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Theorem fta1g 25532
Description: The one-sided fundamental theorem of algebra. A polynomial of degree 𝑛 has at most 𝑛 roots. Unlike the real fundamental theorem fta 26429, 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 2736 . 2 (𝐷𝐹) = (𝐷𝐹)
2 fveqeq2 6851 . . . 4 (𝑓 = 𝐹 → ((𝐷𝑓) = (𝐷𝐹) ↔ (𝐷𝐹) = (𝐷𝐹)))
3 fveq2 6842 . . . . . . . 8 (𝑓 = 𝐹 → (𝑂𝑓) = (𝑂𝐹))
43cnveqd 5831 . . . . . . 7 (𝑓 = 𝐹(𝑂𝑓) = (𝑂𝐹))
54imaeq1d 6012 . . . . . 6 (𝑓 = 𝐹 → ((𝑂𝑓) “ {𝑊}) = ((𝑂𝐹) “ {𝑊}))
65fveq2d 6846 . . . . 5 (𝑓 = 𝐹 → (♯‘((𝑂𝑓) “ {𝑊})) = (♯‘((𝑂𝐹) “ {𝑊})))
7 fveq2 6842 . . . . 5 (𝑓 = 𝐹 → (𝐷𝑓) = (𝐷𝐹))
86, 7breq12d 5118 . . . 4 (𝑓 = 𝐹 → ((♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓) ↔ (♯‘((𝑂𝐹) “ {𝑊})) ≤ (𝐷𝐹)))
92, 8imbi12d 344 . . 3 (𝑓 = 𝐹 → (((𝐷𝑓) = (𝐷𝐹) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝐹) = (𝐷𝐹) → (♯‘((𝑂𝐹) “ {𝑊})) ≤ (𝐷𝐹))))
10 fta1g.1 . . . . . 6 (𝜑𝑅 ∈ IDomn)
11 isidom 20774 . . . . . . 7 (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
1211simplbi 498 . . . . . 6 (𝑅 ∈ IDomn → 𝑅 ∈ CRing)
13 crngring 19976 . . . . . 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 25453 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐹0 ) → (𝐷𝐹) ∈ ℕ0)
2214, 15, 16, 21syl3anc 1371 . . . 4 (𝜑 → (𝐷𝐹) ∈ ℕ0)
23 eqeq2 2748 . . . . . . . 8 (𝑥 = 0 → ((𝐷𝑓) = 𝑥 ↔ (𝐷𝑓) = 0))
2423imbi1d 341 . . . . . . 7 (𝑥 = 0 → (((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑓) = 0 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
2524ralbidv 3174 . . . . . 6 (𝑥 = 0 → (∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑓𝐵 ((𝐷𝑓) = 0 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
2625imbi2d 340 . . . . 5 (𝑥 = 0 → ((𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 0 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
27 eqeq2 2748 . . . . . . . 8 (𝑥 = 𝑑 → ((𝐷𝑓) = 𝑥 ↔ (𝐷𝑓) = 𝑑))
2827imbi1d 341 . . . . . . 7 (𝑥 = 𝑑 → (((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
2928ralbidv 3174 . . . . . 6 (𝑥 = 𝑑 → (∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑓𝐵 ((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
3029imbi2d 340 . . . . 5 (𝑥 = 𝑑 → ((𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
31 eqeq2 2748 . . . . . . . 8 (𝑥 = (𝑑 + 1) → ((𝐷𝑓) = 𝑥 ↔ (𝐷𝑓) = (𝑑 + 1)))
3231imbi1d 341 . . . . . . 7 (𝑥 = (𝑑 + 1) → (((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑓) = (𝑑 + 1) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
3332ralbidv 3174 . . . . . 6 (𝑥 = (𝑑 + 1) → (∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑓𝐵 ((𝐷𝑓) = (𝑑 + 1) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
3433imbi2d 340 . . . . 5 (𝑥 = (𝑑 + 1) → ((𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = (𝑑 + 1) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
35 eqeq2 2748 . . . . . . . 8 (𝑥 = (𝐷𝐹) → ((𝐷𝑓) = 𝑥 ↔ (𝐷𝑓) = (𝐷𝐹)))
3635imbi1d 341 . . . . . . 7 (𝑥 = (𝐷𝐹) → (((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑓) = (𝐷𝐹) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
3736ralbidv 3174 . . . . . 6 (𝑥 = (𝐷𝐹) → (∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑓𝐵 ((𝐷𝑓) = (𝐷𝐹) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
3837imbi2d 340 . . . . 5 (𝑥 = (𝐷𝐹) → ((𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = (𝐷𝐹) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
39 simprr 771 . . . . . . . . . . . . . 14 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝐷𝑓) = 0)
40 0nn0 12428 . . . . . . . . . . . . . 14 0 ∈ ℕ0
4139, 40eqeltrdi 2846 . . . . . . . . . . . . 13 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝐷𝑓) ∈ ℕ0)
4212, 13syl 17 . . . . . . . . . . . . . 14 (𝑅 ∈ IDomn → 𝑅 ∈ Ring)
43 simpl 483 . . . . . . . . . . . . . 14 ((𝑓𝐵 ∧ (𝐷𝑓) = 0) → 𝑓𝐵)
4417, 18, 19, 20deg1nn0clb 25455 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 𝑓𝐵) → (𝑓0 ↔ (𝐷𝑓) ∈ ℕ0))
4542, 43, 44syl2an 596 . . . . . . . . . . . . 13 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑓0 ↔ (𝐷𝑓) ∈ ℕ0))
4641, 45mpbird 256 . . . . . . . . . . . 12 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 𝑓0 )
47 simplrr 776 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝐷𝑓) = 0)
48 0le0 12254 . . . . . . . . . . . . . . . . 17 0 ≤ 0
4947, 48eqbrtrdi 5144 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝐷𝑓) ≤ 0)
5042ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑅 ∈ Ring)
51 simplrl 775 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑓𝐵)
52 eqid 2736 . . . . . . . . . . . . . . . . . 18 (algSc‘𝑃) = (algSc‘𝑃)
5317, 18, 20, 52deg1le0 25476 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝑓𝐵) → ((𝐷𝑓) ≤ 0 ↔ 𝑓 = ((algSc‘𝑃)‘((coe1𝑓)‘0))))
5450, 51, 53syl2anc 584 . . . . . . . . . . . . . . . 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 6846 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝑂𝑓) = (𝑂‘((algSc‘𝑃)‘((coe1𝑓)‘0))))
5712adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 𝑅 ∈ CRing)
5857adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑅 ∈ CRing)
59 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . 23 (coe1𝑓) = (coe1𝑓)
60 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . 23 (Base‘𝑅) = (Base‘𝑅)
6159, 20, 18, 60coe1f 21582 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓𝐵 → (coe1𝑓):ℕ0⟶(Base‘𝑅))
6251, 61syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (coe1𝑓):ℕ0⟶(Base‘𝑅))
63 ffvelcdm 7032 . . . . . . . . . . . . . . . . . . . . 21 (((coe1𝑓):ℕ0⟶(Base‘𝑅) ∧ 0 ∈ ℕ0) → ((coe1𝑓)‘0) ∈ (Base‘𝑅))
6462, 40, 63sylancl 586 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((coe1𝑓)‘0) ∈ (Base‘𝑅))
65 fta1g.o . . . . . . . . . . . . . . . . . . . . 21 𝑂 = (eval1𝑅)
6665, 18, 60, 52evl1sca 21700 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ CRing ∧ ((coe1𝑓)‘0) ∈ (Base‘𝑅)) → (𝑂‘((algSc‘𝑃)‘((coe1𝑓)‘0))) = ((Base‘𝑅) × {((coe1𝑓)‘0)}))
6758, 64, 66syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝑂‘((algSc‘𝑃)‘((coe1𝑓)‘0))) = ((Base‘𝑅) × {((coe1𝑓)‘0)}))
6856, 67eqtrd 2776 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝑂𝑓) = ((Base‘𝑅) × {((coe1𝑓)‘0)}))
6968fveq1d 6844 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((𝑂𝑓)‘𝑥) = (((Base‘𝑅) × {((coe1𝑓)‘0)})‘𝑥))
70 eqid 2736 . . . . . . . . . . . . . . . . . . . 20 (𝑅s (Base‘𝑅)) = (𝑅s (Base‘𝑅))
71 eqid 2736 . . . . . . . . . . . . . . . . . . . 20 (Base‘(𝑅s (Base‘𝑅))) = (Base‘(𝑅s (Base‘𝑅)))
72 simpl 483 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 𝑅 ∈ IDomn)
73 fvexd 6857 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (Base‘𝑅) ∈ V)
7465, 18, 70, 60evl1rhm 21698 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅s (Base‘𝑅))))
7520, 71rhmf 20158 . . . . . . . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 𝑓𝐵)
7876, 77ffvelcdmd 7036 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑂𝑓) ∈ (Base‘(𝑅s (Base‘𝑅))))
7970, 60, 71, 72, 73, 78pwselbas 17371 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑂𝑓):(Base‘𝑅)⟶(Base‘𝑅))
80 ffn 6668 . . . . . . . . . . . . . . . . . . 19 ((𝑂𝑓):(Base‘𝑅)⟶(Base‘𝑅) → (𝑂𝑓) Fn (Base‘𝑅))
81 fniniseg 7010 . . . . . . . . . . . . . . . . . . 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 500 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((𝑂𝑓)‘𝑥) = 𝑊)
8482simprbda 499 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑥 ∈ (Base‘𝑅))
85 fvex 6855 . . . . . . . . . . . . . . . . . . 19 ((coe1𝑓)‘0) ∈ V
8685fvconst2 7153 . . . . . . . . . . . . . . . . . 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 2785 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((coe1𝑓)‘0) = 𝑊)
8988fveq2d 6846 . . . . . . . . . . . . . . 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 21661 . . . . . . . . . . . . . . . 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 2780 . . . . . . . . . . . . . 14 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑓 = 0 )
9493ex 413 . . . . . . . . . . . . 13 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) → 𝑓 = 0 ))
9594necon3ad 2956 . . . . . . . . . . . 12 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑓0 → ¬ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})))
9646, 95mpd 15 . . . . . . . . . . 11 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → ¬ 𝑥 ∈ ((𝑂𝑓) “ {𝑊}))
9796eq0rdv 4364 . . . . . . . . . 10 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → ((𝑂𝑓) “ {𝑊}) = ∅)
9897fveq2d 6846 . . . . . . . . 9 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (♯‘((𝑂𝑓) “ {𝑊})) = (♯‘∅))
99 hash0 14267 . . . . . . . . 9 (♯‘∅) = 0
10098, 99eqtrdi 2792 . . . . . . . 8 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (♯‘((𝑂𝑓) “ {𝑊})) = 0)
10148, 39breqtrrid 5143 . . . . . . . 8 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 0 ≤ (𝐷𝑓))
102100, 101eqbrtrd 5127 . . . . . . 7 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))
103102expr 457 . . . . . 6 ((𝑅 ∈ IDomn ∧ 𝑓𝐵) → ((𝐷𝑓) = 0 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))
104103ralrimiva 3143 . . . . 5 (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 0 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))
105 fveqeq2 6851 . . . . . . . . . 10 (𝑓 = 𝑔 → ((𝐷𝑓) = 𝑑 ↔ (𝐷𝑔) = 𝑑))
106 fveq2 6842 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → (𝑂𝑓) = (𝑂𝑔))
107106cnveqd 5831 . . . . . . . . . . . . 13 (𝑓 = 𝑔(𝑂𝑓) = (𝑂𝑔))
108107imaeq1d 6012 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝑂𝑓) “ {𝑊}) = ((𝑂𝑔) “ {𝑊}))
109108fveq2d 6846 . . . . . . . . . . 11 (𝑓 = 𝑔 → (♯‘((𝑂𝑓) “ {𝑊})) = (♯‘((𝑂𝑔) “ {𝑊})))
110 fveq2 6842 . . . . . . . . . . 11 (𝑓 = 𝑔 → (𝐷𝑓) = (𝐷𝑔))
111109, 110breq12d 5118 . . . . . . . . . 10 (𝑓 = 𝑔 → ((♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓) ↔ (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))
112105, 111imbi12d 344 . . . . . . . . 9 (𝑓 = 𝑔 → (((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔))))
113112cbvralvw 3225 . . . . . . . 8 (∀𝑓𝐵 ((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))
114 simprr 771 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (𝐷𝑓) = (𝑑 + 1))
115 peano2nn0 12453 . . . . . . . . . . . . . . . . 17 (𝑑 ∈ ℕ0 → (𝑑 + 1) ∈ ℕ0)
116115ad2antlr 725 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (𝑑 + 1) ∈ ℕ0)
117114, 116eqeltrd 2838 . . . . . . . . . . . . . . 15 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (𝐷𝑓) ∈ ℕ0)
118117nn0ge0d 12476 . . . . . . . . . . . . . 14 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → 0 ≤ (𝐷𝑓))
119 fveq2 6842 . . . . . . . . . . . . . . . 16 (((𝑂𝑓) “ {𝑊}) = ∅ → (♯‘((𝑂𝑓) “ {𝑊})) = (♯‘∅))
120119, 99eqtrdi 2792 . . . . . . . . . . . . . . 15 (((𝑂𝑓) “ {𝑊}) = ∅ → (♯‘((𝑂𝑓) “ {𝑊})) = 0)
121120breq1d 5115 . . . . . . . . . . . . . 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 4306 . . . . . . . . . . . . 13 (((𝑂𝑓) “ {𝑊}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝑂𝑓) “ {𝑊}))
125 simplll 773 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → 𝑅 ∈ IDomn)
126 simplrl 775 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → 𝑓𝐵)
127 eqid 2736 . . . . . . . . . . . . . . . 16 (var1𝑅) = (var1𝑅)
128 eqid 2736 . . . . . . . . . . . . . . . 16 (-g𝑃) = (-g𝑃)
129 eqid 2736 . . . . . . . . . . . . . . . 16 ((var1𝑅)(-g𝑃)((algSc‘𝑃)‘𝑥)) = ((var1𝑅)(-g𝑃)((algSc‘𝑃)‘𝑥))
130 simpllr 774 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → 𝑑 ∈ ℕ0)
131 simplrr 776 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → (𝐷𝑓) = (𝑑 + 1))
132 simprl 769 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → 𝑥 ∈ ((𝑂𝑓) “ {𝑊}))
133 simprr 771 . . . . . . . . . . . . . . . 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 25531 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))
135134exp32 421 . . . . . . . . . . . . . 14 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
136135exlimdv 1936 . . . . . . . . . . . . 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 3031 . . . . . . . . . . 11 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))
139138expr 457 . . . . . . . . . 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 3151 . . . . . . . 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 414 . . . . . 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 12598 . . . 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 3582 . 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 396   = wceq 1541  wex 1781  wcel 2106  wne 2943  wral 3064  Vcvv 3445  c0 4282  {csn 4586   class class class wbr 5105   × cxp 5631  ccnv 5632  cima 5636   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  0cc0 11051  1c1 11052   + caddc 11054  cle 11190  0cn0 12413  chash 14230  Basecbs 17083  0gc0g 17321  s cpws 17328  -gcsg 18750  Ringcrg 19964  CRingccrg 19965   RingHom crh 20143  Domncdomn 20750  IDomncidom 20751  algSccascl 21258  var1cv1 21547  Poly1cpl1 21548  coe1cco1 21549  eval1ce1 21680   deg1 cdg1 25416
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-xnn0 12486  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-srg 19918  df-ring 19966  df-cring 19967  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-rnghom 20146  df-subrg 20220  df-lmod 20324  df-lss 20393  df-lsp 20433  df-nzr 20728  df-rlreg 20753  df-domn 20754  df-idom 20755  df-cnfld 20797  df-assa 21259  df-asp 21260  df-ascl 21261  df-psr 21311  df-mvr 21312  df-mpl 21313  df-opsr 21315  df-evls 21482  df-evl 21483  df-psr1 21551  df-vr1 21552  df-ply1 21553  df-coe1 21554  df-evl1 21682  df-mdeg 25417  df-deg1 25418  df-mon1 25495  df-uc1p 25496  df-q1p 25497  df-r1p 25498
This theorem is referenced by:  fta1b  25534  lgsqrlem4  26697  idomrootle  41508
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