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Theorem fta1g 26209

Description: The one-sided fundamental theorem of algebra. A polynomial of degree 𝑛 has at most 𝑛 roots. Unlike the real fundamental theorem fta 27123, 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 2737	. 2 ⊢ (𝐷‘𝐹) = (𝐷‘𝐹)
2		fveqeq2 6915	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝐷‘𝑓) = (𝐷‘𝐹) ↔ (𝐷‘𝐹) = (𝐷‘𝐹)))
3		fveq2 6906	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑂‘𝑓) = (𝑂‘𝐹))
4	3	cnveqd 5886	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ^◡(𝑂‘𝑓) = ^◡(𝑂‘𝐹))
5	4	imaeq1d 6077	. . . . . 6 ⊢ (𝑓 = 𝐹 → (^◡(𝑂‘𝑓) “ {𝑊}) = (^◡(𝑂‘𝐹) “ {𝑊}))
6	5	fveq2d 6910	. . . . 5 ⊢ (𝑓 = 𝐹 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = (♯‘(^◡(𝑂‘𝐹) “ {𝑊})))
7		fveq2 6906	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝐷‘𝑓) = (𝐷‘𝐹))
8	6, 7	breq12d 5156	. . . 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 20725	. . . . . . 7 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
12	11	simplbi 497	. . . . . 6 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ CRing)
13		crngring 20242	. . . . . 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 26127	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ₀)
22	14, 15, 16, 21	syl3anc 1373	. . . 4 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℕ₀)
23		eqeq2 2749	. . . . . . . 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 3178	. . . . . 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 2749	. . . . . . . 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 3178	. . . . . 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 2749	. . . . . . . 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 3178	. . . . . 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 2749	. . . . . . . 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 3178	. . . . . 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 773	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝐷‘𝑓) = 0)
40		0nn0 12541	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ₀
41	39, 40	eqeltrdi 2849	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝐷‘𝑓) ∈ ℕ₀)
42	12, 13	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ Ring)
43		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0) → 𝑓 ∈ 𝐵)
44	17, 18, 19, 20	deg1nn0clb 26129	. . . . . . . . . . . . . 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 257	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → 𝑓 ≠ 0 )
47		simplrr 778	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → (𝐷‘𝑓) = 0)
48		0le0 12367	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≤ 0
49	47, 48	eqbrtrdi 5182	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → (𝐷‘𝑓) ≤ 0)
50	42	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑅 ∈ Ring)
51		simplrl 777	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑓 ∈ 𝐵)
52		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
53	17, 18, 20, 52	deg1le0 26150	. . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑓 = ((algSc‘𝑃)‘((coe₁‘𝑓)‘0)))
56	55	fveq2d 6910	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → (𝑂‘𝑓) = (𝑂‘((algSc‘𝑃)‘((coe₁‘𝑓)‘0))))
57	12	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → 𝑅 ∈ CRing)
58	57	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑅 ∈ CRing)
59		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (coe₁‘𝑓) = (coe₁‘𝑓)
60		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Base‘𝑅) = (Base‘𝑅)
61	59, 20, 18, 60	coe1f 22213	. . . . . . . . . . . . . . . . . . . . . 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 7101	. . . . . . . . . . . . . . . . . . . . 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 22338	. . . . . . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → (𝑂‘𝑓) = ((Base‘𝑅) × {((coe₁‘𝑓)‘0)}))
69	68	fveq1d 6908	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → ((𝑂‘𝑓)‘𝑥) = (((Base‘𝑅) × {((coe₁‘𝑓)‘0)})‘𝑥))
70		eqid 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 ↑_s (Base‘𝑅)) = (𝑅 ↑_s (Base‘𝑅))
71		eqid 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Base‘(𝑅 ↑_s (Base‘𝑅))) = (Base‘(𝑅 ↑_s (Base‘𝑅)))
72		simpl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → 𝑅 ∈ IDomn)
73		fvexd 6921	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (Base‘𝑅) ∈ V)
74	65, 18, 70, 60	evl1rhm 22336	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s (Base‘𝑅))))
75	20, 71	rhmf 20485	. . . . . . . . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → 𝑓 ∈ 𝐵)
78	76, 77	ffvelcdmd 7105	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝑂‘𝑓) ∈ (Base‘(𝑅 ↑_s (Base‘𝑅))))
79	70, 60, 71, 72, 73, 78	pwselbas 17534	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝑂‘𝑓):(Base‘𝑅)⟶(Base‘𝑅))
80		ffn 6736	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑂‘𝑓):(Base‘𝑅)⟶(Base‘𝑅) → (𝑂‘𝑓) Fn (Base‘𝑅))
81		fniniseg 7080	. . . . . . . . . . . . . . . . . . 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 499	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → ((𝑂‘𝑓)‘𝑥) = 𝑊)
84	82	simprbda 498	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑥 ∈ (Base‘𝑅))
85		fvex 6919	. . . . . . . . . . . . . . . . . . 19 ⊢ ((coe₁‘𝑓)‘0) ∈ V
86	85	fvconst2 7224	. . . . . . . . . . . . . . . . . 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 2786	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → ((coe₁‘𝑓)‘0) = 𝑊)
89	88	fveq2d 6910	. . . . . . . . . . . . . . 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 22293	. . . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑓 = 0 )
94	93	ex 412	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}) → 𝑓 = 0 ))
95	94	necon3ad 2953	. . . . . . . . . . . 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 4407	. . . . . . . . . 10 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (^◡(𝑂‘𝑓) “ {𝑊}) = ∅)
98	97	fveq2d 6910	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = (♯‘∅))
99		hash0 14406	. . . . . . . . 9 ⊢ (♯‘∅) = 0
100	98, 99	eqtrdi 2793	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = 0)
101	48, 39	breqtrrid 5181	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → 0 ≤ (𝐷‘𝑓))
102	100, 101	eqbrtrd 5165	. . . . . . 7 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))
103	102	expr 456	. . . . . 6 ⊢ ((𝑅 ∈ IDomn ∧ 𝑓 ∈ 𝐵) → ((𝐷‘𝑓) = 0 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)))
104	103	ralrimiva 3146	. . . . 5 ⊢ (𝑅 ∈ IDomn → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 0 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)))
105		fveqeq2 6915	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ((𝐷‘𝑓) = 𝑑 ↔ (𝐷‘𝑔) = 𝑑))
106		fveq2 6906	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → (𝑂‘𝑓) = (𝑂‘𝑔))
107	106	cnveqd 5886	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → ^◡(𝑂‘𝑓) = ^◡(𝑂‘𝑔))
108	107	imaeq1d 6077	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (^◡(𝑂‘𝑓) “ {𝑊}) = (^◡(𝑂‘𝑔) “ {𝑊}))
109	108	fveq2d 6910	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = (♯‘(^◡(𝑂‘𝑔) “ {𝑊})))
110		fveq2 6906	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (𝐷‘𝑓) = (𝐷‘𝑔))
111	109, 110	breq12d 5156	. . . . . . . . . 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 3237	. . . . . . . 8 ⊢ (∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 𝑑 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)) ↔ ∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)))
114		simprr 773	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → (𝐷‘𝑓) = (𝑑 + 1))
115		peano2nn0 12566	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ₀)
116	115	ad2antlr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → (𝑑 + 1) ∈ ℕ₀)
117	114, 116	eqeltrd 2841	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → (𝐷‘𝑓) ∈ ℕ₀)
118	117	nn0ge0d 12590	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → 0 ≤ (𝐷‘𝑓))
119		fveq2 6906	. . . . . . . . . . . . . . . 16 ⊢ ((^◡(𝑂‘𝑓) “ {𝑊}) = ∅ → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = (♯‘∅))
120	119, 99	eqtrdi 2793	. . . . . . . . . . . . . . 15 ⊢ ((^◡(𝑂‘𝑓) “ {𝑊}) = ∅ → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = 0)
121	120	breq1d 5153	. . . . . . . . . . . . . 14 ⊢ ((^◡(𝑂‘𝑓) “ {𝑊}) = ∅ → ((♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓) ↔ 0 ≤ (𝐷‘𝑓)))
122	118, 121	syl5ibrcom 247	. . . . . . . . . . . . 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 4353	. . . . . . . . . . . . 13 ⊢ ((^◡(𝑂‘𝑓) “ {𝑊}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}))
125		simplll 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)))) → 𝑅 ∈ IDomn)
126		simplrl 777	. . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . 16 ⊢ (var₁‘𝑅) = (var₁‘𝑅)
128		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (-_g‘𝑃) = (-_g‘𝑃)
129		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ ((var₁‘𝑅)(-_g‘𝑃)((algSc‘𝑃)‘𝑥)) = ((var₁‘𝑅)(-_g‘𝑃)((algSc‘𝑃)‘𝑥))
130		simpllr 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 ∈ ℕ₀)
131		simplrr 778	. . . . . . . . . . . . . . . 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 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)))) → 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}))
133		simprr 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)))) → ∀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 26208	. . . . . . . . . . . . . . 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 420	. . . . . . . . . . . . . 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 1933	. . . . . . . . . . . . 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 242	. . . . . . . . . . . 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 3028	. . . . . . . . . . 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 456	. . . . . . . . . 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 3154	. . . . . . . 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 242	. . . . . . 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 413	. . . . . 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 12713	. . . 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 3623	. 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 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 Vcvv 3480 ∅c0 4333 {csn 4626 class class class wbr 5143 × cxp 5683 ^◡ccnv 5684 “ cima 5688 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 ≤ cle 11296 ℕ₀cn0 12526 ♯chash 14369 Basecbs 17247 0_gc0g 17484 ↑_s cpws 17491 -_gcsg 18953 Ringcrg 20230 CRingccrg 20231 RingHom crh 20469 Domncdomn 20692 IDomncidom 20693 algSccascl 21872 var₁cv1 22177 Poly₁cpl1 22178 coe₁cco1 22179 eval₁ce1 22318 deg₁cdg1 26093

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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-xnn0 12600 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-srg 20184 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-rhm 20472 df-nzr 20513 df-subrng 20546 df-subrg 20570 df-rlreg 20694 df-domn 20695 df-idom 20696 df-lmod 20860 df-lss 20930 df-lsp 20970 df-cnfld 21365 df-assa 21873 df-asp 21874 df-ascl 21875 df-psr 21929 df-mvr 21930 df-mpl 21931 df-opsr 21933 df-evls 22098 df-evl 22099 df-psr1 22181 df-vr1 22182 df-ply1 22183 df-coe1 22184 df-evl1 22320 df-mdeg 26094 df-deg1 26095 df-mon1 26170 df-uc1p 26171 df-q1p 26172 df-r1p 26173

This theorem is referenced by: fta1b 26211 idomrootle 26212 lgsqrlem4 27393 aks6d1c2lem4 42128 aks6d1c6lem3 42173

W3C validator