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Theorem fta1g 24764
 Description: The one-sided fundamental theorem of algebra. A polynomial of degree 𝑛 has at most 𝑛 roots. Unlike the real fundamental theorem fta 25660, which is only true in ℂ 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 2824 . 2 (𝐷𝐹) = (𝐷𝐹)
2 fveqeq2 6682 . . . 4 (𝑓 = 𝐹 → ((𝐷𝑓) = (𝐷𝐹) ↔ (𝐷𝐹) = (𝐷𝐹)))
3 fveq2 6673 . . . . . . . 8 (𝑓 = 𝐹 → (𝑂𝑓) = (𝑂𝐹))
43cnveqd 5749 . . . . . . 7 (𝑓 = 𝐹(𝑂𝑓) = (𝑂𝐹))
54imaeq1d 5931 . . . . . 6 (𝑓 = 𝐹 → ((𝑂𝑓) “ {𝑊}) = ((𝑂𝐹) “ {𝑊}))
65fveq2d 6677 . . . . 5 (𝑓 = 𝐹 → (♯‘((𝑂𝑓) “ {𝑊})) = (♯‘((𝑂𝐹) “ {𝑊})))
7 fveq2 6673 . . . . 5 (𝑓 = 𝐹 → (𝐷𝑓) = (𝐷𝐹))
86, 7breq12d 5082 . . . 4 (𝑓 = 𝐹 → ((♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓) ↔ (♯‘((𝑂𝐹) “ {𝑊})) ≤ (𝐷𝐹)))
92, 8imbi12d 347 . . 3 (𝑓 = 𝐹 → (((𝐷𝑓) = (𝐷𝐹) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝐹) = (𝐷𝐹) → (♯‘((𝑂𝐹) “ {𝑊})) ≤ (𝐷𝐹))))
10 fta1g.1 . . . . . 6 (𝜑𝑅 ∈ IDomn)
11 isidom 20080 . . . . . . 7 (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
1211simplbi 500 . . . . . 6 (𝑅 ∈ IDomn → 𝑅 ∈ CRing)
13 crngring 19311 . . . . . 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 24685 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐹0 ) → (𝐷𝐹) ∈ ℕ0)
2214, 15, 16, 21syl3anc 1367 . . . 4 (𝜑 → (𝐷𝐹) ∈ ℕ0)
23 eqeq2 2836 . . . . . . . 8 (𝑥 = 0 → ((𝐷𝑓) = 𝑥 ↔ (𝐷𝑓) = 0))
2423imbi1d 344 . . . . . . 7 (𝑥 = 0 → (((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑓) = 0 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
2524ralbidv 3200 . . . . . 6 (𝑥 = 0 → (∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑓𝐵 ((𝐷𝑓) = 0 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
2625imbi2d 343 . . . . 5 (𝑥 = 0 → ((𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 0 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
27 eqeq2 2836 . . . . . . . 8 (𝑥 = 𝑑 → ((𝐷𝑓) = 𝑥 ↔ (𝐷𝑓) = 𝑑))
2827imbi1d 344 . . . . . . 7 (𝑥 = 𝑑 → (((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
2928ralbidv 3200 . . . . . 6 (𝑥 = 𝑑 → (∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑓𝐵 ((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
3029imbi2d 343 . . . . 5 (𝑥 = 𝑑 → ((𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
31 eqeq2 2836 . . . . . . . 8 (𝑥 = (𝑑 + 1) → ((𝐷𝑓) = 𝑥 ↔ (𝐷𝑓) = (𝑑 + 1)))
3231imbi1d 344 . . . . . . 7 (𝑥 = (𝑑 + 1) → (((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑓) = (𝑑 + 1) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
3332ralbidv 3200 . . . . . 6 (𝑥 = (𝑑 + 1) → (∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑓𝐵 ((𝐷𝑓) = (𝑑 + 1) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
3433imbi2d 343 . . . . 5 (𝑥 = (𝑑 + 1) → ((𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = (𝑑 + 1) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
35 eqeq2 2836 . . . . . . . 8 (𝑥 = (𝐷𝐹) → ((𝐷𝑓) = 𝑥 ↔ (𝐷𝑓) = (𝐷𝐹)))
3635imbi1d 344 . . . . . . 7 (𝑥 = (𝐷𝐹) → (((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑓) = (𝐷𝐹) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
3736ralbidv 3200 . . . . . 6 (𝑥 = (𝐷𝐹) → (∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑓𝐵 ((𝐷𝑓) = (𝐷𝐹) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
3837imbi2d 343 . . . . 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 11915 . . . . . . . . . . . . . 14 0 ∈ ℕ0
4139, 40eqeltrdi 2924 . . . . . . . . . . . . 13 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝐷𝑓) ∈ ℕ0)
4212, 13syl 17 . . . . . . . . . . . . . 14 (𝑅 ∈ IDomn → 𝑅 ∈ Ring)
43 simpl 485 . . . . . . . . . . . . . 14 ((𝑓𝐵 ∧ (𝐷𝑓) = 0) → 𝑓𝐵)
4417, 18, 19, 20deg1nn0clb 24687 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 𝑓𝐵) → (𝑓0 ↔ (𝐷𝑓) ∈ ℕ0))
4542, 43, 44syl2an 597 . . . . . . . . . . . . 13 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑓0 ↔ (𝐷𝑓) ∈ ℕ0))
4641, 45mpbird 259 . . . . . . . . . . . 12 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 𝑓0 )
47 simplrr 776 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝐷𝑓) = 0)
48 0le0 11741 . . . . . . . . . . . . . . . . 17 0 ≤ 0
4947, 48eqbrtrdi 5108 . . . . . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . . . . . 18 (algSc‘𝑃) = (algSc‘𝑃)
5317, 18, 20, 52deg1le0 24708 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝑓𝐵) → ((𝐷𝑓) ≤ 0 ↔ 𝑓 = ((algSc‘𝑃)‘((coe1𝑓)‘0))))
5450, 51, 53syl2anc 586 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((𝐷𝑓) ≤ 0 ↔ 𝑓 = ((algSc‘𝑃)‘((coe1𝑓)‘0))))
5549, 54mpbid 234 . . . . . . . . . . . . . . 15 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑓 = ((algSc‘𝑃)‘((coe1𝑓)‘0)))
5655fveq2d 6677 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝑂𝑓) = (𝑂‘((algSc‘𝑃)‘((coe1𝑓)‘0))))
5712adantr 483 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 𝑅 ∈ CRing)
5857adantr 483 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑅 ∈ CRing)
59 eqid 2824 . . . . . . . . . . . . . . . . . . . . . . 23 (coe1𝑓) = (coe1𝑓)
60 eqid 2824 . . . . . . . . . . . . . . . . . . . . . . 23 (Base‘𝑅) = (Base‘𝑅)
6159, 20, 18, 60coe1f 20382 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓𝐵 → (coe1𝑓):ℕ0⟶(Base‘𝑅))
6251, 61syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (coe1𝑓):ℕ0⟶(Base‘𝑅))
63 ffvelrn 6852 . . . . . . . . . . . . . . . . . . . . 21 (((coe1𝑓):ℕ0⟶(Base‘𝑅) ∧ 0 ∈ ℕ0) → ((coe1𝑓)‘0) ∈ (Base‘𝑅))
6462, 40, 63sylancl 588 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((coe1𝑓)‘0) ∈ (Base‘𝑅))
65 fta1g.o . . . . . . . . . . . . . . . . . . . . 21 𝑂 = (eval1𝑅)
6665, 18, 60, 52evl1sca 20500 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ CRing ∧ ((coe1𝑓)‘0) ∈ (Base‘𝑅)) → (𝑂‘((algSc‘𝑃)‘((coe1𝑓)‘0))) = ((Base‘𝑅) × {((coe1𝑓)‘0)}))
6758, 64, 66syl2anc 586 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝑂‘((algSc‘𝑃)‘((coe1𝑓)‘0))) = ((Base‘𝑅) × {((coe1𝑓)‘0)}))
6856, 67eqtrd 2859 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝑂𝑓) = ((Base‘𝑅) × {((coe1𝑓)‘0)}))
6968fveq1d 6675 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((𝑂𝑓)‘𝑥) = (((Base‘𝑅) × {((coe1𝑓)‘0)})‘𝑥))
70 eqid 2824 . . . . . . . . . . . . . . . . . . . 20 (𝑅s (Base‘𝑅)) = (𝑅s (Base‘𝑅))
71 eqid 2824 . . . . . . . . . . . . . . . . . . . 20 (Base‘(𝑅s (Base‘𝑅))) = (Base‘(𝑅s (Base‘𝑅)))
72 simpl 485 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 𝑅 ∈ IDomn)
73 fvexd 6688 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (Base‘𝑅) ∈ V)
7465, 18, 70, 60evl1rhm 20498 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅s (Base‘𝑅))))
7520, 71rhmf 19481 . . . . . . . . . . . . . . . . . . . . . 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, 77ffvelrnd 6855 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑂𝑓) ∈ (Base‘(𝑅s (Base‘𝑅))))
7970, 60, 71, 72, 73, 78pwselbas 16765 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑂𝑓):(Base‘𝑅)⟶(Base‘𝑅))
80 ffn 6517 . . . . . . . . . . . . . . . . . . 19 ((𝑂𝑓):(Base‘𝑅)⟶(Base‘𝑅) → (𝑂𝑓) Fn (Base‘𝑅))
81 fniniseg 6833 . . . . . . . . . . . . . . . . . . 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 502 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((𝑂𝑓)‘𝑥) = 𝑊)
8482simprbda 501 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑥 ∈ (Base‘𝑅))
85 fvex 6686 . . . . . . . . . . . . . . . . . . 19 ((coe1𝑓)‘0) ∈ V
8685fvconst2 6969 . . . . . . . . . . . . . . . . . 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 2868 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((coe1𝑓)‘0) = 𝑊)
8988fveq2d 6677 . . . . . . . . . . . . . . 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 20461 . . . . . . . . . . . . . . . 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 2863 . . . . . . . . . . . . . 14 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑓 = 0 )
9493ex 415 . . . . . . . . . . . . 13 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) → 𝑓 = 0 ))
9594necon3ad 3032 . . . . . . . . . . . 12 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑓0 → ¬ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})))
9646, 95mpd 15 . . . . . . . . . . 11 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → ¬ 𝑥 ∈ ((𝑂𝑓) “ {𝑊}))
9796eq0rdv 4360 . . . . . . . . . 10 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → ((𝑂𝑓) “ {𝑊}) = ∅)
9897fveq2d 6677 . . . . . . . . 9 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (♯‘((𝑂𝑓) “ {𝑊})) = (♯‘∅))
99 hash0 13731 . . . . . . . . 9 (♯‘∅) = 0
10098, 99syl6eq 2875 . . . . . . . 8 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (♯‘((𝑂𝑓) “ {𝑊})) = 0)
10148, 39breqtrrid 5107 . . . . . . . 8 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 0 ≤ (𝐷𝑓))
102100, 101eqbrtrd 5091 . . . . . . 7 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))
103102expr 459 . . . . . 6 ((𝑅 ∈ IDomn ∧ 𝑓𝐵) → ((𝐷𝑓) = 0 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))
104103ralrimiva 3185 . . . . 5 (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 0 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))
105 fveqeq2 6682 . . . . . . . . . 10 (𝑓 = 𝑔 → ((𝐷𝑓) = 𝑑 ↔ (𝐷𝑔) = 𝑑))
106 fveq2 6673 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → (𝑂𝑓) = (𝑂𝑔))
107106cnveqd 5749 . . . . . . . . . . . . 13 (𝑓 = 𝑔(𝑂𝑓) = (𝑂𝑔))
108107imaeq1d 5931 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝑂𝑓) “ {𝑊}) = ((𝑂𝑔) “ {𝑊}))
109108fveq2d 6677 . . . . . . . . . . 11 (𝑓 = 𝑔 → (♯‘((𝑂𝑓) “ {𝑊})) = (♯‘((𝑂𝑔) “ {𝑊})))
110 fveq2 6673 . . . . . . . . . . 11 (𝑓 = 𝑔 → (𝐷𝑓) = (𝐷𝑔))
111109, 110breq12d 5082 . . . . . . . . . 10 (𝑓 = 𝑔 → ((♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓) ↔ (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))
112105, 111imbi12d 347 . . . . . . . . 9 (𝑓 = 𝑔 → (((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔))))
113112cbvralvw 3452 . . . . . . . 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 11940 . . . . . . . . . . . . . . . . 17 (𝑑 ∈ ℕ0 → (𝑑 + 1) ∈ ℕ0)
116115ad2antlr 725 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (𝑑 + 1) ∈ ℕ0)
117114, 116eqeltrd 2916 . . . . . . . . . . . . . . 15 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (𝐷𝑓) ∈ ℕ0)
118117nn0ge0d 11961 . . . . . . . . . . . . . 14 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → 0 ≤ (𝐷𝑓))
119 fveq2 6673 . . . . . . . . . . . . . . . 16 (((𝑂𝑓) “ {𝑊}) = ∅ → (♯‘((𝑂𝑓) “ {𝑊})) = (♯‘∅))
120119, 99syl6eq 2875 . . . . . . . . . . . . . . 15 (((𝑂𝑓) “ {𝑊}) = ∅ → (♯‘((𝑂𝑓) “ {𝑊})) = 0)
121120breq1d 5079 . . . . . . . . . . . . . 14 (((𝑂𝑓) “ {𝑊}) = ∅ → ((♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓) ↔ 0 ≤ (𝐷𝑓)))
122118, 121syl5ibrcom 249 . . . . . . . . . . . . 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 4313 . . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . . . 16 (var1𝑅) = (var1𝑅)
128 eqid 2824 . . . . . . . . . . . . . . . 16 (-g𝑃) = (-g𝑃)
129 eqid 2824 . . . . . . . . . . . . . . . 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 24763 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))
135134exp32 423 . . . . . . . . . . . . . 14 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
136135exlimdv 1933 . . . . . . . . . . . . 13 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (∃𝑥 𝑥 ∈ ((𝑂𝑓) “ {𝑊}) → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
137124, 136syl5bi 244 . . . . . . . . . . . 12 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (((𝑂𝑓) “ {𝑊}) ≠ ∅ → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
138123, 137pm2.61dne 3106 . . . . . . . . . . 11 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))
139138expr 459 . . . . . . . . . 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 3192 . . . . . . . 8 ((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (♯‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → ∀𝑓𝐵 ((𝐷𝑓) = (𝑑 + 1) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
142113, 141syl5bi 244 . . . . . . 7 ((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) → (∀𝑓𝐵 ((𝐷𝑓) = 𝑑 → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) → ∀𝑓𝐵 ((𝐷𝑓) = (𝑑 + 1) → (♯‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
143142expcom 416 . . . . . 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 12080 . . . 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 3628 . 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 208   ∧ wa 398   = wceq 1536  ∃wex 1779   ∈ wcel 2113   ≠ wne 3019  ∀wral 3141  Vcvv 3497  ∅c0 4294  {csn 4570   class class class wbr 5069   × cxp 5556  ◡ccnv 5557   “ cima 5561   Fn wfn 6353  ⟶wf 6354  ‘cfv 6358  (class class class)co 7159  0cc0 10540  1c1 10541   + caddc 10543   ≤ cle 10679  ℕ0cn0 11900  ♯chash 13693  Basecbs 16486  0gc0g 16716   ↑s cpws 16723  -gcsg 18108  Ringcrg 19300  CRingccrg 19301   RingHom crh 19467  Domncdomn 20056  IDomncidom 20057  algSccascl 20087  var1cv1 20347  Poly1cpl1 20348  coe1cco1 20349  eval1ce1 20480   deg1 cdg1 24651 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617  ax-pre-sup 10618  ax-addf 10619  ax-mulf 10620 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-iin 4925  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-se 5518  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-isom 6367  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-of 7412  df-ofr 7413  df-om 7584  df-1st 7692  df-2nd 7693  df-supp 7834  df-tpos 7895  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-2o 8106  df-oadd 8109  df-er 8292  df-map 8411  df-pm 8412  df-ixp 8465  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-fsupp 8837  df-sup 8909  df-oi 8977  df-dju 9333  df-card 9371  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-nn 11642  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-xnn0 11971  df-z 11985  df-dec 12102  df-uz 12247  df-fz 12896  df-fzo 13037  df-seq 13373  df-hash 13694  df-struct 16488  df-ndx 16489  df-slot 16490  df-base 16492  df-sets 16493  df-ress 16494  df-plusg 16581  df-mulr 16582  df-starv 16583  df-sca 16584  df-vsca 16585  df-ip 16586  df-tset 16587  df-ple 16588  df-ds 16590  df-unif 16591  df-hom 16592  df-cco 16593  df-0g 16718  df-gsum 16719  df-prds 16724  df-pws 16726  df-mre 16860  df-mrc 16861  df-acs 16863  df-mgm 17855  df-sgrp 17904  df-mnd 17915  df-mhm 17959  df-submnd 17960  df-grp 18109  df-minusg 18110  df-sbg 18111  df-mulg 18228  df-subg 18279  df-ghm 18359  df-cntz 18450  df-cmn 18911  df-abl 18912  df-mgp 19243  df-ur 19255  df-srg 19259  df-ring 19302  df-cring 19303  df-oppr 19376  df-dvdsr 19394  df-unit 19395  df-invr 19425  df-rnghom 19470  df-subrg 19536  df-lmod 19639  df-lss 19707  df-lsp 19747  df-nzr 20034  df-rlreg 20059  df-domn 20060  df-idom 20061  df-assa 20088  df-asp 20089  df-ascl 20090  df-psr 20139  df-mvr 20140  df-mpl 20141  df-opsr 20143  df-evls 20289  df-evl 20290  df-psr1 20351  df-vr1 20352  df-ply1 20353  df-coe1 20354  df-evl1 20482  df-cnfld 20549  df-mdeg 24652  df-deg1 24653  df-mon1 24727  df-uc1p 24728  df-q1p 24729  df-r1p 24730 This theorem is referenced by:  fta1b  24766  lgsqrlem4  25928  idomrootle  39801
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