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Theorem fta1g 26364

Description: The one-sided fundamental theorem of algebra. A polynomial of degree 𝑛 has at most 𝑛 roots. Unlike the real fundamental theorem fta 27281, 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 2766	. 2 ⊢ (𝐷‘𝐹) = (𝐷‘𝐹)
2		fveqeq2 6897	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝐷‘𝑓) = (𝐷‘𝐹) ↔ (𝐷‘𝐹) = (𝐷‘𝐹)))
3		fveq2 6888	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑂‘𝑓) = (𝑂‘𝐹))
4	3	cnveqd 5866	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ^◡(𝑂‘𝑓) = ^◡(𝑂‘𝐹))
5	4	imaeq1d 6066	. . . . . 6 ⊢ (𝑓 = 𝐹 → (^◡(𝑂‘𝑓) “ {𝑊}) = (^◡(𝑂‘𝐹) “ {𝑊}))
6	5	fveq2d 6892	. . . . 5 ⊢ (𝑓 = 𝐹 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = (♯‘(^◡(𝑂‘𝐹) “ {𝑊})))
7		fveq2 6888	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝐷‘𝑓) = (𝐷‘𝐹))
8	6, 7	breq12d 5127	. . . 4 ⊢ (𝑓 = 𝐹 → ((♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓) ↔ (♯‘(^◡(𝑂‘𝐹) “ {𝑊})) ≤ (𝐷‘𝐹)))
9	2, 8	imbi12d 347	. . 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 20860	. . . . . . 7 ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
12	11	simplbi 502	. . . . . 6 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ CRing)
13		crngring 20358	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
14	10, 12, 13	3syl 19	. . . . 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 26282	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ) → (𝐷‘𝐹) ∈ ℕ₀)
22	14, 15, 16, 21	syl3anc 1398	. . . 4 ⊢ (𝜑 → (𝐷‘𝐹) ∈ ℕ₀)
23		eqeq2 2778	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝐷‘𝑓) = 𝑥 ↔ (𝐷‘𝑓) = 0))
24	23	imbi1d 344	. . . . . . 7 ⊢ (𝑥 = 0 → (((𝐷‘𝑓) = 𝑥 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)) ↔ ((𝐷‘𝑓) = 0 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))))
25	24	ralbidv 3191	. . . . . 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 343	. . . . 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 2778	. . . . . . . 8 ⊢ (𝑥 = 𝑑 → ((𝐷‘𝑓) = 𝑥 ↔ (𝐷‘𝑓) = 𝑑))
28	27	imbi1d 344	. . . . . . 7 ⊢ (𝑥 = 𝑑 → (((𝐷‘𝑓) = 𝑥 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)) ↔ ((𝐷‘𝑓) = 𝑑 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))))
29	28	ralbidv 3191	. . . . . 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 343	. . . . 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 2778	. . . . . . . 8 ⊢ (𝑥 = (𝑑 + 1) → ((𝐷‘𝑓) = 𝑥 ↔ (𝐷‘𝑓) = (𝑑 + 1)))
32	31	imbi1d 344	. . . . . . 7 ⊢ (𝑥 = (𝑑 + 1) → (((𝐷‘𝑓) = 𝑥 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)) ↔ ((𝐷‘𝑓) = (𝑑 + 1) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))))
33	32	ralbidv 3191	. . . . . 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 343	. . . . 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 2778	. . . . . . . 8 ⊢ (𝑥 = (𝐷‘𝐹) → ((𝐷‘𝑓) = 𝑥 ↔ (𝐷‘𝑓) = (𝐷‘𝐹)))
36	35	imbi1d 344	. . . . . . 7 ⊢ (𝑥 = (𝐷‘𝐹) → (((𝐷‘𝑓) = 𝑥 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)) ↔ ((𝐷‘𝑓) = (𝐷‘𝐹) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))))
37	36	ralbidv 3191	. . . . . 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 343	. . . . 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 785	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝐷‘𝑓) = 0)
40		0nn0 12537	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ₀
41	39, 40	eqeltrdi 2874	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝐷‘𝑓) ∈ ℕ₀)
42	12, 13	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ IDomn → 𝑅 ∈ Ring)
43		simpl 488	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0) → 𝑓 ∈ 𝐵)
44	17, 18, 19, 20	deg1nn0clb 26284	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ 𝐵) → (𝑓 ≠ 0 ↔ (𝐷‘𝑓) ∈ ℕ₀))
45	42, 43, 44	syl2an 608	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝑓 ≠ 0 ↔ (𝐷‘𝑓) ∈ ℕ₀))
46	41, 45	mpbird 260	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → 𝑓 ≠ 0 )
47		simplrr 790	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → (𝐷‘𝑓) = 0)
48		0le0 12360	. . . . . . . . . . . . . . . . 17 ⊢ 0 ≤ 0
49	47, 48	eqbrtrdi 5155	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → (𝐷‘𝑓) ≤ 0)
50	42	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑅 ∈ Ring)
51		simplrl 789	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑓 ∈ 𝐵)
52		eqid 2766	. . . . . . . . . . . . . . . . . 18 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
53	17, 18, 20, 52	deg1le0 26305	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝑓 ∈ 𝐵) → ((𝐷‘𝑓) ≤ 0 ↔ 𝑓 = ((algSc‘𝑃)‘((coe₁‘𝑓)‘0))))
54	50, 51, 53	syl2anc 596	. . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑓 = ((algSc‘𝑃)‘((coe₁‘𝑓)‘0)))
56	55	fveq2d 6892	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → (𝑂‘𝑓) = (𝑂‘((algSc‘𝑃)‘((coe₁‘𝑓)‘0))))
57	12	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → 𝑅 ∈ CRing)
58	57	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑅 ∈ CRing)
59		eqid 2766	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (coe₁‘𝑓) = (coe₁‘𝑓)
60		eqid 2766	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Base‘𝑅) = (Base‘𝑅)
61	59, 20, 18, 60	coe1f 22408	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 ∈ 𝐵 → (coe₁‘𝑓):ℕ₀⟶(Base‘𝑅))
62	51, 61	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → (coe₁‘𝑓):ℕ₀⟶(Base‘𝑅))
63		ffvelcdm 7083	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((coe₁‘𝑓):ℕ₀⟶(Base‘𝑅) ∧ 0 ∈ ℕ₀) → ((coe₁‘𝑓)‘0) ∈ (Base‘𝑅))
64	62, 40, 63	sylancl 598	. . . . . . . . . . . . . . . . . . . 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 22531	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ CRing ∧ ((coe₁‘𝑓)‘0) ∈ (Base‘𝑅)) → (𝑂‘((algSc‘𝑃)‘((coe₁‘𝑓)‘0))) = ((Base‘𝑅) × {((coe₁‘𝑓)‘0)}))
67	58, 64, 66	syl2anc 596	. . . . . . . . . . . . . . . . . . 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 2801	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → (𝑂‘𝑓) = ((Base‘𝑅) × {((coe₁‘𝑓)‘0)}))
69	68	fveq1d 6890	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → ((𝑂‘𝑓)‘𝑥) = (((Base‘𝑅) × {((coe₁‘𝑓)‘0)})‘𝑥))
70		eqid 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 ↑_s (Base‘𝑅)) = (𝑅 ↑_s (Base‘𝑅))
71		eqid 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Base‘(𝑅 ↑_s (Base‘𝑅))) = (Base‘(𝑅 ↑_s (Base‘𝑅)))
72		simpl 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → 𝑅 ∈ IDomn)
73		fvexd 6903	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (Base‘𝑅) ∈ V)
74	65, 18, 70, 60	evl1rhm 22529	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅 ∈ CRing → 𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s (Base‘𝑅))))
75	20, 71	rhmf 20600	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑂 ∈ (𝑃 RingHom (𝑅 ↑_s (Base‘𝑅))) → 𝑂:𝐵⟶(Base‘(𝑅 ↑_s (Base‘𝑅))))
76	57, 74, 75	3syl 19	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → 𝑂:𝐵⟶(Base‘(𝑅 ↑_s (Base‘𝑅))))
77		simprl 783	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → 𝑓 ∈ 𝐵)
78	76, 77	ffvelcdmd 7087	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝑂‘𝑓) ∈ (Base‘(𝑅 ↑_s (Base‘𝑅))))
79	70, 60, 71, 72, 73, 78	pwselbas 17567	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝑂‘𝑓):(Base‘𝑅)⟶(Base‘𝑅))
80		ffn 6712	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑂‘𝑓):(Base‘𝑅)⟶(Base‘𝑅) → (𝑂‘𝑓) Fn (Base‘𝑅))
81		fniniseg 7062	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑂‘𝑓) Fn (Base‘𝑅) → (𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ((𝑂‘𝑓)‘𝑥) = 𝑊)))
82	79, 80, 81	3syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ((𝑂‘𝑓)‘𝑥) = 𝑊)))
83	82	simplbda 505	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → ((𝑂‘𝑓)‘𝑥) = 𝑊)
84	82	simprbda 504	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑥 ∈ (Base‘𝑅))
85		fvex 6901	. . . . . . . . . . . . . . . . . . 19 ⊢ ((coe₁‘𝑓)‘0) ∈ V
86	85	fvconst2 7209	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (Base‘𝑅) → (((Base‘𝑅) × {((coe₁‘𝑓)‘0)})‘𝑥) = ((coe₁‘𝑓)‘0))
87	84, 86	syl 18	. . . . . . . . . . . . . . . . 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 2810	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → ((coe₁‘𝑓)‘0) = 𝑊)
89	88	fveq2d 6892	. . . . . . . . . . . . . . 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 22488	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → ((algSc‘𝑃)‘𝑊) = 0 )
92	50, 91	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → ((algSc‘𝑃)‘𝑊) = 0 )
93	55, 89, 92	3eqtrd 2805	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) ∧ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})) → 𝑓 = 0 )
94	93	ex 418	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}) → 𝑓 = 0 ))
95	94	necon3ad 2974	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (𝑓 ≠ 0 → ¬ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊})))
96	46, 95	mpd 16	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → ¬ 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}))
97	96	eq0rdv 4375	. . . . . . . . . 10 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (^◡(𝑂‘𝑓) “ {𝑊}) = ∅)
98	97	fveq2d 6892	. . . . . . . . 9 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = (♯‘∅))
99		hash0 14423	. . . . . . . . 9 ⊢ (♯‘∅) = 0
100	98, 99	eqtrdi 2817	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = 0)
101	48, 39	breqtrrid 5154	. . . . . . . 8 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → 0 ≤ (𝐷‘𝑓))
102	100, 101	eqbrtrd 5138	. . . . . . 7 ⊢ ((𝑅 ∈ IDomn ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = 0)) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))
103	102	expr 462	. . . . . 6 ⊢ ((𝑅 ∈ IDomn ∧ 𝑓 ∈ 𝐵) → ((𝐷‘𝑓) = 0 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)))
104	103	ralrimiva 3160	. . . . 5 ⊢ (𝑅 ∈ IDomn → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 0 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)))
105		fveqeq2 6897	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ((𝐷‘𝑓) = 𝑑 ↔ (𝐷‘𝑔) = 𝑑))
106		fveq2 6888	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → (𝑂‘𝑓) = (𝑂‘𝑔))
107	106	cnveqd 5866	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → ^◡(𝑂‘𝑓) = ^◡(𝑂‘𝑔))
108	107	imaeq1d 6066	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (^◡(𝑂‘𝑓) “ {𝑊}) = (^◡(𝑂‘𝑔) “ {𝑊}))
109	108	fveq2d 6892	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = (♯‘(^◡(𝑂‘𝑔) “ {𝑊})))
110		fveq2 6888	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (𝐷‘𝑓) = (𝐷‘𝑔))
111	109, 110	breq12d 5127	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ((♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓) ↔ (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)))
112	105, 111	imbi12d 347	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (((𝐷‘𝑓) = 𝑑 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)) ↔ ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔))))
113	112	cbvralvw 3246	. . . . . . . 8 ⊢ (∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = 𝑑 → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)) ↔ ∀𝑔 ∈ 𝐵 ((𝐷‘𝑔) = 𝑑 → (♯‘(^◡(𝑂‘𝑔) “ {𝑊})) ≤ (𝐷‘𝑔)))
114		simprr 785	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → (𝐷‘𝑓) = (𝑑 + 1))
115		peano2nn0 12562	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ ℕ₀ → (𝑑 + 1) ∈ ℕ₀)
116	115	ad2antlr 740	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → (𝑑 + 1) ∈ ℕ₀)
117	114, 116	eqeltrd 2866	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → (𝐷‘𝑓) ∈ ℕ₀)
118	117	nn0ge0d 12586	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → 0 ≤ (𝐷‘𝑓))
119		fveq2 6888	. . . . . . . . . . . . . . . 16 ⊢ ((^◡(𝑂‘𝑓) “ {𝑊}) = ∅ → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = (♯‘∅))
120	119, 99	eqtrdi 2817	. . . . . . . . . . . . . . 15 ⊢ ((^◡(𝑂‘𝑓) “ {𝑊}) = ∅ → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) = 0)
121	120	breq1d 5124	. . . . . . . . . . . . . 14 ⊢ ((^◡(𝑂‘𝑓) “ {𝑊}) = ∅ → ((♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓) ↔ 0 ≤ (𝐷‘𝑓)))
122	118, 121	syl5ibrcom 250	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ IDomn ∧ 𝑑 ∈ ℕ₀) ∧ (𝑓 ∈ 𝐵 ∧ (𝐷‘𝑓) = (𝑑 + 1))) → ((^◡(𝑂‘𝑓) “ {𝑊}) = ∅ → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)))
123	122	a1dd 51	. . . . . . . . . . . 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 4310	. . . . . . . . . . . . 13 ⊢ ((^◡(𝑂‘𝑓) “ {𝑊}) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (^◡(𝑂‘𝑓) “ {𝑊}))
125		simplll 787	. . . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . 16 ⊢ (var₁‘𝑅) = (var₁‘𝑅)
128		eqid 2766	. . . . . . . . . . . . . . . 16 ⊢ (-_g‘𝑃) = (-_g‘𝑃)
129		eqid 2766	. . . . . . . . . . . . . . . 16 ⊢ ((var₁‘𝑅)(-_g‘𝑃)((algSc‘𝑃)‘𝑥)) = ((var₁‘𝑅)(-_g‘𝑃)((algSc‘𝑃)‘𝑥))
130		simpllr 788	. . . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . 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 26363	. . . . . . . . . . . . . . 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 426	. . . . . . . . . . . . . 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 1966	. . . . . . . . . . . . 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 245	. . . . . . . . . . . 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 3047	. . . . . . . . . . 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 462	. . . . . . . . . 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 87	. . . . . . . . 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 3168	. . . . . . . 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 245	. . . . . . 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 419	. . . . . 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 30	. . . . 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 12709	. . . 4 ⊢ ((𝐷‘𝐹) ∈ ℕ₀ → (𝑅 ∈ IDomn → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = (𝐷‘𝐹) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓))))
146	22, 10, 145	sylc 66	. . 3 ⊢ (𝜑 → ∀𝑓 ∈ 𝐵 ((𝐷‘𝑓) = (𝐷‘𝐹) → (♯‘(^◡(𝑂‘𝑓) “ {𝑊})) ≤ (𝐷‘𝑓)))
147	9, 146, 15	rspcdva 3585	. 2 ⊢ (𝜑 → ((𝐷‘𝐹) = (𝐷‘𝐹) → (♯‘(^◡(𝑂‘𝐹) “ {𝑊})) ≤ (𝐷‘𝐹)))
148	1, 147	mpi 21	1 ⊢ (𝜑 → (♯‘(^◡(𝑂‘𝐹) “ {𝑊})) ≤ (𝐷‘𝐹))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∃wex 1812 ∈ wcel 2146 ≠ wne 2961 ∀wral 3082 Vcvv 3458 ∅c0 4289 {csn 4594 class class class wbr 5114 × cxp 5664 ^◡ccnv 5665 “ cima 5669 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 (class class class)co 7423 0cc0 11118 1c1 11119 + caddc 11121 ≤ cle 11262 ℕ₀cn0 12522 ♯chash 14386 Basecbs 17294 0_gc0g 17517 ↑_s cpws 17524 -_gcsg 19033 Ringcrg 20346 CRingccrg 20347 RingHom crh 20584 Domncdomn 20828 IDomncidom 20829 algSccascl 22039 var₁cv1 22373 Poly₁cpl1 22374 coe₁cco1 22375 eval₁ce1 22511 deg₁cdg1 26248

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196 ax-addf 11197

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7687 df-ofr 7688 df-om 7872 df-1st 7995 df-2nd 7996 df-supp 8166 df-tpos 8231 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8703 df-map 8835 df-pm 8836 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-sup 9412 df-oi 9482 df-dju 9906 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-xnn0 12596 df-z 12610 df-dec 12730 df-uz 12881 df-fz 13554 df-fzo 13702 df-seq 14058 df-hash 14387 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-starv 17350 df-sca 17351 df-vsca 17352 df-ip 17353 df-tset 17354 df-ple 17355 df-ds 17357 df-unif 17358 df-hom 17359 df-cco 17360 df-0g 17519 df-gsum 17520 df-prds 17525 df-pws 17527 df-mre 17663 df-mrc 17664 df-acs 17666 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-mhm 18872 df-submnd 18873 df-grp 19034 df-minusg 19035 df-sbg 19036 df-mulg 19165 df-subg 19220 df-ghm 19315 df-cntz 19418 df-cmn 19883 df-abl 19884 df-mgp 20248 df-rng 20262 df-ur 20295 df-srg 20300 df-ring 20348 df-cring 20349 df-oppr 20452 df-dvdsr 20472 df-unit 20473 df-invr 20503 df-rhm 20587 df-nzr 20647 df-subrng 20682 df-subrg 20706 df-rlreg 20830 df-domn 20831 df-idom 20832 df-lmod 21020 df-lss 21090 df-lsp 21130 df-cnfld 21560 df-assa 22040 df-asp 22041 df-ascl 22042 df-psr 22096 df-mvr 22097 df-mpl 22098 df-opsr 22100 df-evls 22262 df-evl 22263 df-psr1 22377 df-vr1 22378 df-ply1 22379 df-coe1 22380 df-evl1 22513 df-mdeg 26249 df-deg1 26250 df-mon1 26325 df-uc1p 26326 df-q1p 26327 df-r1p 26328

This theorem is used by: fta1b 26366 idomrootle 26367 lgsqrlem4 27550 aks6d1c2lem4 42935 aks6d1c6lem3 42980

W3C validator