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Theorem fta1g 23972
Description: The one-sided fundamental theorem of algebra. A polynomial of degree 𝑛 has at most 𝑛 roots. Unlike the real fundamental theorem fta 24851, 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 2651 . 2 (𝐷𝐹) = (𝐷𝐹)
2 fta1g.2 . . 3 (𝜑𝐹𝐵)
3 fta1g.1 . . . . . 6 (𝜑𝑅 ∈ IDomn)
4 isidom 19352 . . . . . . 7 (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
54simplbi 475 . . . . . 6 (𝑅 ∈ IDomn → 𝑅 ∈ CRing)
6 crngring 18604 . . . . . 6 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
73, 5, 63syl 18 . . . . 5 (𝜑𝑅 ∈ Ring)
8 fta1g.3 . . . . 5 (𝜑𝐹0 )
9 fta1g.d . . . . . 6 𝐷 = ( deg1𝑅)
10 fta1g.p . . . . . 6 𝑃 = (Poly1𝑅)
11 fta1g.z . . . . . 6 0 = (0g𝑃)
12 fta1g.b . . . . . 6 𝐵 = (Base‘𝑃)
139, 10, 11, 12deg1nn0cl 23893 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐹𝐵𝐹0 ) → (𝐷𝐹) ∈ ℕ0)
147, 2, 8, 13syl3anc 1366 . . . 4 (𝜑 → (𝐷𝐹) ∈ ℕ0)
15 eqeq2 2662 . . . . . . . 8 (𝑥 = 0 → ((𝐷𝑓) = 𝑥 ↔ (𝐷𝑓) = 0))
1615imbi1d 330 . . . . . . 7 (𝑥 = 0 → (((𝐷𝑓) = 𝑥 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑓) = 0 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
1716ralbidv 3015 . . . . . 6 (𝑥 = 0 → (∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑓𝐵 ((𝐷𝑓) = 0 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
1817imbi2d 329 . . . . 5 (𝑥 = 0 → ((𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 0 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
19 eqeq2 2662 . . . . . . . 8 (𝑥 = 𝑑 → ((𝐷𝑓) = 𝑥 ↔ (𝐷𝑓) = 𝑑))
2019imbi1d 330 . . . . . . 7 (𝑥 = 𝑑 → (((𝐷𝑓) = 𝑥 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑓) = 𝑑 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
2120ralbidv 3015 . . . . . 6 (𝑥 = 𝑑 → (∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑓𝐵 ((𝐷𝑓) = 𝑑 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
2221imbi2d 329 . . . . 5 (𝑥 = 𝑑 → ((𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑑 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
23 eqeq2 2662 . . . . . . . 8 (𝑥 = (𝑑 + 1) → ((𝐷𝑓) = 𝑥 ↔ (𝐷𝑓) = (𝑑 + 1)))
2423imbi1d 330 . . . . . . 7 (𝑥 = (𝑑 + 1) → (((𝐷𝑓) = 𝑥 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑓) = (𝑑 + 1) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
2524ralbidv 3015 . . . . . 6 (𝑥 = (𝑑 + 1) → (∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑓𝐵 ((𝐷𝑓) = (𝑑 + 1) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
2625imbi2d 329 . . . . 5 (𝑥 = (𝑑 + 1) → ((𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = (𝑑 + 1) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
27 eqeq2 2662 . . . . . . . 8 (𝑥 = (𝐷𝐹) → ((𝐷𝑓) = 𝑥 ↔ (𝐷𝑓) = (𝐷𝐹)))
2827imbi1d 330 . . . . . . 7 (𝑥 = (𝐷𝐹) → (((𝐷𝑓) = 𝑥 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑓) = (𝐷𝐹) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
2928ralbidv 3015 . . . . . 6 (𝑥 = (𝐷𝐹) → (∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑓𝐵 ((𝐷𝑓) = (𝐷𝐹) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
3029imbi2d 329 . . . . 5 (𝑥 = (𝐷𝐹) → ((𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑥 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = (𝐷𝐹) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
31 simprr 811 . . . . . . . . . . . . . 14 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝐷𝑓) = 0)
32 0nn0 11345 . . . . . . . . . . . . . 14 0 ∈ ℕ0
3331, 32syl6eqel 2738 . . . . . . . . . . . . 13 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝐷𝑓) ∈ ℕ0)
345, 6syl 17 . . . . . . . . . . . . . 14 (𝑅 ∈ IDomn → 𝑅 ∈ Ring)
35 simpl 472 . . . . . . . . . . . . . 14 ((𝑓𝐵 ∧ (𝐷𝑓) = 0) → 𝑓𝐵)
369, 10, 11, 12deg1nn0clb 23895 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 𝑓𝐵) → (𝑓0 ↔ (𝐷𝑓) ∈ ℕ0))
3734, 35, 36syl2an 493 . . . . . . . . . . . . 13 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑓0 ↔ (𝐷𝑓) ∈ ℕ0))
3833, 37mpbird 247 . . . . . . . . . . . 12 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 𝑓0 )
39 simplrr 818 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝐷𝑓) = 0)
40 0le0 11148 . . . . . . . . . . . . . . . . 17 0 ≤ 0
4139, 40syl6eqbr 4724 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝐷𝑓) ≤ 0)
4234ad2antrr 762 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑅 ∈ Ring)
43 simplrl 817 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑓𝐵)
44 eqid 2651 . . . . . . . . . . . . . . . . . 18 (algSc‘𝑃) = (algSc‘𝑃)
459, 10, 12, 44deg1le0 23916 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝑓𝐵) → ((𝐷𝑓) ≤ 0 ↔ 𝑓 = ((algSc‘𝑃)‘((coe1𝑓)‘0))))
4642, 43, 45syl2anc 694 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((𝐷𝑓) ≤ 0 ↔ 𝑓 = ((algSc‘𝑃)‘((coe1𝑓)‘0))))
4741, 46mpbid 222 . . . . . . . . . . . . . . 15 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑓 = ((algSc‘𝑃)‘((coe1𝑓)‘0)))
4847fveq2d 6233 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝑂𝑓) = (𝑂‘((algSc‘𝑃)‘((coe1𝑓)‘0))))
495adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 𝑅 ∈ CRing)
5049adantr 480 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑅 ∈ CRing)
51 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . 23 (coe1𝑓) = (coe1𝑓)
52 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . 23 (Base‘𝑅) = (Base‘𝑅)
5351, 12, 10, 52coe1f 19629 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓𝐵 → (coe1𝑓):ℕ0⟶(Base‘𝑅))
5443, 53syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (coe1𝑓):ℕ0⟶(Base‘𝑅))
55 ffvelrn 6397 . . . . . . . . . . . . . . . . . . . . 21 (((coe1𝑓):ℕ0⟶(Base‘𝑅) ∧ 0 ∈ ℕ0) → ((coe1𝑓)‘0) ∈ (Base‘𝑅))
5654, 32, 55sylancl 695 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((coe1𝑓)‘0) ∈ (Base‘𝑅))
57 fta1g.o . . . . . . . . . . . . . . . . . . . . 21 𝑂 = (eval1𝑅)
5857, 10, 52, 44evl1sca 19746 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ CRing ∧ ((coe1𝑓)‘0) ∈ (Base‘𝑅)) → (𝑂‘((algSc‘𝑃)‘((coe1𝑓)‘0))) = ((Base‘𝑅) × {((coe1𝑓)‘0)}))
5950, 56, 58syl2anc 694 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝑂‘((algSc‘𝑃)‘((coe1𝑓)‘0))) = ((Base‘𝑅) × {((coe1𝑓)‘0)}))
6048, 59eqtrd 2685 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (𝑂𝑓) = ((Base‘𝑅) × {((coe1𝑓)‘0)}))
6160fveq1d 6231 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((𝑂𝑓)‘𝑥) = (((Base‘𝑅) × {((coe1𝑓)‘0)})‘𝑥))
62 eqid 2651 . . . . . . . . . . . . . . . . . . . 20 (𝑅s (Base‘𝑅)) = (𝑅s (Base‘𝑅))
63 eqid 2651 . . . . . . . . . . . . . . . . . . . 20 (Base‘(𝑅s (Base‘𝑅))) = (Base‘(𝑅s (Base‘𝑅)))
64 simpl 472 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 𝑅 ∈ IDomn)
65 fvexd 6241 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (Base‘𝑅) ∈ V)
6657, 10, 62, 52evl1rhm 19744 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅s (Base‘𝑅))))
6712, 63rhmf 18774 . . . . . . . . . . . . . . . . . . . . . 22 (𝑂 ∈ (𝑃 RingHom (𝑅s (Base‘𝑅))) → 𝑂:𝐵⟶(Base‘(𝑅s (Base‘𝑅))))
6849, 66, 673syl 18 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 𝑂:𝐵⟶(Base‘(𝑅s (Base‘𝑅))))
69 simprl 809 . . . . . . . . . . . . . . . . . . . . 21 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 𝑓𝐵)
7068, 69ffvelrnd 6400 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑂𝑓) ∈ (Base‘(𝑅s (Base‘𝑅))))
7162, 52, 63, 64, 65, 70pwselbas 16196 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑂𝑓):(Base‘𝑅)⟶(Base‘𝑅))
72 ffn 6083 . . . . . . . . . . . . . . . . . . 19 ((𝑂𝑓):(Base‘𝑅)⟶(Base‘𝑅) → (𝑂𝑓) Fn (Base‘𝑅))
73 fniniseg 6378 . . . . . . . . . . . . . . . . . . 19 ((𝑂𝑓) Fn (Base‘𝑅) → (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ((𝑂𝑓)‘𝑥) = 𝑊)))
7471, 72, 733syl 18 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ((𝑂𝑓)‘𝑥) = 𝑊)))
7574simplbda 653 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((𝑂𝑓)‘𝑥) = 𝑊)
7674simprbda 652 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑥 ∈ (Base‘𝑅))
77 fvex 6239 . . . . . . . . . . . . . . . . . . 19 ((coe1𝑓)‘0) ∈ V
7877fvconst2 6510 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (Base‘𝑅) → (((Base‘𝑅) × {((coe1𝑓)‘0)})‘𝑥) = ((coe1𝑓)‘0))
7976, 78syl 17 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → (((Base‘𝑅) × {((coe1𝑓)‘0)})‘𝑥) = ((coe1𝑓)‘0))
8061, 75, 793eqtr3rd 2694 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((coe1𝑓)‘0) = 𝑊)
8180fveq2d 6233 . . . . . . . . . . . . . . 15 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((algSc‘𝑃)‘((coe1𝑓)‘0)) = ((algSc‘𝑃)‘𝑊))
82 fta1g.w . . . . . . . . . . . . . . . . 17 𝑊 = (0g𝑅)
8310, 44, 82, 11ply1scl0 19708 . . . . . . . . . . . . . . . 16 (𝑅 ∈ Ring → ((algSc‘𝑃)‘𝑊) = 0 )
8442, 83syl 17 . . . . . . . . . . . . . . 15 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → ((algSc‘𝑃)‘𝑊) = 0 )
8547, 81, 843eqtrd 2689 . . . . . . . . . . . . . 14 (((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) ∧ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})) → 𝑓 = 0 )
8685ex 449 . . . . . . . . . . . . 13 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) → 𝑓 = 0 ))
8786necon3ad 2836 . . . . . . . . . . . 12 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (𝑓0 → ¬ 𝑥 ∈ ((𝑂𝑓) “ {𝑊})))
8838, 87mpd 15 . . . . . . . . . . 11 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → ¬ 𝑥 ∈ ((𝑂𝑓) “ {𝑊}))
8988eq0rdv 4012 . . . . . . . . . 10 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → ((𝑂𝑓) “ {𝑊}) = ∅)
9089fveq2d 6233 . . . . . . . . 9 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (#‘((𝑂𝑓) “ {𝑊})) = (#‘∅))
91 hash0 13196 . . . . . . . . 9 (#‘∅) = 0
9290, 91syl6eq 2701 . . . . . . . 8 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (#‘((𝑂𝑓) “ {𝑊})) = 0)
9340, 31syl5breqr 4723 . . . . . . . 8 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → 0 ≤ (𝐷𝑓))
9492, 93eqbrtrd 4707 . . . . . . 7 ((𝑅 ∈ IDomn ∧ (𝑓𝐵 ∧ (𝐷𝑓) = 0)) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))
9594expr 642 . . . . . 6 ((𝑅 ∈ IDomn ∧ 𝑓𝐵) → ((𝐷𝑓) = 0 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))
9695ralrimiva 2995 . . . . 5 (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 0 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))
97 fveq2 6229 . . . . . . . . . . 11 (𝑓 = 𝑔 → (𝐷𝑓) = (𝐷𝑔))
9897eqeq1d 2653 . . . . . . . . . 10 (𝑓 = 𝑔 → ((𝐷𝑓) = 𝑑 ↔ (𝐷𝑔) = 𝑑))
99 fveq2 6229 . . . . . . . . . . . . . 14 (𝑓 = 𝑔 → (𝑂𝑓) = (𝑂𝑔))
10099cnveqd 5330 . . . . . . . . . . . . 13 (𝑓 = 𝑔(𝑂𝑓) = (𝑂𝑔))
101100imaeq1d 5500 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝑂𝑓) “ {𝑊}) = ((𝑂𝑔) “ {𝑊}))
102101fveq2d 6233 . . . . . . . . . . 11 (𝑓 = 𝑔 → (#‘((𝑂𝑓) “ {𝑊})) = (#‘((𝑂𝑔) “ {𝑊})))
103102, 97breq12d 4698 . . . . . . . . . 10 (𝑓 = 𝑔 → ((#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓) ↔ (#‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))
10498, 103imbi12d 333 . . . . . . . . 9 (𝑓 = 𝑔 → (((𝐷𝑓) = 𝑑 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝑔) = 𝑑 → (#‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔))))
105104cbvralv 3201 . . . . . . . 8 (∀𝑓𝐵 ((𝐷𝑓) = 𝑑 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (#‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))
106 simprr 811 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (𝐷𝑓) = (𝑑 + 1))
107 peano2nn0 11371 . . . . . . . . . . . . . . . . 17 (𝑑 ∈ ℕ0 → (𝑑 + 1) ∈ ℕ0)
108107ad2antlr 763 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (𝑑 + 1) ∈ ℕ0)
109106, 108eqeltrd 2730 . . . . . . . . . . . . . . 15 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (𝐷𝑓) ∈ ℕ0)
110109nn0ge0d 11392 . . . . . . . . . . . . . 14 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → 0 ≤ (𝐷𝑓))
111 fveq2 6229 . . . . . . . . . . . . . . . 16 (((𝑂𝑓) “ {𝑊}) = ∅ → (#‘((𝑂𝑓) “ {𝑊})) = (#‘∅))
112111, 91syl6eq 2701 . . . . . . . . . . . . . . 15 (((𝑂𝑓) “ {𝑊}) = ∅ → (#‘((𝑂𝑓) “ {𝑊})) = 0)
113112breq1d 4695 . . . . . . . . . . . . . 14 (((𝑂𝑓) “ {𝑊}) = ∅ → ((#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓) ↔ 0 ≤ (𝐷𝑓)))
114110, 113syl5ibrcom 237 . . . . . . . . . . . . 13 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (((𝑂𝑓) “ {𝑊}) = ∅ → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))
115114a1dd 50 . . . . . . . . . . . 12 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (((𝑂𝑓) “ {𝑊}) = ∅ → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (#‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
116 n0 3964 . . . . . . . . . . . . 13 (((𝑂𝑓) “ {𝑊}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝑂𝑓) “ {𝑊}))
117 simplll 813 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (#‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → 𝑅 ∈ IDomn)
118 simplrl 817 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (#‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → 𝑓𝐵)
119 eqid 2651 . . . . . . . . . . . . . . . 16 (var1𝑅) = (var1𝑅)
120 eqid 2651 . . . . . . . . . . . . . . . 16 (-g𝑃) = (-g𝑃)
121 eqid 2651 . . . . . . . . . . . . . . . 16 ((var1𝑅)(-g𝑃)((algSc‘𝑃)‘𝑥)) = ((var1𝑅)(-g𝑃)((algSc‘𝑃)‘𝑥))
122 simpllr 815 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (#‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → 𝑑 ∈ ℕ0)
123 simplrr 818 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (#‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → (𝐷𝑓) = (𝑑 + 1))
124 simprl 809 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (#‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → 𝑥 ∈ ((𝑂𝑓) “ {𝑊}))
125 simprr 811 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (#‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (#‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))
12610, 12, 9, 57, 82, 11, 117, 118, 52, 119, 120, 44, 121, 122, 123, 124, 125fta1glem2 23971 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) ∧ ∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (#‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)))) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))
127126exp32 630 . . . . . . . . . . . . . 14 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (𝑥 ∈ ((𝑂𝑓) “ {𝑊}) → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (#‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
128127exlimdv 1901 . . . . . . . . . . . . 13 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (∃𝑥 𝑥 ∈ ((𝑂𝑓) “ {𝑊}) → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (#‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
129116, 128syl5bi 232 . . . . . . . . . . . 12 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (((𝑂𝑓) “ {𝑊}) ≠ ∅ → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (#‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
130115, 129pm2.61dne 2909 . . . . . . . . . . 11 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ (𝑓𝐵 ∧ (𝐷𝑓) = (𝑑 + 1))) → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (#‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))
131130expr 642 . . . . . . . . . 10 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ 𝑓𝐵) → ((𝐷𝑓) = (𝑑 + 1) → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (#‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
132131com23 86 . . . . . . . . 9 (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) ∧ 𝑓𝐵) → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (#‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → ((𝐷𝑓) = (𝑑 + 1) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
133132ralrimdva 2998 . . . . . . . 8 ((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) → (∀𝑔𝐵 ((𝐷𝑔) = 𝑑 → (#‘((𝑂𝑔) “ {𝑊})) ≤ (𝐷𝑔)) → ∀𝑓𝐵 ((𝐷𝑓) = (𝑑 + 1) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
134105, 133syl5bi 232 . . . . . . 7 ((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ0) → (∀𝑓𝐵 ((𝐷𝑓) = 𝑑 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) → ∀𝑓𝐵 ((𝐷𝑓) = (𝑑 + 1) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
135134expcom 450 . . . . . 6 (𝑑 ∈ ℕ0 → (𝑅 ∈ IDomn → (∀𝑓𝐵 ((𝐷𝑓) = 𝑑 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) → ∀𝑓𝐵 ((𝐷𝑓) = (𝑑 + 1) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
136135a2d 29 . . . . 5 (𝑑 ∈ ℕ0 → ((𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = 𝑑 → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))) → (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = (𝑑 + 1) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))))
13718, 22, 26, 30, 96, 136nn0ind 11510 . . . 4 ((𝐷𝐹) ∈ ℕ0 → (𝑅 ∈ IDomn → ∀𝑓𝐵 ((𝐷𝑓) = (𝐷𝐹) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓))))
13814, 3, 137sylc 65 . . 3 (𝜑 → ∀𝑓𝐵 ((𝐷𝑓) = (𝐷𝐹) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)))
139 fveq2 6229 . . . . . 6 (𝑓 = 𝐹 → (𝐷𝑓) = (𝐷𝐹))
140139eqeq1d 2653 . . . . 5 (𝑓 = 𝐹 → ((𝐷𝑓) = (𝐷𝐹) ↔ (𝐷𝐹) = (𝐷𝐹)))
141 fveq2 6229 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑂𝑓) = (𝑂𝐹))
142141cnveqd 5330 . . . . . . . 8 (𝑓 = 𝐹(𝑂𝑓) = (𝑂𝐹))
143142imaeq1d 5500 . . . . . . 7 (𝑓 = 𝐹 → ((𝑂𝑓) “ {𝑊}) = ((𝑂𝐹) “ {𝑊}))
144143fveq2d 6233 . . . . . 6 (𝑓 = 𝐹 → (#‘((𝑂𝑓) “ {𝑊})) = (#‘((𝑂𝐹) “ {𝑊})))
145144, 139breq12d 4698 . . . . 5 (𝑓 = 𝐹 → ((#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓) ↔ (#‘((𝑂𝐹) “ {𝑊})) ≤ (𝐷𝐹)))
146140, 145imbi12d 333 . . . 4 (𝑓 = 𝐹 → (((𝐷𝑓) = (𝐷𝐹) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) ↔ ((𝐷𝐹) = (𝐷𝐹) → (#‘((𝑂𝐹) “ {𝑊})) ≤ (𝐷𝐹))))
147146rspcv 3336 . . 3 (𝐹𝐵 → (∀𝑓𝐵 ((𝐷𝑓) = (𝐷𝐹) → (#‘((𝑂𝑓) “ {𝑊})) ≤ (𝐷𝑓)) → ((𝐷𝐹) = (𝐷𝐹) → (#‘((𝑂𝐹) “ {𝑊})) ≤ (𝐷𝐹))))
1482, 138, 147sylc 65 . 2 (𝜑 → ((𝐷𝐹) = (𝐷𝐹) → (#‘((𝑂𝐹) “ {𝑊})) ≤ (𝐷𝐹)))
1491, 148mpi 20 1 (𝜑 → (#‘((𝑂𝐹) “ {𝑊})) ≤ (𝐷𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wne 2823  wral 2941  Vcvv 3231  c0 3948  {csn 4210   class class class wbr 4685   × cxp 5141  ccnv 5142  cima 5146   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  0cc0 9974  1c1 9975   + caddc 9977  cle 10113  0cn0 11330  #chash 13157  Basecbs 15904  0gc0g 16147  s cpws 16154  -gcsg 17471  Ringcrg 18593  CRingccrg 18594   RingHom crh 18760  Domncdomn 19328  IDomncidom 19329  algSccascl 19359  var1cv1 19594  Poly1cpl1 19595  coe1cco1 19596  eval1ce1 19727   deg1 cdg1 23859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofr 6940  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-tpos 7397  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-sup 8389  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-xnn0 11402  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-0g 16149  df-gsum 16150  df-prds 16155  df-pws 16157  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-subg 17638  df-ghm 17705  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-srg 18552  df-ring 18595  df-cring 18596  df-oppr 18669  df-dvdsr 18687  df-unit 18688  df-invr 18718  df-rnghom 18763  df-subrg 18826  df-lmod 18913  df-lss 18981  df-lsp 19020  df-nzr 19306  df-rlreg 19331  df-domn 19332  df-idom 19333  df-assa 19360  df-asp 19361  df-ascl 19362  df-psr 19404  df-mvr 19405  df-mpl 19406  df-opsr 19408  df-evls 19554  df-evl 19555  df-psr1 19598  df-vr1 19599  df-ply1 19600  df-coe1 19601  df-evl1 19729  df-cnfld 19795  df-mdeg 23860  df-deg1 23861  df-mon1 23935  df-uc1p 23936  df-q1p 23937  df-r1p 23938
This theorem is referenced by:  fta1b  23974  lgsqrlem4  25119  idomrootle  38090
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