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Theorem fta1g 25677

Description: The one-sided fundamental theorem of algebra. A polynomial of degree 𝑛 has at most 𝑛 roots. Unlike the real fundamental theorem fta 26574, 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 2733	. 2 ⊢ (𝐷‘𝐹) = (𝐷‘𝐹)
2		fveqeq2 6898	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝐷‘𝑓) = (𝐷‘𝐹) ↔ (𝐷‘𝐹) = (𝐷‘𝐹)))
3		fveq2 6889	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑂‘𝑓) = (𝑂‘𝐹))
4	3	cnveqd 5874	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ^◡(𝑂‘𝑓) = ^◡(𝑂‘𝐹))
5	4	imaeq1d 6057	. . . . . 6 ⊢ (𝑓 = 𝐹 → (^◡(𝑂‘𝑓) “ {𝑊}) = (^◡(𝑂‘𝐹) “ {𝑊}))
6	5	fveq2d 6893	. . . . 5 ⊢ (𝑓 = 𝐹 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = (♯‘(^◡(𝑂‘𝐹) “ {𝑊})))
7		fveq2 6889	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝐷‘𝑓) = (𝐷‘𝐹))
8	6, 7	breq12d 5161	. . . 4 ⊢ (𝑓 = 𝐹 → ((♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓) ↔ (♯‘(^◡(𝑂‘𝐹) “ {𝑊})) ≤ (𝐷‘𝐹)))
9	2, 8	imbi12d 345	. . 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 20915	. . . . . . 7 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
12	11	simplbi 499	. . . . . 6 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ CRing)
13		crngring 20062	. . . . . 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 25598	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ₀)
22	14, 15, 16, 21	syl3anc 1372	. . . 4 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℕ₀)
23		eqeq2 2745	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝐷‘𝑓) = 𝑥 ↔ (𝐷‘𝑓) = 0))
24	23	imbi1d 342	. . . . . . 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 341	. . . . 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 2745	. . . . . . . 8 ⊢ (𝑥 = 𝑑 → ((𝐷‘𝑓) = 𝑥 ↔ (𝐷‘𝑓) = 𝑑))
28	27	imbi1d 342	. . . . . . 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 341	. . . . 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 2745	. . . . . . . 8 ⊢ (𝑥 = (𝑑 + 1) → ((𝐷‘𝑓) = 𝑥 ↔ (𝐷‘𝑓) = (𝑑 + 1)))
32	31	imbi1d 342	. . . . . . 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 341	. . . . 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 2745	. . . . . . . 8 ⊢ (𝑥 = (𝐷‘𝐹) → ((𝐷‘𝑓) = 𝑥 ↔ (𝐷‘𝑓) = (𝐷‘𝐹)))
36	35	imbi1d 342	. . . . . . 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 341	. . . . 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 772	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝐷‘𝑓) = 0)
40		0nn0 12484	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ₀
41	39, 40	eqeltrdi 2842	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝐷‘𝑓) ∈ ℕ₀)
42	12, 13	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ Ring)
43		simpl 484	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0) → 𝑓 ∈ 𝐵)
44	17, 18, 19, 20	deg1nn0clb 25600	. . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → (𝐷‘𝑓) = 0)
48		0le0 12310	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≤ 0
49	47, 48	eqbrtrdi 5187	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → (𝐷‘𝑓) ≤ 0)
50	42	ad2antrr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑅 ∈ Ring)
51		simplrl 776	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑓 ∈ 𝐵)
52		eqid 2733	. . . . . . . . . . . . . . . . . 18 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
53	17, 18, 20, 52	deg1le0 25621	. . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑓 = ((algSc‘𝑃)‘((coe₁‘𝑓)‘0)))
56	55	fveq2d 6893	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → (𝑂‘𝑓) = (𝑂‘((algSc‘𝑃)‘((coe₁‘𝑓)‘0))))
57	12	adantr 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → 𝑅 ∈ CRing)
58	57	adantr 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑅 ∈ CRing)
59		eqid 2733	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (coe₁‘𝑓) = (coe₁‘𝑓)
60		eqid 2733	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Base‘𝑅) = (Base‘𝑅)
61	59, 20, 18, 60	coe1f 21727	. . . . . . . . . . . . . . . . . . . . . 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 7081	. . . . . . . . . . . . . . . . . . . . 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 21845	. . . . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → (𝑂‘𝑓) = ((Base‘𝑅) × {((coe₁‘𝑓)‘0)}))
69	68	fveq1d 6891	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → ((𝑂‘𝑓)‘𝑥) = (((Base‘𝑅) × {((coe₁‘𝑓)‘0)})‘𝑥))
70		eqid 2733	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 ↑_s (Base‘𝑅)) = (𝑅 ↑_s (Base‘𝑅))
71		eqid 2733	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Base‘(𝑅 ↑_s (Base‘𝑅))) = (Base‘(𝑅 ↑_s (Base‘𝑅)))
72		simpl 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → 𝑅 ∈ IDomn)
73		fvexd 6904	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (Base‘𝑅) ∈ V)
74	65, 18, 70, 60	evl1rhm 21843	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s (Base‘𝑅))))
75	20, 71	rhmf 20256	. . . . . . . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → 𝑓 ∈ 𝐵)
78	76, 77	ffvelcdmd 7085	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝑂‘𝑓) ∈ (Base‘(𝑅 ↑_s (Base‘𝑅))))
79	70, 60, 71, 72, 73, 78	pwselbas 17432	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝑂‘𝑓):(Base‘𝑅)⟶(Base‘𝑅))
80		ffn 6715	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑂‘𝑓):(Base‘𝑅)⟶(Base‘𝑅) → (𝑂‘𝑓) Fn (Base‘𝑅))
81		fniniseg 7059	. . . . . . . . . . . . . . . . . . 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 501	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → ((𝑂‘𝑓)‘𝑥) = 𝑊)
84	82	simprbda 500	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑥 ∈ (Base‘𝑅))
85		fvex 6902	. . . . . . . . . . . . . . . . . . 19 ⊢ ((coe₁‘𝑓)‘0) ∈ V
86	85	fvconst2 7202	. . . . . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → ((coe₁‘𝑓)‘0) = 𝑊)
89	88	fveq2d 6893	. . . . . . . . . . . . . . 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 21804	. . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑓 = 0 )
94	93	ex 414	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}) → 𝑓 = 0 ))
95	94	necon3ad 2954	. . . . . . . . . . . 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 4404	. . . . . . . . . 10 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (^◡(𝑂‘𝑓) “ {𝑊}) = ∅)
98	97	fveq2d 6893	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = (♯‘∅))
99		hash0 14324	. . . . . . . . 9 ⊢ (♯‘∅) = 0
100	98, 99	eqtrdi 2789	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = 0)
101	48, 39	breqtrrid 5186	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → 0 ≤ (𝐷‘𝑓))
102	100, 101	eqbrtrd 5170	. . . . . . 7 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))
103	102	expr 458	. . . . . 6 ⊢ ((𝑅 ∈ IDomn ∧ 𝑓 ∈ 𝐵) → ((𝐷‘𝑓) = 0 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)))
104	103	ralrimiva 3147	. . . . 5 ⊢ (𝑅 ∈ IDomn → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 0 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)))
105		fveqeq2 6898	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ((𝐷‘𝑓) = 𝑑 ↔ (𝐷‘𝑔) = 𝑑))
106		fveq2 6889	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → (𝑂‘𝑓) = (𝑂‘𝑔))
107	106	cnveqd 5874	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → ^◡(𝑂‘𝑓) = ^◡(𝑂‘𝑔))
108	107	imaeq1d 6057	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (^◡(𝑂‘𝑓) “ {𝑊}) = (^◡(𝑂‘𝑔) “ {𝑊}))
109	108	fveq2d 6893	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = (♯‘(^◡(𝑂‘𝑔) “ {𝑊})))
110		fveq2 6889	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (𝐷‘𝑓) = (𝐷‘𝑔))
111	109, 110	breq12d 5161	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ((♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓) ↔ (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)))
112	105, 111	imbi12d 345	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (((𝐷‘𝑓) = 𝑑 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)) ↔ ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔))))
113	112	cbvralvw 3235	. . . . . . . 8 ⊢ (∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 𝑑 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)) ↔ ∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)))
114		simprr 772	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → (𝐷‘𝑓) = (𝑑 + 1))
115		peano2nn0 12509	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ₀)
116	115	ad2antlr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → (𝑑 + 1) ∈ ℕ₀)
117	114, 116	eqeltrd 2834	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → (𝐷‘𝑓) ∈ ℕ₀)
118	117	nn0ge0d 12532	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → 0 ≤ (𝐷‘𝑓))
119		fveq2 6889	. . . . . . . . . . . . . . . 16 ⊢ ((^◡(𝑂‘𝑓) “ {𝑊}) = ∅ → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = (♯‘∅))
120	119, 99	eqtrdi 2789	. . . . . . . . . . . . . . 15 ⊢ ((^◡(𝑂‘𝑓) “ {𝑊}) = ∅ → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = 0)
121	120	breq1d 5158	. . . . . . . . . . . . . 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 4346	. . . . . . . . . . . . 13 ⊢ ((^◡(𝑂‘𝑓) “ {𝑊}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}))
125		simplll 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)))) → 𝑅 ∈ IDomn)
126		simplrl 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)))) → 𝑓 ∈ 𝐵)
127		eqid 2733	. . . . . . . . . . . . . . . 16 ⊢ (var₁‘𝑅) = (var₁‘𝑅)
128		eqid 2733	. . . . . . . . . . . . . . . 16 ⊢ (-_g‘𝑃) = (-_g‘𝑃)
129		eqid 2733	. . . . . . . . . . . . . . . 16 ⊢ ((var₁‘𝑅)(-_g‘𝑃)((algSc‘𝑃)‘𝑥)) = ((var₁‘𝑅)(-_g‘𝑃)((algSc‘𝑃)‘𝑥))
130		simpllr 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)))) → 𝑑 ∈ ℕ₀)
131		simplrr 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)))) → (𝐷‘𝑓) = (𝑑 + 1))
132		simprl 770	. . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . 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 25676	. . . . . . . . . . . . . . 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 422	. . . . . . . . . . . . . 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 1937	. . . . . . . . . . . . 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 3029	. . . . . . . . . . 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 458	. . . . . . . . . 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 3155	. . . . . . . 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 415	. . . . . 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 12654	. . . 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 3614	. 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 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 Vcvv 3475 ∅c0 4322 {csn 4628 class class class wbr 5148 × cxp 5674 ^◡ccnv 5675 “ cima 5679 Fn wfn 6536 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 0cc0 11107 1c1 11108 + caddc 11110 ≤ cle 11246 ℕ₀cn0 12469 ♯chash 14287 Basecbs 17141 0_gc0g 17382 ↑_s cpws 17389 -_gcsg 18818 Ringcrg 20050 CRingccrg 20051 RingHom crh 20241 Domncdomn 20889 IDomncidom 20890 algSccascl 21399 var₁cv1 21692 Poly₁cpl1 21693 coe₁cco1 21694 eval₁ce1 21825 deg₁ cdg1 25561

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-ofr 7668 df-om 7853 df-1st 7972 df-2nd 7973 df-supp 8144 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-oadd 8467 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-sup 9434 df-oi 9502 df-dju 9893 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-xnn0 12542 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-fzo 13625 df-seq 13964 df-hash 14288 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-starv 17209 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-hom 17218 df-cco 17219 df-0g 17384 df-gsum 17385 df-prds 17390 df-pws 17392 df-mre 17527 df-mrc 17528 df-acs 17530 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-mhm 18668 df-submnd 18669 df-grp 18819 df-minusg 18820 df-sbg 18821 df-mulg 18946 df-subg 18998 df-ghm 19085 df-cntz 19176 df-cmn 19645 df-abl 19646 df-mgp 19983 df-ur 20000 df-srg 20004 df-ring 20052 df-cring 20053 df-oppr 20143 df-dvdsr 20164 df-unit 20165 df-invr 20195 df-rnghom 20244 df-nzr 20285 df-subrg 20354 df-lmod 20466 df-lss 20536 df-lsp 20576 df-rlreg 20892 df-domn 20893 df-idom 20894 df-cnfld 20938 df-assa 21400 df-asp 21401 df-ascl 21402 df-psr 21454 df-mvr 21455 df-mpl 21456 df-opsr 21458 df-evls 21627 df-evl 21628 df-psr1 21696 df-vr1 21697 df-ply1 21698 df-coe1 21699 df-evl1 21827 df-mdeg 25562 df-deg1 25563 df-mon1 25640 df-uc1p 25641 df-q1p 25642 df-r1p 25643

This theorem is referenced by: fta1b 25679 lgsqrlem4 26842 idomrootle 41923

W3C validator