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Theorem fta1g 26145

Description: The one-sided fundamental theorem of algebra. A polynomial of degree 𝑛 has at most 𝑛 roots. Unlike the real fundamental theorem fta 27057, 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 6843	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝐷‘𝑓) = (𝐷‘𝐹) ↔ (𝐷‘𝐹) = (𝐷‘𝐹)))
3		fveq2 6834	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑂‘𝑓) = (𝑂‘𝐹))
4	3	cnveqd 5824	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ^◡(𝑂‘𝑓) = ^◡(𝑂‘𝐹))
5	4	imaeq1d 6018	. . . . . 6 ⊢ (𝑓 = 𝐹 → (^◡(𝑂‘𝑓) “ {𝑊}) = (^◡(𝑂‘𝐹) “ {𝑊}))
6	5	fveq2d 6838	. . . . 5 ⊢ (𝑓 = 𝐹 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = (♯‘(^◡(𝑂‘𝐹) “ {𝑊})))
7		fveq2 6834	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝐷‘𝑓) = (𝐷‘𝐹))
8	6, 7	breq12d 5099	. . . 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 20693	. . . . . . 7 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
12	11	simplbi 496	. . . . . 6 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ CRing)
13		crngring 20217	. . . . . 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 26063	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ₀)
22	14, 15, 16, 21	syl3anc 1374	. . . 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 3161	. . . . . 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 3161	. . . . . 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 3161	. . . . . 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 3161	. . . . . 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 12443	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ₀
41	39, 40	eqeltrdi 2845	. . . . . . . . . . . . 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 26065	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ 𝐵) → (𝑓 ≠ 0 ↔ (𝐷‘𝑓) ∈ ℕ₀))
45	42, 43, 44	syl2an 597	. . . . . . . . . . . . 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 12273	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≤ 0
49	47, 48	eqbrtrdi 5125	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → (𝐷‘𝑓) ≤ 0)
50	42	ad2antrr 727	. . . . . . . . . . . . . . . . 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 26086	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ 𝐵) → ((𝐷‘𝑓) ≤ 0 ↔ 𝑓 = ((algSc‘𝑃)‘((coe₁‘𝑓)‘0))))
54	50, 51, 53	syl2anc 585	. . . . . . . . . . . . . . . 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 6838	. . . . . . . . . . . . . . . . . . 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 22185	. . . . . . . . . . . . . . . . . . . . . 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 7027	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((coe₁‘𝑓):ℕ₀⟶(Base‘𝑅) ∧ 0 ∈ ℕ₀) → ((coe₁‘𝑓)‘0) ∈ (Base‘𝑅))
64	62, 40, 63	sylancl 587	. . . . . . . . . . . . . . . . . . . 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 22309	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ CRing ∧ ((coe₁‘𝑓)‘0) ∈ (Base‘𝑅)) → (𝑂‘((algSc‘𝑃)‘((coe₁‘𝑓)‘0))) = ((Base‘𝑅) × {((coe₁‘𝑓)‘0)}))
67	58, 64, 66	syl2anc 585	. . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → (𝑂‘𝑓) = ((Base‘𝑅) × {((coe₁‘𝑓)‘0)}))
69	68	fveq1d 6836	. . . . . . . . . . . . . . . . 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 6849	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (Base‘𝑅) ∈ V)
74	65, 18, 70, 60	evl1rhm 22307	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s (Base‘𝑅))))
75	20, 71	rhmf 20455	. . . . . . . . . . . . . . . . . . . . . 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 7031	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝑂‘𝑓) ∈ (Base‘(𝑅 ↑_s (Base‘𝑅))))
79	70, 60, 71, 72, 73, 78	pwselbas 17443	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝑂‘𝑓):(Base‘𝑅)⟶(Base‘𝑅))
80		ffn 6662	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑂‘𝑓):(Base‘𝑅)⟶(Base‘𝑅) → (𝑂‘𝑓) Fn (Base‘𝑅))
81		fniniseg 7006	. . . . . . . . . . . . . . . . . . 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 6847	. . . . . . . . . . . . . . . . . . 19 ⊢ ((coe₁‘𝑓)‘0) ∈ V
86	85	fvconst2 7152	. . . . . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → ((coe₁‘𝑓)‘0) = 𝑊)
89	88	fveq2d 6838	. . . . . . . . . . . . . . 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 22265	. . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . 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 2946	. . . . . . . . . . . 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 4348	. . . . . . . . . 10 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (^◡(𝑂‘𝑓) “ {𝑊}) = ∅)
98	97	fveq2d 6838	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = (♯‘∅))
99		hash0 14320	. . . . . . . . 9 ⊢ (♯‘∅) = 0
100	98, 99	eqtrdi 2788	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = 0)
101	48, 39	breqtrrid 5124	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → 0 ≤ (𝐷‘𝑓))
102	100, 101	eqbrtrd 5108	. . . . . . 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 3130	. . . . 5 ⊢ (𝑅 ∈ IDomn → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 0 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)))
105		fveqeq2 6843	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ((𝐷‘𝑓) = 𝑑 ↔ (𝐷‘𝑔) = 𝑑))
106		fveq2 6834	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → (𝑂‘𝑓) = (𝑂‘𝑔))
107	106	cnveqd 5824	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → ^◡(𝑂‘𝑓) = ^◡(𝑂‘𝑔))
108	107	imaeq1d 6018	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (^◡(𝑂‘𝑓) “ {𝑊}) = (^◡(𝑂‘𝑔) “ {𝑊}))
109	108	fveq2d 6838	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = (♯‘(^◡(𝑂‘𝑔) “ {𝑊})))
110		fveq2 6834	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (𝐷‘𝑓) = (𝐷‘𝑔))
111	109, 110	breq12d 5099	. . . . . . . . . 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 3216	. . . . . . . 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 12468	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ₀)
116	115	ad2antlr 728	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → (𝑑 + 1) ∈ ℕ₀)
117	114, 116	eqeltrd 2837	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → (𝐷‘𝑓) ∈ ℕ₀)
118	117	nn0ge0d 12492	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → 0 ≤ (𝐷‘𝑓))
119		fveq2 6834	. . . . . . . . . . . . . . . 16 ⊢ ((^◡(𝑂‘𝑓) “ {𝑊}) = ∅ → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = (♯‘∅))
120	119, 99	eqtrdi 2788	. . . . . . . . . . . . . . 15 ⊢ ((^◡(𝑂‘𝑓) “ {𝑊}) = ∅ → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = 0)
121	120	breq1d 5096	. . . . . . . . . . . . . 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 4294	. . . . . . . . . . . . 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 26144	. . . . . . . . . . . . . . 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 1935	. . . . . . . . . . . . 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 3019	. . . . . . . . . . 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 3138	. . . . . . . 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 12615	. . . 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 3566	. 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 1542 ∃wex 1781 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 Vcvv 3430 ∅c0 4274 {csn 4568 class class class wbr 5086 × cxp 5622 ^◡ccnv 5623 “ cima 5627 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 0cc0 11029 1c1 11030 + caddc 11032 ≤ cle 11171 ℕ₀cn0 12428 ♯chash 14283 Basecbs 17170 0_gc0g 17393 ↑_s cpws 17400 -_gcsg 18902 Ringcrg 20205 CRingccrg 20206 RingHom crh 20440 Domncdomn 20660 IDomncidom 20661 algSccascl 21842 var₁cv1 22149 Poly₁cpl1 22150 coe₁cco1 22151 eval₁ce1 22289 deg₁cdg1 26029

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-ofr 7625 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-tpos 8169 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-oadd 8402 df-er 8636 df-map 8768 df-pm 8769 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-sup 9348 df-oi 9418 df-dju 9816 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-xnn0 12502 df-z 12516 df-dec 12636 df-uz 12780 df-fz 13453 df-fzo 13600 df-seq 13955 df-hash 14284 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-0g 17395 df-gsum 17396 df-prds 17401 df-pws 17403 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-mulg 19035 df-subg 19090 df-ghm 19179 df-cntz 19283 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-srg 20159 df-ring 20207 df-cring 20208 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-rhm 20443 df-nzr 20481 df-subrng 20514 df-subrg 20538 df-rlreg 20662 df-domn 20663 df-idom 20664 df-lmod 20848 df-lss 20918 df-lsp 20958 df-cnfld 21345 df-assa 21843 df-asp 21844 df-ascl 21845 df-psr 21899 df-mvr 21900 df-mpl 21901 df-opsr 21903 df-evls 22062 df-evl 22063 df-psr1 22153 df-vr1 22154 df-ply1 22155 df-coe1 22156 df-evl1 22291 df-mdeg 26030 df-deg1 26031 df-mon1 26106 df-uc1p 26107 df-q1p 26108 df-r1p 26109

This theorem is referenced by: fta1b 26147 idomrootle 26148 lgsqrlem4 27326 aks6d1c2lem4 42580 aks6d1c6lem3 42625

W3C validator