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Theorem fta1g 25666
Description: The one-sided fundamental theorem of algebra. A polynomial of degree 𝑛 has at most 𝑛 roots. Unlike the real fundamental theorem fta 26563, which is only true in and other algebraically closed fields, this is true in any integral domain. (Contributed by Mario Carneiro, 12-Jun-2015.)
Hypotheses
Ref Expression
fta1g.p 𝑃 = (Poly1𝑅)
fta1g.b 𝐵 = (Base‘𝑃)
fta1g.d 𝐷 = ( deg1𝑅)
fta1g.o 𝑂 = (eval1𝑅)
fta1g.w 𝑊 = (0g𝑅)
fta1g.z 0 = (0g𝑃)
fta1g.1 (𝜑𝑅 ∈ IDomn)
fta1g.2 (𝜑𝐹𝐵)
fta1g.3 (𝜑𝐹0 )
Assertion
Ref Expression
fta1g (𝜑 → (♯‘((𝑂𝐹) “ {𝑊})) ≤ (𝐷𝐹))

Proof of Theorem fta1g
Dummy variables 𝑓 𝑑 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . 2 (𝐷𝐹) = (𝐷𝐹)
2 fveqeq2 6896 . . . 4 (𝑓 = 𝐹 → ((𝐷𝑓) = (𝐷𝐹) ↔ (𝐷𝐹) = (𝐷𝐹)))
3 fveq2 6887 . . . . . . . 8 (𝑓 = 𝐹 → (𝑂𝑓) = (𝑂𝐹))
43cnveqd 5872 . . . . . . 7 (𝑓 = 𝐹(𝑂𝑓) = (𝑂𝐹))
54imaeq1d 6055 . . . . . 6 (𝑓 = 𝐹 → ((𝑂𝑓) “ {𝑊}) = ((𝑂𝐹) “ {𝑊}))
65fveq2d 6891 . . . . 5 (𝑓 = 𝐹 → (♯‘((𝑂𝑓) “ {𝑊})) = (♯‘((𝑂𝐹) “ {𝑊})))
7 fveq2 6887 . . . . 5 (𝑓 = 𝐹 → (𝐷𝑓) = (𝐷𝐹))
86, 7breq12d 5159 . . . 4 (𝑓 = 𝐹 → ((♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓) ↔ (♯‘((𝑂𝐹) “ {𝑊})) ≤ (𝐷𝐹)))
92, 8imbi12d 345 . . 3 (𝑓 = 𝐹 → (((𝐷𝑓) = (𝐷𝐹) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝐹) = (𝐷𝐹) → (♯‘((𝑂𝐹) “ {𝑊})) ≤ (𝐷𝐹))))
10 fta1g.1 . . . . . 6 (𝜑𝑅 ∈ IDomn)
11 isidom 20906 . . . . . . 7 (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
1211simplbi 499 . . . . . 6 (𝑅 ∈ IDomn → 𝑅 ∈ CRing)
13 crngring 20058 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
1410, 12, 133syl 18 . . . . 5 (𝜑𝑅 ∈ Ring)
15 fta1g.2 . . . . 5 (𝜑𝐹𝐵)
16 fta1g.3 . . . . 5 (𝜑𝐹0 )
17 fta1g.d . . . . . 6 𝐷 = ( deg1𝑅)
18 fta1g.p . . . . . 6 𝑃 = (Poly1𝑅)
19 fta1g.z . . . . . 6 0 = (0g𝑃)
20 fta1g.b . . . . . 6 𝐵 = (Base‘𝑃)
2117, 18, 19, 20deg1nn0cl 25587 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐹0 ) → (𝐷𝐹) ∈ ℕ0)
2214, 15, 16, 21syl3anc 1372 . . . 4 (𝜑 → (𝐷𝐹) ∈ ℕ0)
23 eqeq2 2745 . . . . . . . 8 (𝑥 = 0 → ((𝐷𝑓) = 𝑥 ↔ (𝐷𝑓) = 0))
2423imbi1d 342 . . . . . . 7 (𝑥 = 0 → (((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑓) = 0 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
2524ralbidv 3178 . . . . . 6 (𝑥 = 0 → (∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑓𝐵 ((𝐷𝑓) = 0 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
2625imbi2d 341 . . . . 5 (𝑥 = 0 → ((𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 0 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
27 eqeq2 2745 . . . . . . . 8 (𝑥 = 𝑑 → ((𝐷𝑓) = 𝑥 ↔ (𝐷𝑓) = 𝑑))
2827imbi1d 342 . . . . . . 7 (𝑥 = 𝑑 → (((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
2928ralbidv 3178 . . . . . 6 (𝑥 = 𝑑 → (∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑓𝐵 ((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
3029imbi2d 341 . . . . 5 (𝑥 = 𝑑 → ((𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
31 eqeq2 2745 . . . . . . . 8 (𝑥 = (𝑑 + 1) → ((𝐷𝑓) = 𝑥 ↔ (𝐷𝑓) = (𝑑 + 1)))
3231imbi1d 342 . . . . . . 7 (𝑥 = (𝑑 + 1) → (((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑓) = (𝑑 + 1) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
3332ralbidv 3178 . . . . . 6 (𝑥 = (𝑑 + 1) → (∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑓𝐵 ((𝐷𝑓) = (𝑑 + 1) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
3433imbi2d 341 . . . . 5 (𝑥 = (𝑑 + 1) → ((𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = (𝑑 + 1) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
35 eqeq2 2745 . . . . . . . 8 (𝑥 = (𝐷𝐹) → ((𝐷𝑓) = 𝑥 ↔ (𝐷𝑓) = (𝐷𝐹)))
3635imbi1d 342 . . . . . . 7 (𝑥 = (𝐷𝐹) → (((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑓) = (𝐷𝐹) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
3736ralbidv 3178 . . . . . 6 (𝑥 = (𝐷𝐹) → (∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑓𝐵 ((𝐷𝑓) = (𝐷𝐹) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
3837imbi2d 341 . . . . 5 (𝑥 = (𝐷𝐹) → ((𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = (𝐷𝐹) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
39 simprr 772 . . . . . . . . . . . . . 14 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝐷𝑓) = 0)
40 0nn0 12482 . . . . . . . . . . . . . 14 0 ∈ ℕ0
4139, 40eqeltrdi 2842 . . . . . . . . . . . . 13 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝐷𝑓) ∈ ℕ0)
4212, 13syl 17 . . . . . . . . . . . . . 14 (𝑅 ∈ IDomn → 𝑅 ∈ Ring)
43 simpl 484 . . . . . . . . . . . . . 14 ((𝑓𝐵 ∧ (𝐷𝑓) = 0) → 𝑓𝐵)
4417, 18, 19, 20deg1nn0clb 25589 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 𝑓𝐵) → (𝑓0 ↔ (𝐷𝑓) ∈ ℕ0))
4542, 43, 44syl2an 597 . . . . . . . . . . . . 13 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑓0 ↔ (𝐷𝑓) ∈ ℕ0))
4641, 45mpbird 257 . . . . . . . . . . . 12 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 𝑓0 )
47 simplrr 777 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝐷𝑓) = 0)
48 0le0 12308 . . . . . . . . . . . . . . . . 17 0 ≤ 0
4947, 48eqbrtrdi 5185 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝐷𝑓) ≤ 0)
5042ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑅 ∈ Ring)
51 simplrl 776 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑓𝐵)
52 eqid 2733 . . . . . . . . . . . . . . . . . 18 (algSc‘𝑃) = (algSc‘𝑃)
5317, 18, 20, 52deg1le0 25610 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝑓𝐵) → ((𝐷𝑓) ≤ 0 ↔ 𝑓 = ((algSc‘𝑃)‘((coe1𝑓)‘0))))
5450, 51, 53syl2anc 585 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((𝐷𝑓) ≤ 0 ↔ 𝑓 = ((algSc‘𝑃)‘((coe1𝑓)‘0))))
5549, 54mpbid 231 . . . . . . . . . . . . . . 15 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑓 = ((algSc‘𝑃)‘((coe1𝑓)‘0)))
5655fveq2d 6891 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝑂𝑓) = (𝑂‘((algSc‘𝑃)‘((coe1𝑓)‘0))))
5712adantr 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 𝑅 ∈ CRing)
5857adantr 482 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑅 ∈ CRing)
59 eqid 2733 . . . . . . . . . . . . . . . . . . . . . . 23 (coe1𝑓) = (coe1𝑓)
60 eqid 2733 . . . . . . . . . . . . . . . . . . . . . . 23 (Base‘𝑅) = (Base‘𝑅)
6159, 20, 18, 60coe1f 21716 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓𝐵 → (coe1𝑓):ℕ0⟶(Base‘𝑅))
6251, 61syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (coe1𝑓):ℕ0⟶(Base‘𝑅))
63 ffvelcdm 7078 . . . . . . . . . . . . . . . . . . . . 21 (((coe1𝑓):ℕ0⟶(Base‘𝑅) ∧ 0 ∈ ℕ0) → ((coe1𝑓)‘0) ∈ (Base‘𝑅))
6462, 40, 63sylancl 587 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((coe1𝑓)‘0) ∈ (Base‘𝑅))
65 fta1g.o . . . . . . . . . . . . . . . . . . . . 21 𝑂 = (eval1𝑅)
6665, 18, 60, 52evl1sca 21834 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ CRing ∧ ((coe1𝑓)‘0) ∈ (Base‘𝑅)) → (𝑂‘((algSc‘𝑃)‘((coe1𝑓)‘0))) = ((Base‘𝑅) × {((coe1𝑓)‘0)}))
6758, 64, 66syl2anc 585 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝑂‘((algSc‘𝑃)‘((coe1𝑓)‘0))) = ((Base‘𝑅) × {((coe1𝑓)‘0)}))
6856, 67eqtrd 2773 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝑂𝑓) = ((Base‘𝑅) × {((coe1𝑓)‘0)}))
6968fveq1d 6889 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((𝑂𝑓)‘𝑥) = (((Base‘𝑅) × {((coe1𝑓)‘0)})‘𝑥))
70 eqid 2733 . . . . . . . . . . . . . . . . . . . 20 (𝑅s (Base‘𝑅)) = (𝑅s (Base‘𝑅))
71 eqid 2733 . . . . . . . . . . . . . . . . . . . 20 (Base‘(𝑅s (Base‘𝑅))) = (Base‘(𝑅s (Base‘𝑅)))
72 simpl 484 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 𝑅 ∈ IDomn)
73 fvexd 6902 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (Base‘𝑅) ∈ V)
7465, 18, 70, 60evl1rhm 21832 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅s (Base‘𝑅))))
7520, 71rhmf 20251 . . . . . . . . . . . . . . . . . . . . . 22 (𝑂 ∈ (𝑃 RingHom (𝑅s (Base‘𝑅))) → 𝑂:𝐵⟶(Base‘(𝑅s (Base‘𝑅))))
7657, 74, 753syl 18 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 𝑂:𝐵⟶(Base‘(𝑅s (Base‘𝑅))))
77 simprl 770 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 𝑓𝐵)
7876, 77ffvelcdmd 7082 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑂𝑓) ∈ (Base‘(𝑅s (Base‘𝑅))))
7970, 60, 71, 72, 73, 78pwselbas 17430 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑂𝑓):(Base‘𝑅)⟶(Base‘𝑅))
80 ffn 6713 . . . . . . . . . . . . . . . . . . 19 ((𝑂𝑓):(Base‘𝑅)⟶(Base‘𝑅) → (𝑂𝑓) Fn (Base‘𝑅))
81 fniniseg 7056 . . . . . . . . . . . . . . . . . . 19 ((𝑂𝑓) Fn (Base‘𝑅) → (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ((𝑂𝑓)‘𝑥) = 𝑊)))
8279, 80, 813syl 18 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ((𝑂𝑓)‘𝑥) = 𝑊)))
8382simplbda 501 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((𝑂𝑓)‘𝑥) = 𝑊)
8482simprbda 500 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑥 ∈ (Base‘𝑅))
85 fvex 6900 . . . . . . . . . . . . . . . . . . 19 ((coe1𝑓)‘0) ∈ V
8685fvconst2 7199 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (Base‘𝑅) → (((Base‘𝑅) × {((coe1𝑓)‘0)})‘𝑥) = ((coe1𝑓)‘0))
8784, 86syl 17 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (((Base‘𝑅) × {((coe1𝑓)‘0)})‘𝑥) = ((coe1𝑓)‘0))
8869, 83, 873eqtr3rd 2782 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((coe1𝑓)‘0) = 𝑊)
8988fveq2d 6891 . . . . . . . . . . . . . . 15 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((algSc‘𝑃)‘((coe1𝑓)‘0)) = ((algSc‘𝑃)‘𝑊))
90 fta1g.w . . . . . . . . . . . . . . . . 17 𝑊 = (0g𝑅)
9118, 52, 90, 19ply1scl0 21793 . . . . . . . . . . . . . . . 16 (𝑅 ∈ Ring → ((algSc‘𝑃)‘𝑊) = 0 )
9250, 91syl 17 . . . . . . . . . . . . . . 15 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((algSc‘𝑃)‘𝑊) = 0 )
9355, 89, 923eqtrd 2777 . . . . . . . . . . . . . 14 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑓 = 0 )
9493ex 414 . . . . . . . . . . . . 13 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) → 𝑓 = 0 ))
9594necon3ad 2954 . . . . . . . . . . . 12 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑓0 → ¬ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})))
9646, 95mpd 15 . . . . . . . . . . 11 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → ¬ 𝑥 ∈ ((𝑂𝑓) “ {𝑊}))
9796eq0rdv 4402 . . . . . . . . . 10 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → ((𝑂𝑓) “ {𝑊}) = ∅)
9897fveq2d 6891 . . . . . . . . 9 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (♯‘((𝑂𝑓) “ {𝑊})) = (♯‘∅))
99 hash0 14322 . . . . . . . . 9 (♯‘∅) = 0
10098, 99eqtrdi 2789 . . . . . . . 8 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (♯‘((𝑂𝑓) “ {𝑊})) = 0)
10148, 39breqtrrid 5184 . . . . . . . 8 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 0 ≤ (𝐷𝑓))
102100, 101eqbrtrd 5168 . . . . . . 7 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))
103102expr 458 . . . . . 6 ((𝑅 ∈ IDomn ∧ 𝑓𝐵) → ((𝐷𝑓) = 0 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))
104103ralrimiva 3147 . . . . 5 (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 0 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))
105 fveqeq2 6896 . . . . . . . . . 10 (𝑓 = 𝑔 → ((𝐷𝑓) = 𝑑 ↔ (𝐷𝑔) = 𝑑))
106 fveq2 6887 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → (𝑂𝑓) = (𝑂𝑔))
107106cnveqd 5872 . . . . . . . . . . . . 13 (𝑓 = 𝑔(𝑂𝑓) = (𝑂𝑔))
108107imaeq1d 6055 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝑂𝑓) “ {𝑊}) = ((𝑂𝑔) “ {𝑊}))
109108fveq2d 6891 . . . . . . . . . . 11 (𝑓 = 𝑔 → (♯‘((𝑂𝑓) “ {𝑊})) = (♯‘((𝑂𝑔) “ {𝑊})))
110 fveq2 6887 . . . . . . . . . . 11 (𝑓 = 𝑔 → (𝐷𝑓) = (𝐷𝑔))
111109, 110breq12d 5159 . . . . . . . . . 10 (𝑓 = 𝑔 → ((♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓) ↔ (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))
112105, 111imbi12d 345 . . . . . . . . 9 (𝑓 = 𝑔 → (((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔))))
113112cbvralvw 3235 . . . . . . . 8 (∀𝑓𝐵 ((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))
114 simprr 772 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (𝐷𝑓) = (𝑑 + 1))
115 peano2nn0 12507 . . . . . . . . . . . . . . . . 17 (𝑑 ∈ ℕ0 → (𝑑 + 1) ∈ ℕ0)
116115ad2antlr 726 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (𝑑 + 1) ∈ ℕ0)
117114, 116eqeltrd 2834 . . . . . . . . . . . . . . 15 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (𝐷𝑓) ∈ ℕ0)
118117nn0ge0d 12530 . . . . . . . . . . . . . 14 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → 0 ≤ (𝐷𝑓))
119 fveq2 6887 . . . . . . . . . . . . . . . 16 (((𝑂𝑓) “ {𝑊}) = ∅ → (♯‘((𝑂𝑓) “ {𝑊})) = (♯‘∅))
120119, 99eqtrdi 2789 . . . . . . . . . . . . . . 15 (((𝑂𝑓) “ {𝑊}) = ∅ → (♯‘((𝑂𝑓) “ {𝑊})) = 0)
121120breq1d 5156 . . . . . . . . . . . . . 14 (((𝑂𝑓) “ {𝑊}) = ∅ → ((♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓) ↔ 0 ≤ (𝐷𝑓)))
122118, 121syl5ibrcom 246 . . . . . . . . . . . . 13 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (((𝑂𝑓) “ {𝑊}) = ∅ → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))
123122a1dd 50 . . . . . . . . . . . 12 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (((𝑂𝑓) “ {𝑊}) = ∅ → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
124 n0 4344 . . . . . . . . . . . . 13 (((𝑂𝑓) “ {𝑊}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝑂𝑓) “ {𝑊}))
125 simplll 774 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → 𝑅 ∈ IDomn)
126 simplrl 776 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → 𝑓𝐵)
127 eqid 2733 . . . . . . . . . . . . . . . 16 (var1𝑅) = (var1𝑅)
128 eqid 2733 . . . . . . . . . . . . . . . 16 (-g𝑃) = (-g𝑃)
129 eqid 2733 . . . . . . . . . . . . . . . 16 ((var1𝑅)(-g𝑃)((algSc‘𝑃)‘𝑥)) = ((var1𝑅)(-g𝑃)((algSc‘𝑃)‘𝑥))
130 simpllr 775 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → 𝑑 ∈ ℕ0)
131 simplrr 777 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → (𝐷𝑓) = (𝑑 + 1))
132 simprl 770 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → 𝑥 ∈ ((𝑂𝑓) “ {𝑊}))
133 simprr 772 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))
13418, 20, 17, 65, 90, 19, 125, 126, 60, 127, 128, 52, 129, 130, 131, 132, 133fta1glem2 25665 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))
135134exp32 422 . . . . . . . . . . . . . 14 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
136135exlimdv 1937 . . . . . . . . . . . . 13 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (∃𝑥 𝑥 ∈ ((𝑂𝑓) “ {𝑊}) → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
137124, 136biimtrid 241 . . . . . . . . . . . 12 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (((𝑂𝑓) “ {𝑊}) ≠ ∅ → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
138123, 137pm2.61dne 3029 . . . . . . . . . . 11 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))
139138expr 458 . . . . . . . . . 10 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ 𝑓𝐵) → ((𝐷𝑓) = (𝑑 + 1) → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
140139com23 86 . . . . . . . . 9 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ 𝑓𝐵) → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → ((𝐷𝑓) = (𝑑 + 1) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
141140ralrimdva 3155 . . . . . . . 8 ((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → ∀𝑓𝐵 ((𝐷𝑓) = (𝑑 + 1) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
142113, 141biimtrid 241 . . . . . . 7 ((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) → (∀𝑓𝐵 ((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) → ∀𝑓𝐵 ((𝐷𝑓) = (𝑑 + 1) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
143142expcom 415 . . . . . 6 (𝑑 ∈ ℕ0 → (𝑅 ∈ IDomn → (∀𝑓𝐵 ((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) → ∀𝑓𝐵 ((𝐷𝑓) = (𝑑 + 1) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
144143a2d 29 . . . . 5 (𝑑 ∈ ℕ0 → ((𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))) → (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = (𝑑 + 1) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
14526, 30, 34, 38, 104, 144nn0ind 12652 . . . 4 ((𝐷𝐹) ∈ ℕ0 → (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = (𝐷𝐹) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
14622, 10, 145sylc 65 . . 3 (𝜑 → ∀𝑓𝐵 ((𝐷𝑓) = (𝐷𝐹) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))
1479, 146, 15rspcdva 3612 . 2 (𝜑 → ((𝐷𝐹) = (𝐷𝐹) → (♯‘((𝑂𝐹) “ {𝑊})) ≤ (𝐷𝐹)))
1481, 147mpi 20 1 (𝜑 → (♯‘((𝑂𝐹) “ {𝑊})) ≤ (𝐷𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wne 2941  wral 3062  Vcvv 3475  c0 4320  {csn 4626   class class class wbr 5146   × cxp 5672  ccnv 5673  cima 5677   Fn wfn 6534  wf 6535  cfv 6539  (class class class)co 7403  0cc0 11105  1c1 11106   + caddc 11108  cle 11244  0cn0 12467  chash 14285  Basecbs 17139  0gc0g 17380  s cpws 17387  -gcsg 18816  Ringcrg 20046  CRingccrg 20047   RingHom crh 20236  Domncdomn 20882  IDomncidom 20883  algSccascl 21390  var1cv1 21681  Poly1cpl1 21682  coe1cco1 21683  eval1ce1 21814   deg1 cdg1 25550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5283  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182  ax-pre-sup 11183  ax-addf 11184  ax-mulf 11185
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4907  df-int 4949  df-iun 4997  df-iin 4998  df-br 5147  df-opab 5209  df-mpt 5230  df-tr 5264  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-se 5630  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6296  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-isom 6548  df-riota 7359  df-ov 7406  df-oprab 7407  df-mpo 7408  df-of 7664  df-ofr 7665  df-om 7850  df-1st 7969  df-2nd 7970  df-supp 8141  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8365  df-rdg 8404  df-1o 8460  df-oadd 8464  df-er 8698  df-map 8817  df-pm 8818  df-ixp 8887  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-fsupp 9357  df-sup 9432  df-oi 9500  df-dju 9891  df-card 9929  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11441  df-neg 11442  df-nn 12208  df-2 12270  df-3 12271  df-4 12272  df-5 12273  df-6 12274  df-7 12275  df-8 12276  df-9 12277  df-n0 12468  df-xnn0 12540  df-z 12554  df-dec 12673  df-uz 12818  df-fz 13480  df-fzo 13623  df-seq 13962  df-hash 14286  df-struct 17075  df-sets 17092  df-slot 17110  df-ndx 17122  df-base 17140  df-ress 17169  df-plusg 17205  df-mulr 17206  df-starv 17207  df-sca 17208  df-vsca 17209  df-ip 17210  df-tset 17211  df-ple 17212  df-ds 17214  df-unif 17215  df-hom 17216  df-cco 17217  df-0g 17382  df-gsum 17383  df-prds 17388  df-pws 17390  df-mre 17525  df-mrc 17526  df-acs 17528  df-mgm 18556  df-sgrp 18605  df-mnd 18621  df-mhm 18666  df-submnd 18667  df-grp 18817  df-minusg 18818  df-sbg 18819  df-mulg 18944  df-subg 18996  df-ghm 19083  df-cntz 19174  df-cmn 19642  df-abl 19643  df-mgp 19979  df-ur 19996  df-srg 20000  df-ring 20048  df-cring 20049  df-oppr 20138  df-dvdsr 20159  df-unit 20160  df-invr 20190  df-rnghom 20239  df-nzr 20280  df-subrg 20348  df-lmod 20460  df-lss 20530  df-lsp 20570  df-rlreg 20885  df-domn 20886  df-idom 20887  df-cnfld 20929  df-assa 21391  df-asp 21392  df-ascl 21393  df-psr 21443  df-mvr 21444  df-mpl 21445  df-opsr 21447  df-evls 21616  df-evl 21617  df-psr1 21685  df-vr1 21686  df-ply1 21687  df-coe1 21688  df-evl1 21816  df-mdeg 25551  df-deg1 25552  df-mon1 25629  df-uc1p 25630  df-q1p 25631  df-r1p 25632
This theorem is referenced by:  fta1b  25668  lgsqrlem4  26831  idomrootle  41869
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