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Theorem fta1g 24218
Description: The one-sided fundamental theorem of algebra. A polynomial of degree 𝑛 has at most 𝑛 roots. Unlike the real fundamental theorem fta 25097, 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 2765 . 2 (𝐷𝐹) = (𝐷𝐹)
2 fveqeq2 6384 . . . 4 (𝑓 = 𝐹 → ((𝐷𝑓) = (𝐷𝐹) ↔ (𝐷𝐹) = (𝐷𝐹)))
3 fveq2 6375 . . . . . . . 8 (𝑓 = 𝐹 → (𝑂𝑓) = (𝑂𝐹))
43cnveqd 5466 . . . . . . 7 (𝑓 = 𝐹(𝑂𝑓) = (𝑂𝐹))
54imaeq1d 5647 . . . . . 6 (𝑓 = 𝐹 → ((𝑂𝑓) “ {𝑊}) = ((𝑂𝐹) “ {𝑊}))
65fveq2d 6379 . . . . 5 (𝑓 = 𝐹 → (♯‘((𝑂𝑓) “ {𝑊})) = (♯‘((𝑂𝐹) “ {𝑊})))
7 fveq2 6375 . . . . 5 (𝑓 = 𝐹 → (𝐷𝑓) = (𝐷𝐹))
86, 7breq12d 4822 . . . 4 (𝑓 = 𝐹 → ((♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓) ↔ (♯‘((𝑂𝐹) “ {𝑊})) ≤ (𝐷𝐹)))
92, 8imbi12d 335 . . 3 (𝑓 = 𝐹 → (((𝐷𝑓) = (𝐷𝐹) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝐹) = (𝐷𝐹) → (♯‘((𝑂𝐹) “ {𝑊})) ≤ (𝐷𝐹))))
10 fta1g.1 . . . . . 6 (𝜑𝑅 ∈ IDomn)
11 isidom 19578 . . . . . . 7 (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
1211simplbi 491 . . . . . 6 (𝑅 ∈ IDomn → 𝑅 ∈ CRing)
13 crngring 18825 . . . . . 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 24139 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐹0 ) → (𝐷𝐹) ∈ ℕ0)
2214, 15, 16, 21syl3anc 1490 . . . 4 (𝜑 → (𝐷𝐹) ∈ ℕ0)
23 eqeq2 2776 . . . . . . . 8 (𝑥 = 0 → ((𝐷𝑓) = 𝑥 ↔ (𝐷𝑓) = 0))
2423imbi1d 332 . . . . . . 7 (𝑥 = 0 → (((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑓) = 0 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
2524ralbidv 3133 . . . . . 6 (𝑥 = 0 → (∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑓𝐵 ((𝐷𝑓) = 0 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
2625imbi2d 331 . . . . 5 (𝑥 = 0 → ((𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 0 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
27 eqeq2 2776 . . . . . . . 8 (𝑥 = 𝑑 → ((𝐷𝑓) = 𝑥 ↔ (𝐷𝑓) = 𝑑))
2827imbi1d 332 . . . . . . 7 (𝑥 = 𝑑 → (((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
2928ralbidv 3133 . . . . . 6 (𝑥 = 𝑑 → (∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑓𝐵 ((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
3029imbi2d 331 . . . . 5 (𝑥 = 𝑑 → ((𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
31 eqeq2 2776 . . . . . . . 8 (𝑥 = (𝑑 + 1) → ((𝐷𝑓) = 𝑥 ↔ (𝐷𝑓) = (𝑑 + 1)))
3231imbi1d 332 . . . . . . 7 (𝑥 = (𝑑 + 1) → (((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑓) = (𝑑 + 1) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
3332ralbidv 3133 . . . . . 6 (𝑥 = (𝑑 + 1) → (∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑓𝐵 ((𝐷𝑓) = (𝑑 + 1) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
3433imbi2d 331 . . . . 5 (𝑥 = (𝑑 + 1) → ((𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = (𝑑 + 1) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
35 eqeq2 2776 . . . . . . . 8 (𝑥 = (𝐷𝐹) → ((𝐷𝑓) = 𝑥 ↔ (𝐷𝑓) = (𝐷𝐹)))
3635imbi1d 332 . . . . . . 7 (𝑥 = (𝐷𝐹) → (((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑓) = (𝐷𝐹) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
3736ralbidv 3133 . . . . . 6 (𝑥 = (𝐷𝐹) → (∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑓𝐵 ((𝐷𝑓) = (𝐷𝐹) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
3837imbi2d 331 . . . . 5 (𝑥 = (𝐷𝐹) → ((𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = (𝐷𝐹) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
39 simprr 789 . . . . . . . . . . . . . 14 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝐷𝑓) = 0)
40 0nn0 11555 . . . . . . . . . . . . . 14 0 ∈ ℕ0
4139, 40syl6eqel 2852 . . . . . . . . . . . . 13 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝐷𝑓) ∈ ℕ0)
4212, 13syl 17 . . . . . . . . . . . . . 14 (𝑅 ∈ IDomn → 𝑅 ∈ Ring)
43 simpl 474 . . . . . . . . . . . . . 14 ((𝑓𝐵 ∧ (𝐷𝑓) = 0) → 𝑓𝐵)
4417, 18, 19, 20deg1nn0clb 24141 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 𝑓𝐵) → (𝑓0 ↔ (𝐷𝑓) ∈ ℕ0))
4542, 43, 44syl2an 589 . . . . . . . . . . . . 13 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑓0 ↔ (𝐷𝑓) ∈ ℕ0))
4641, 45mpbird 248 . . . . . . . . . . . 12 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 𝑓0 )
47 simplrr 796 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝐷𝑓) = 0)
48 0le0 11380 . . . . . . . . . . . . . . . . 17 0 ≤ 0
4947, 48syl6eqbr 4848 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝐷𝑓) ≤ 0)
5042ad2antrr 717 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑅 ∈ Ring)
51 simplrl 795 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑓𝐵)
52 eqid 2765 . . . . . . . . . . . . . . . . . 18 (algSc‘𝑃) = (algSc‘𝑃)
5317, 18, 20, 52deg1le0 24162 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝑓𝐵) → ((𝐷𝑓) ≤ 0 ↔ 𝑓 = ((algSc‘𝑃)‘((coe1𝑓)‘0))))
5450, 51, 53syl2anc 579 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((𝐷𝑓) ≤ 0 ↔ 𝑓 = ((algSc‘𝑃)‘((coe1𝑓)‘0))))
5549, 54mpbid 223 . . . . . . . . . . . . . . 15 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑓 = ((algSc‘𝑃)‘((coe1𝑓)‘0)))
5655fveq2d 6379 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝑂𝑓) = (𝑂‘((algSc‘𝑃)‘((coe1𝑓)‘0))))
5712adantr 472 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 𝑅 ∈ CRing)
5857adantr 472 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑅 ∈ CRing)
59 eqid 2765 . . . . . . . . . . . . . . . . . . . . . . 23 (coe1𝑓) = (coe1𝑓)
60 eqid 2765 . . . . . . . . . . . . . . . . . . . . . . 23 (Base‘𝑅) = (Base‘𝑅)
6159, 20, 18, 60coe1f 19854 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓𝐵 → (coe1𝑓):ℕ0⟶(Base‘𝑅))
6251, 61syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (coe1𝑓):ℕ0⟶(Base‘𝑅))
63 ffvelrn 6547 . . . . . . . . . . . . . . . . . . . . 21 (((coe1𝑓):ℕ0⟶(Base‘𝑅) ∧ 0 ∈ ℕ0) → ((coe1𝑓)‘0) ∈ (Base‘𝑅))
6462, 40, 63sylancl 580 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((coe1𝑓)‘0) ∈ (Base‘𝑅))
65 fta1g.o . . . . . . . . . . . . . . . . . . . . 21 𝑂 = (eval1𝑅)
6665, 18, 60, 52evl1sca 19971 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ CRing ∧ ((coe1𝑓)‘0) ∈ (Base‘𝑅)) → (𝑂‘((algSc‘𝑃)‘((coe1𝑓)‘0))) = ((Base‘𝑅) × {((coe1𝑓)‘0)}))
6758, 64, 66syl2anc 579 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝑂‘((algSc‘𝑃)‘((coe1𝑓)‘0))) = ((Base‘𝑅) × {((coe1𝑓)‘0)}))
6856, 67eqtrd 2799 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝑂𝑓) = ((Base‘𝑅) × {((coe1𝑓)‘0)}))
6968fveq1d 6377 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((𝑂𝑓)‘𝑥) = (((Base‘𝑅) × {((coe1𝑓)‘0)})‘𝑥))
70 eqid 2765 . . . . . . . . . . . . . . . . . . . 20 (𝑅s (Base‘𝑅)) = (𝑅s (Base‘𝑅))
71 eqid 2765 . . . . . . . . . . . . . . . . . . . 20 (Base‘(𝑅s (Base‘𝑅))) = (Base‘(𝑅s (Base‘𝑅)))
72 simpl 474 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 𝑅 ∈ IDomn)
73 fvexd 6390 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (Base‘𝑅) ∈ V)
7465, 18, 70, 60evl1rhm 19969 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅s (Base‘𝑅))))
7520, 71rhmf 18995 . . . . . . . . . . . . . . . . . . . . . 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 787 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 𝑓𝐵)
7876, 77ffvelrnd 6550 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑂𝑓) ∈ (Base‘(𝑅s (Base‘𝑅))))
7970, 60, 71, 72, 73, 78pwselbas 16417 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑂𝑓):(Base‘𝑅)⟶(Base‘𝑅))
80 ffn 6223 . . . . . . . . . . . . . . . . . . 19 ((𝑂𝑓):(Base‘𝑅)⟶(Base‘𝑅) → (𝑂𝑓) Fn (Base‘𝑅))
81 fniniseg 6528 . . . . . . . . . . . . . . . . . . 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 493 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((𝑂𝑓)‘𝑥) = 𝑊)
8482simprbda 492 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑥 ∈ (Base‘𝑅))
85 fvex 6388 . . . . . . . . . . . . . . . . . . 19 ((coe1𝑓)‘0) ∈ V
8685fvconst2 6662 . . . . . . . . . . . . . . . . . 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 2808 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((coe1𝑓)‘0) = 𝑊)
8988fveq2d 6379 . . . . . . . . . . . . . . 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 19933 . . . . . . . . . . . . . . . 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 2803 . . . . . . . . . . . . . 14 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑓 = 0 )
9493ex 401 . . . . . . . . . . . . 13 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) → 𝑓 = 0 ))
9594necon3ad 2950 . . . . . . . . . . . 12 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑓0 → ¬ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})))
9646, 95mpd 15 . . . . . . . . . . 11 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → ¬ 𝑥 ∈ ((𝑂𝑓) “ {𝑊}))
9796eq0rdv 4141 . . . . . . . . . 10 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → ((𝑂𝑓) “ {𝑊}) = ∅)
9897fveq2d 6379 . . . . . . . . 9 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (♯‘((𝑂𝑓) “ {𝑊})) = (♯‘∅))
99 hash0 13360 . . . . . . . . 9 (♯‘∅) = 0
10098, 99syl6eq 2815 . . . . . . . 8 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (♯‘((𝑂𝑓) “ {𝑊})) = 0)
10148, 39syl5breqr 4847 . . . . . . . 8 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 0 ≤ (𝐷𝑓))
102100, 101eqbrtrd 4831 . . . . . . 7 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))
103102expr 448 . . . . . 6 ((𝑅 ∈ IDomn ∧ 𝑓𝐵) → ((𝐷𝑓) = 0 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))
104103ralrimiva 3113 . . . . 5 (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 0 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))
105 fveqeq2 6384 . . . . . . . . . 10 (𝑓 = 𝑔 → ((𝐷𝑓) = 𝑑 ↔ (𝐷𝑔) = 𝑑))
106 fveq2 6375 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → (𝑂𝑓) = (𝑂𝑔))
107106cnveqd 5466 . . . . . . . . . . . . 13 (𝑓 = 𝑔(𝑂𝑓) = (𝑂𝑔))
108107imaeq1d 5647 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝑂𝑓) “ {𝑊}) = ((𝑂𝑔) “ {𝑊}))
109108fveq2d 6379 . . . . . . . . . . 11 (𝑓 = 𝑔 → (♯‘((𝑂𝑓) “ {𝑊})) = (♯‘((𝑂𝑔) “ {𝑊})))
110 fveq2 6375 . . . . . . . . . . 11 (𝑓 = 𝑔 → (𝐷𝑓) = (𝐷𝑔))
111109, 110breq12d 4822 . . . . . . . . . 10 (𝑓 = 𝑔 → ((♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓) ↔ (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))
112105, 111imbi12d 335 . . . . . . . . 9 (𝑓 = 𝑔 → (((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔))))
113112cbvralv 3319 . . . . . . . 8 (∀𝑓𝐵 ((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))
114 simprr 789 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (𝐷𝑓) = (𝑑 + 1))
115 peano2nn0 11580 . . . . . . . . . . . . . . . . 17 (𝑑 ∈ ℕ0 → (𝑑 + 1) ∈ ℕ0)
116115ad2antlr 718 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (𝑑 + 1) ∈ ℕ0)
117114, 116eqeltrd 2844 . . . . . . . . . . . . . . 15 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (𝐷𝑓) ∈ ℕ0)
118117nn0ge0d 11601 . . . . . . . . . . . . . 14 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → 0 ≤ (𝐷𝑓))
119 fveq2 6375 . . . . . . . . . . . . . . . 16 (((𝑂𝑓) “ {𝑊}) = ∅ → (♯‘((𝑂𝑓) “ {𝑊})) = (♯‘∅))
120119, 99syl6eq 2815 . . . . . . . . . . . . . . 15 (((𝑂𝑓) “ {𝑊}) = ∅ → (♯‘((𝑂𝑓) “ {𝑊})) = 0)
121120breq1d 4819 . . . . . . . . . . . . . 14 (((𝑂𝑓) “ {𝑊}) = ∅ → ((♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓) ↔ 0 ≤ (𝐷𝑓)))
122118, 121syl5ibrcom 238 . . . . . . . . . . . . 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 4095 . . . . . . . . . . . . 13 (((𝑂𝑓) “ {𝑊}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝑂𝑓) “ {𝑊}))
125 simplll 791 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → 𝑅 ∈ IDomn)
126 simplrl 795 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → 𝑓𝐵)
127 eqid 2765 . . . . . . . . . . . . . . . 16 (var1𝑅) = (var1𝑅)
128 eqid 2765 . . . . . . . . . . . . . . . 16 (-g𝑃) = (-g𝑃)
129 eqid 2765 . . . . . . . . . . . . . . . 16 ((var1𝑅)(-g𝑃)((algSc‘𝑃)‘𝑥)) = ((var1𝑅)(-g𝑃)((algSc‘𝑃)‘𝑥))
130 simpllr 793 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → 𝑑 ∈ ℕ0)
131 simplrr 796 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → (𝐷𝑓) = (𝑑 + 1))
132 simprl 787 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → 𝑥 ∈ ((𝑂𝑓) “ {𝑊}))
133 simprr 789 . . . . . . . . . . . . . . . 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 24217 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))
135134exp32 411 . . . . . . . . . . . . . 14 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
136135exlimdv 2028 . . . . . . . . . . . . 13 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (∃𝑥 𝑥 ∈ ((𝑂𝑓) “ {𝑊}) → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
137124, 136syl5bi 233 . . . . . . . . . . . 12 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (((𝑂𝑓) “ {𝑊}) ≠ ∅ → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
138123, 137pm2.61dne 3023 . . . . . . . . . . 11 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))
139138expr 448 . . . . . . . . . 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 3116 . . . . . . . 8 ((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → ∀𝑓𝐵 ((𝐷𝑓) = (𝑑 + 1) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
142113, 141syl5bi 233 . . . . . . 7 ((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) → (∀𝑓𝐵 ((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) → ∀𝑓𝐵 ((𝐷𝑓) = (𝑑 + 1) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
143142expcom 402 . . . . . 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 11719 . . . 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 3467 . 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 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wne 2937  wral 3055  Vcvv 3350  c0 4079  {csn 4334   class class class wbr 4809   × cxp 5275  ccnv 5276  cima 5280   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842  0cc0 10189  1c1 10190   + caddc 10192  cle 10329  0cn0 11538  chash 13321  Basecbs 16132  0gc0g 16368  s cpws 16375  -gcsg 17693  Ringcrg 18814  CRingccrg 18815   RingHom crh 18981  Domncdomn 19554  IDomncidom 19555  algSccascl 19585  var1cv1 19819  Poly1cpl1 19820  coe1cco1 19821  eval1ce1 19952   deg1 cdg1 24105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267  ax-addf 10268  ax-mulf 10269
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-ofr 7096  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-tpos 7555  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-sup 8555  df-oi 8622  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-xnn0 11611  df-z 11625  df-dec 11741  df-uz 11887  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-struct 16134  df-ndx 16135  df-slot 16136  df-base 16138  df-sets 16139  df-ress 16140  df-plusg 16229  df-mulr 16230  df-starv 16231  df-sca 16232  df-vsca 16233  df-ip 16234  df-tset 16235  df-ple 16236  df-ds 16238  df-unif 16239  df-hom 16240  df-cco 16241  df-0g 16370  df-gsum 16371  df-prds 16376  df-pws 16378  df-mre 16514  df-mrc 16515  df-acs 16517  df-mgm 17510  df-sgrp 17552  df-mnd 17563  df-mhm 17603  df-submnd 17604  df-grp 17694  df-minusg 17695  df-sbg 17696  df-mulg 17810  df-subg 17857  df-ghm 17924  df-cntz 18015  df-cmn 18461  df-abl 18462  df-mgp 18757  df-ur 18769  df-srg 18773  df-ring 18816  df-cring 18817  df-oppr 18890  df-dvdsr 18908  df-unit 18909  df-invr 18939  df-rnghom 18984  df-subrg 19047  df-lmod 19134  df-lss 19202  df-lsp 19244  df-nzr 19532  df-rlreg 19557  df-domn 19558  df-idom 19559  df-assa 19586  df-asp 19587  df-ascl 19588  df-psr 19630  df-mvr 19631  df-mpl 19632  df-opsr 19634  df-evls 19779  df-evl 19780  df-psr1 19823  df-vr1 19824  df-ply1 19825  df-coe1 19826  df-evl1 19954  df-cnfld 20020  df-mdeg 24106  df-deg1 24107  df-mon1 24181  df-uc1p 24182  df-q1p 24183  df-r1p 24184
This theorem is referenced by:  fta1b  24220  lgsqrlem4  25365  idomrootle  38382
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