1‘(𝐸s 𝐹))
290174, 282, 61, 5, 6, 7, 114, 4, 289, 120, 12, 276, 46, 132minplymindeg 33716 . . 3 (𝜑 → ((deg1𝐾)‘((𝐸 minPoly 𝐹)‘𝑋)) ≤ ((deg1𝐾)‘𝐺))
291288, 290eqbrtrd 5170 . 2 (𝜑 → (𝐿[:]𝐾) ≤ ((deg1𝐾)‘𝐺))
292291, 168breqtrd 5174 1 (𝜑 → (𝐿[:]𝐾) ≤ 2)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  wrex 3068  {crab 3433  Vcvv 3478  cun 3961  cin 3962  wss 3963  ifcif 4531  {csn 4631   class class class wbr 5148  cmpt 5231  dom cdm 5689  cres 5691  cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154  *cxr 11292   < clt 11293  cle 11294  2c2 12319  0cn0 12524  Basecbs 17245  s cress 17274  +gcplusg 17298  .rcmulr 17299  0gc0g 17486  Mndcmnd 18760  Grpcgrp 18964  .gcmg 19098  SubGrpcsubg 19151  mulGrpcmgp 20152  1rcur 20199  Ringcrg 20251  CRingccrg 20252  NzRingcnzr 20529  SubRingcsubrg 20586  DivRingcdr 20746  Fieldcfield 20747  SubDRingcsdrg 20804  RSpancrsp 21235  algSccascl 21890  PwSer1cps1 22192  var1cv1 22193  Poly1cpl1 22194  coe1cco1 22195   evalSub1 ces1 22333  eval1ce1 22334  deg1cdg1 26108  Monic1pcmn1 26180  idlGen1pcig1p 26184   fldGen cfldgen 33292  [:]cextdg 33669   IntgRing cirng 33698   minPoly cminply 33707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679  ax-ac2 10501  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-rpss 7742  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-ec 8746  df-qs 8750  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-inf 9481  df-oi 9548  df-r1 9802  df-rank 9803  df-dju 9939  df-card 9977  df-acn 9980  df-ac 10154  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-xnn0 12598  df-z 12612  df-dec 12732  df-uz 12877  df-ico 13390  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ocomp 17319  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-imas 17555  df-qus 17556  df-mre 17631  df-mrc 17632  df-mri 17633  df-acs 17634  df-proset 18352  df-drs 18353  df-poset 18371  df-ipo 18586  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-nsg 19155  df-eqg 19156  df-ghm 19244  df-gim 19290  df-cntz 19348  df-oppg 19377  df-lsm 19669  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-srg 20205  df-ring 20253  df-cring 20254  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-irred 20376  df-invr 20405  df-dvr 20418  df-rhm 20489  df-nzr 20530  df-subrng 20563  df-subrg 20587  df-rlreg 20711  df-domn 20712  df-idom 20713  df-drng 20748  df-field 20749  df-sdrg 20805  df-lmod 20877  df-lss 20948  df-lsp 20988  df-lmhm 21039  df-lmim 21040  df-lmic 21041  df-lbs 21092  df-lvec 21120  df-sra 21190  df-rgmod 21191  df-lidl 21236  df-rsp 21237  df-2idl 21278  df-lpidl 21350  df-lpir 21351  df-pid 21365  df-cnfld 21383  df-dsmm 21770  df-frlm 21785  df-uvc 21821  df-lindf 21844  df-linds 21845  df-assa 21891  df-asp 21892  df-ascl 21893  df-psr 21947  df-mvr 21948  df-mpl 21949  df-opsr 21951  df-evls 22116  df-evl 22117  df-psr1 22197  df-vr1 22198  df-ply1 22199  df-coe1 22200  df-evls1 22335  df-evl1 22336  df-mdeg 26109  df-deg1 26110  df-mon1 26185  df-uc1p 26186  df-q1p 26187  df-r1p 26188  df-ig1p 26189  df-fldgen 33293  df-mxidl 33468  df-dim 33627  df-fldext 33670  df-extdg 33671  df-irng 33699  df-minply 33708
This theorem is referenced by:  rtelextdg2  33733
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