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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 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 = (Poly₁‘𝑅)
fta1g.b	⊢ 𝐵 = (Base‘𝑃)
fta1g.d	⊢ 𝐷 = ( deg₁ ‘𝑅)
fta1g.o	⊢ 𝑂 = (eval₁‘𝑅)
fta1g.w	⊢ 𝑊 = (0_g‘𝑅)
fta1g.z	⊢ 0 = (0_g‘𝑃)
fta1g.1	⊢ (𝜑 → 𝑅 ∈ IDomn)
fta1g.2	⊢ (𝜑 → 𝐹 ∈ 𝐵)
fta1g.3	⊢ (𝜑 → 𝐹 ≠ 0 )

**Assertion**
Ref	Expression
fta1g	⊢ (𝜑 → (♯‘(^◡(𝑂‘𝐹) “ {𝑊})) ≤ (𝐷‘𝐹))

Proof of Theorem fta1g Dummy variables 𝑓 𝑑 𝑔 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. 2 ⊢ (𝐷‘𝐹) = (𝐷‘𝐹)
2		fveqeq2 6851	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝐷‘𝑓) = (𝐷‘𝐹) ↔ (𝐷‘𝐹) = (𝐷‘𝐹)))
3		fveq2 6842	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑂‘𝑓) = (𝑂‘𝐹))
4	3	cnveqd 5831	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ^◡(𝑂‘𝑓) = ^◡(𝑂‘𝐹))
5	4	imaeq1d 6012	. . . . . 6 ⊢ (𝑓 = 𝐹 → (^◡(𝑂‘𝑓) “ {𝑊}) = (^◡(𝑂‘𝐹) “ {𝑊}))
6	5	fveq2d 6846	. . . . 5 ⊢ (𝑓 = 𝐹 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = (♯‘(^◡(𝑂‘𝐹) “ {𝑊})))
7		fveq2 6842	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝐷‘𝑓) = (𝐷‘𝐹))
8	6, 7	breq12d 5118	. . . 4 ⊢ (𝑓 = 𝐹 → ((♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓) ↔ (♯‘(^◡(𝑂‘𝐹) “ {𝑊})) ≤ (𝐷‘𝐹)))
9	2, 8	imbi12d 344	. . 3 ⊢ (𝑓 = 𝐹 → (((𝐷‘𝑓) = (𝐷‘𝐹) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)) ↔ ((𝐷‘𝐹) = (𝐷‘𝐹) → (♯‘(^◡(𝑂‘𝐹) “ {𝑊})) ≤ (𝐷‘𝐹))))
10		fta1g.1	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ IDomn)
11		isidom 20774	. . . . . . 7 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
12	11	simplbi 498	. . . . . 6 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ CRing)
13		crngring 19976	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
14	10, 12, 13	3syl 18	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
15		fta1g.2	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
16		fta1g.3	. . . . 5 ⊢ (𝜑 → 𝐹 ≠ 0 )
17		fta1g.d	. . . . . 6 ⊢ 𝐷 = ( deg₁ ‘𝑅)
18		fta1g.p	. . . . . 6 ⊢ 𝑃 = (Poly₁‘𝑅)
19		fta1g.z	. . . . . 6 ⊢ 0 = (0_g‘𝑃)
20		fta1g.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑃)
21	17, 18, 19, 20	deg1nn0cl 25453	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ₀)
22	14, 15, 16, 21	syl3anc 1371	. . . 4 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℕ₀)
23		eqeq2 2748	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝐷‘𝑓) = 𝑥 ↔ (𝐷‘𝑓) = 0))
24	23	imbi1d 341	. . . . . . 7 ⊢ (𝑥 = 0 → (((𝐷‘𝑓) = 𝑥 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)) ↔ ((𝐷‘𝑓) = 0 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))))
25	24	ralbidv 3174	. . . . . 6 ⊢ (𝑥 = 0 → (∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 𝑥 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)) ↔ ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 0 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))))
26	25	imbi2d 340	. . . . 5 ⊢ (𝑥 = 0 → ((𝑅 ∈ IDomn → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 𝑥 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 0 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)))))
27		eqeq2 2748	. . . . . . . 8 ⊢ (𝑥 = 𝑑 → ((𝐷‘𝑓) = 𝑥 ↔ (𝐷‘𝑓) = 𝑑))
28	27	imbi1d 341	. . . . . . 7 ⊢ (𝑥 = 𝑑 → (((𝐷‘𝑓) = 𝑥 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)) ↔ ((𝐷‘𝑓) = 𝑑 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))))
29	28	ralbidv 3174	. . . . . 6 ⊢ (𝑥 = 𝑑 → (∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 𝑥 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)) ↔ ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 𝑑 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))))
30	29	imbi2d 340	. . . . 5 ⊢ (𝑥 = 𝑑 → ((𝑅 ∈ IDomn → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 𝑥 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 𝑑 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)))))
31		eqeq2 2748	. . . . . . . 8 ⊢ (𝑥 = (𝑑 + 1) → ((𝐷‘𝑓) = 𝑥 ↔ (𝐷‘𝑓) = (𝑑 + 1)))
32	31	imbi1d 341	. . . . . . 7 ⊢ (𝑥 = (𝑑 + 1) → (((𝐷‘𝑓) = 𝑥 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)) ↔ ((𝐷‘𝑓) = (𝑑 + 1) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))))
33	32	ralbidv 3174	. . . . . 6 ⊢ (𝑥 = (𝑑 + 1) → (∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 𝑥 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)) ↔ ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = (𝑑 + 1) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))))
34	33	imbi2d 340	. . . . 5 ⊢ (𝑥 = (𝑑 + 1) → ((𝑅 ∈ IDomn → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 𝑥 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = (𝑑 + 1) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)))))
35		eqeq2 2748	. . . . . . . 8 ⊢ (𝑥 = (𝐷‘𝐹) → ((𝐷‘𝑓) = 𝑥 ↔ (𝐷‘𝑓) = (𝐷‘𝐹)))
36	35	imbi1d 341	. . . . . . 7 ⊢ (𝑥 = (𝐷‘𝐹) → (((𝐷‘𝑓) = 𝑥 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)) ↔ ((𝐷‘𝑓) = (𝐷‘𝐹) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))))
37	36	ralbidv 3174	. . . . . 6 ⊢ (𝑥 = (𝐷‘𝐹) → (∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 𝑥 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)) ↔ ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = (𝐷‘𝐹) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))))
38	37	imbi2d 340	. . . . 5 ⊢ (𝑥 = (𝐷‘𝐹) → ((𝑅 ∈ IDomn → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 𝑥 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))) ↔ (𝑅 ∈ IDomn → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = (𝐷‘𝐹) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)))))
39		simprr 771	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝐷‘𝑓) = 0)
40		0nn0 12428	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ₀
41	39, 40	eqeltrdi 2846	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝐷‘𝑓) ∈ ℕ₀)
42	12, 13	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ Ring)
43		simpl 483	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0) → 𝑓 ∈ 𝐵)
44	17, 18, 19, 20	deg1nn0clb 25455	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ 𝐵) → (𝑓 ≠ 0 ↔ (𝐷‘𝑓) ∈ ℕ₀))
45	42, 43, 44	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝑓 ≠ 0 ↔ (𝐷‘𝑓) ∈ ℕ₀))
46	41, 45	mpbird 256	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → 𝑓 ≠ 0 )
47		simplrr 776	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → (𝐷‘𝑓) = 0)
48		0le0 12254	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≤ 0
49	47, 48	eqbrtrdi 5144	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → (𝐷‘𝑓) ≤ 0)
50	42	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑅 ∈ Ring)
51		simplrl 775	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑓 ∈ 𝐵)
52		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
53	17, 18, 20, 52	deg1le0 25476	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ 𝐵) → ((𝐷‘𝑓) ≤ 0 ↔ 𝑓 = ((algSc‘𝑃)‘((coe₁‘𝑓)‘0))))
54	50, 51, 53	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → ((𝐷‘𝑓) ≤ 0 ↔ 𝑓 = ((algSc‘𝑃)‘((coe₁‘𝑓)‘0))))
55	49, 54	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑓 = ((algSc‘𝑃)‘((coe₁‘𝑓)‘0)))
56	55	fveq2d 6846	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → (𝑂‘𝑓) = (𝑂‘((algSc‘𝑃)‘((coe₁‘𝑓)‘0))))
57	12	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → 𝑅 ∈ CRing)
58	57	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑅 ∈ CRing)
59		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (coe₁‘𝑓) = (coe₁‘𝑓)
60		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Base‘𝑅) = (Base‘𝑅)
61	59, 20, 18, 60	coe1f 21582	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 ∈ 𝐵 → (coe₁‘𝑓):ℕ₀⟶(Base‘𝑅))
62	51, 61	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → (coe₁‘𝑓):ℕ₀⟶(Base‘𝑅))
63		ffvelcdm 7032	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((coe₁‘𝑓):ℕ₀⟶(Base‘𝑅) ∧ 0 ∈ ℕ₀) → ((coe₁‘𝑓)‘0) ∈ (Base‘𝑅))
64	62, 40, 63	sylancl 586	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → ((coe₁‘𝑓)‘0) ∈ (Base‘𝑅))
65		fta1g.o	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑂 = (eval₁‘𝑅)
66	65, 18, 60, 52	evl1sca 21700	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ CRing ∧ ((coe₁‘𝑓)‘0) ∈ (Base‘𝑅)) → (𝑂‘((algSc‘𝑃)‘((coe₁‘𝑓)‘0))) = ((Base‘𝑅) × {((coe₁‘𝑓)‘0)}))
67	58, 64, 66	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → (𝑂‘((algSc‘𝑃)‘((coe₁‘𝑓)‘0))) = ((Base‘𝑅) × {((coe₁‘𝑓)‘0)}))
68	56, 67	eqtrd 2776	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → (𝑂‘𝑓) = ((Base‘𝑅) × {((coe₁‘𝑓)‘0)}))
69	68	fveq1d 6844	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → ((𝑂‘𝑓)‘𝑥) = (((Base‘𝑅) × {((coe₁‘𝑓)‘0)})‘𝑥))
70		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 ↑_s (Base‘𝑅)) = (𝑅 ↑_s (Base‘𝑅))
71		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Base‘(𝑅 ↑_s (Base‘𝑅))) = (Base‘(𝑅 ↑_s (Base‘𝑅)))
72		simpl 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → 𝑅 ∈ IDomn)
73		fvexd 6857	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (Base‘𝑅) ∈ V)
74	65, 18, 70, 60	evl1rhm 21698	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s (Base‘𝑅))))
75	20, 71	rhmf 20158	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s (Base‘𝑅))) → 𝑂:𝐵⟶(Base‘(𝑅 ↑_s (Base‘𝑅))))
76	57, 74, 75	3syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → 𝑂:𝐵⟶(Base‘(𝑅 ↑_s (Base‘𝑅))))
77		simprl 769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → 𝑓 ∈ 𝐵)
78	76, 77	ffvelcdmd 7036	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝑂‘𝑓) ∈ (Base‘(𝑅 ↑_s (Base‘𝑅))))
79	70, 60, 71, 72, 73, 78	pwselbas 17371	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝑂‘𝑓):(Base‘𝑅)⟶(Base‘𝑅))
80		ffn 6668	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑂‘𝑓):(Base‘𝑅)⟶(Base‘𝑅) → (𝑂‘𝑓) Fn (Base‘𝑅))
81		fniniseg 7010	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑂‘𝑓) Fn (Base‘𝑅) → (𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ((𝑂‘𝑓)‘𝑥) = 𝑊)))
82	79, 80, 81	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ((𝑂‘𝑓)‘𝑥) = 𝑊)))
83	82	simplbda 500	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → ((𝑂‘𝑓)‘𝑥) = 𝑊)
84	82	simprbda 499	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑥 ∈ (Base‘𝑅))
85		fvex 6855	. . . . . . . . . . . . . . . . . . 19 ⊢ ((coe₁‘𝑓)‘0) ∈ V
86	85	fvconst2 7153	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (Base‘𝑅) → (((Base‘𝑅) × {((coe₁‘𝑓)‘0)})‘𝑥) = ((coe₁‘𝑓)‘0))
87	84, 86	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → (((Base‘𝑅) × {((coe₁‘𝑓)‘0)})‘𝑥) = ((coe₁‘𝑓)‘0))
88	69, 83, 87	3eqtr3rd 2785	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → ((coe₁‘𝑓)‘0) = 𝑊)
89	88	fveq2d 6846	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → ((algSc‘𝑃)‘((coe₁‘𝑓)‘0)) = ((algSc‘𝑃)‘𝑊))
90		fta1g.w	. . . . . . . . . . . . . . . . 17 ⊢ 𝑊 = (0_g‘𝑅)
91	18, 52, 90, 19	ply1scl0 21661	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → ((algSc‘𝑃)‘𝑊) = 0 )
92	50, 91	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → ((algSc‘𝑃)‘𝑊) = 0 )
93	55, 89, 92	3eqtrd 2780	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑓 = 0 )
94	93	ex 413	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}) → 𝑓 = 0 ))
95	94	necon3ad 2956	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝑓 ≠ 0 → ¬ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})))
96	46, 95	mpd 15	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → ¬ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}))
97	96	eq0rdv 4364	. . . . . . . . . 10 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (^◡(𝑂‘𝑓) “ {𝑊}) = ∅)
98	97	fveq2d 6846	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = (♯‘∅))
99		hash0 14267	. . . . . . . . 9 ⊢ (♯‘∅) = 0
100	98, 99	eqtrdi 2792	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = 0)
101	48, 39	breqtrrid 5143	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → 0 ≤ (𝐷‘𝑓))
102	100, 101	eqbrtrd 5127	. . . . . . 7 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))
103	102	expr 457	. . . . . 6 ⊢ ((𝑅 ∈ IDomn ∧ 𝑓 ∈ 𝐵) → ((𝐷‘𝑓) = 0 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)))
104	103	ralrimiva 3143	. . . . 5 ⊢ (𝑅 ∈ IDomn → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 0 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)))
105		fveqeq2 6851	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ((𝐷‘𝑓) = 𝑑 ↔ (𝐷‘𝑔) = 𝑑))
106		fveq2 6842	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → (𝑂‘𝑓) = (𝑂‘𝑔))
107	106	cnveqd 5831	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → ^◡(𝑂‘𝑓) = ^◡(𝑂‘𝑔))
108	107	imaeq1d 6012	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (^◡(𝑂‘𝑓) “ {𝑊}) = (^◡(𝑂‘𝑔) “ {𝑊}))
109	108	fveq2d 6846	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = (♯‘(^◡(𝑂‘𝑔) “ {𝑊})))
110		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (𝐷‘𝑓) = (𝐷‘𝑔))
111	109, 110	breq12d 5118	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ((♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓) ↔ (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)))
112	105, 111	imbi12d 344	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (((𝐷‘𝑓) = 𝑑 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)) ↔ ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔))))
113	112	cbvralvw 3225	. . . . . . . 8 ⊢ (∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 𝑑 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)) ↔ ∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)))
114		simprr 771	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → (𝐷‘𝑓) = (𝑑 + 1))
115		peano2nn0 12453	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ₀)
116	115	ad2antlr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → (𝑑 + 1) ∈ ℕ₀)
117	114, 116	eqeltrd 2838	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → (𝐷‘𝑓) ∈ ℕ₀)
118	117	nn0ge0d 12476	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → 0 ≤ (𝐷‘𝑓))
119		fveq2 6842	. . . . . . . . . . . . . . . 16 ⊢ ((^◡(𝑂‘𝑓) “ {𝑊}) = ∅ → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = (♯‘∅))
120	119, 99	eqtrdi 2792	. . . . . . . . . . . . . . 15 ⊢ ((^◡(𝑂‘𝑓) “ {𝑊}) = ∅ → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = 0)
121	120	breq1d 5115	. . . . . . . . . . . . . 14 ⊢ ((^◡(𝑂‘𝑓) “ {𝑊}) = ∅ → ((♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓) ↔ 0 ≤ (𝐷‘𝑓)))
122	118, 121	syl5ibrcom 246	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → ((^◡(𝑂‘𝑓) “ {𝑊}) = ∅ → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)))
123	122	a1dd 50	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → ((^◡(𝑂‘𝑓) “ {𝑊}) = ∅ → (∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))))
124		n0 4306	. . . . . . . . . . . . 13 ⊢ ((^◡(𝑂‘𝑓) “ {𝑊}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}))
125		simplll 773	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}) ∧ ∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)))) → 𝑅 ∈ IDomn)
126		simplrl 775	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}) ∧ ∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)))) → 𝑓 ∈ 𝐵)
127		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (var₁‘𝑅) = (var₁‘𝑅)
128		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (-_g‘𝑃) = (-_g‘𝑃)
129		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ ((var₁‘𝑅)(-_g‘𝑃)((algSc‘𝑃)‘𝑥)) = ((var₁‘𝑅)(-_g‘𝑃)((algSc‘𝑃)‘𝑥))
130		simpllr 774	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}) ∧ ∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)))) → 𝑑 ∈ ℕ₀)
131		simplrr 776	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}) ∧ ∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)))) → (𝐷‘𝑓) = (𝑑 + 1))
132		simprl 769	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}) ∧ ∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)))) → 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}))
133		simprr 771	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}) ∧ ∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)))) → ∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)))
134	18, 20, 17, 65, 90, 19, 125, 126, 60, 127, 128, 52, 129, 130, 131, 132, 133	fta1glem2 25531	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) ∧ (𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}) ∧ ∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)))) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))
135	134	exp32 421	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → (𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}) → (∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))))
136	135	exlimdv 1936	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → (∃𝑥 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}) → (∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))))
137	124, 136	biimtrid 241	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → ((^◡(𝑂‘𝑓) “ {𝑊}) ≠ ∅ → (∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))))
138	123, 137	pm2.61dne 3031	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → (∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)))
139	138	expr 457	. . . . . . . . . 10 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ 𝑓 ∈ 𝐵) → ((𝐷‘𝑓) = (𝑑 + 1) → (∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))))
140	139	com23 86	. . . . . . . . 9 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ 𝑓 ∈ 𝐵) → (∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)) → ((𝐷‘𝑓) = (𝑑 + 1) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))))
141	140	ralrimdva 3151	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) → (∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)) → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = (𝑑 + 1) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))))
142	113, 141	biimtrid 241	. . . . . . 7 ⊢ ((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) → (∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 𝑑 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)) → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = (𝑑 + 1) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))))
143	142	expcom 414	. . . . . 6 ⊢ (𝑑 ∈ ℕ₀ → (𝑅 ∈ IDomn → (∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 𝑑 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)) → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = (𝑑 + 1) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)))))
144	143	a2d 29	. . . . 5 ⊢ (𝑑 ∈ ℕ₀ → ((𝑅 ∈ IDomn → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 𝑑 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))) → (𝑅 ∈ IDomn → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = (𝑑 + 1) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)))))
145	26, 30, 34, 38, 104, 144	nn0ind 12598	. . . 4 ⊢ ((𝐷‘𝐹) ∈ ℕ₀ → (𝑅 ∈ IDomn → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = (𝐷‘𝐹) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))))
146	22, 10, 145	sylc 65	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = (𝐷‘𝐹) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)))
147	9, 146, 15	rspcdva 3582	. 2 ⊢ (𝜑 → ((𝐷‘𝐹) = (𝐷‘𝐹) → (♯‘(^◡(𝑂‘𝐹) “ {𝑊})) ≤ (𝐷‘𝐹)))
148	1, 147	mpi 20	1 ⊢ (𝜑 → (♯‘(^◡(𝑂‘𝐹) “ {𝑊})) ≤ (𝐷‘𝐹))

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 ℕ₀cn0 12413 ♯chash 14230 Basecbs 17083 0_gc0g 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

W3C validator