hbt - Mathbox for Stefan O'Rear

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Theorem hbt 43575

Description: The Hilbert Basis Theorem - the ring of univariate polynomials over a Noetherian ring is a Noetherian ring. (Contributed by Stefan O'Rear, 4-Apr-2015.)

**Hypothesis**
Ref	Expression
hbt.p	⊢ 𝑃 = (Poly₁‘𝑅)

**Assertion**
Ref	Expression
hbt	⊢ (𝑅 ∈ LNoeR → 𝑃 ∈ LNoeR)

Proof of Theorem hbt Dummy variables 𝑎 𝑏 𝑐 𝑒 𝑓 𝑔 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lnrring 43557	. . 3 ⊢ (𝑅 ∈ LNoeR → 𝑅 ∈ Ring)
2		hbt.p	. . . 4 ⊢ 𝑃 = (Poly₁‘𝑅)
3	2	ply1ring 22232	. . 3 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
4	1, 3	syl 17	. 2 ⊢ (𝑅 ∈ LNoeR → 𝑃 ∈ Ring)
5		eqid 2739	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
6		eqid 2739	. . . . . . . 8 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
7	5, 6	islnr3 43560	. . . . . . 7 ⊢ (𝑅 ∈ LNoeR ↔ (𝑅 ∈ Ring ∧ (LIdeal‘𝑅) ∈ (NoeACS‘(Base‘𝑅))))
8	7	simprbi 498	. . . . . 6 ⊢ (𝑅 ∈ LNoeR → (LIdeal‘𝑅) ∈ (NoeACS‘(Base‘𝑅)))
9	8	adantr 481	. . . . 5 ⊢ ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → (LIdeal‘𝑅) ∈ (NoeACS‘(Base‘𝑅)))
10		eqid 2739	. . . . . . 7 ⊢ (LIdeal‘𝑃) = (LIdeal‘𝑃)
11		eqid 2739	. . . . . . 7 ⊢ (ldgIdlSeq‘𝑅) = (ldgIdlSeq‘𝑅)
12	2, 10, 11, 6	hbtlem7 43570	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ((ldgIdlSeq‘𝑅)‘𝑎):ℕ₀⟶(LIdeal‘𝑅))
13	1, 12	sylan 586	. . . . 5 ⊢ ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ((ldgIdlSeq‘𝑅)‘𝑎):ℕ₀⟶(LIdeal‘𝑅))
14	1	ad2antrr 732	. . . . . . 7 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ₀) → 𝑅 ∈ Ring)
15		simplr 774	. . . . . . 7 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ₀) → 𝑎 ∈ (LIdeal‘𝑃))
16		simpr 485	. . . . . . 7 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ₀) → 𝑏 ∈ ℕ₀)
17		peano2nn0 12468	. . . . . . . 8 ⊢ (𝑏 ∈ ℕ₀ → (𝑏 + 1) ∈ ℕ₀)
18	17	adantl 482	. . . . . . 7 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ₀) → (𝑏 + 1) ∈ ℕ₀)
19		nn0re 12437	. . . . . . . . 9 ⊢ (𝑏 ∈ ℕ₀ → 𝑏 ∈ ℝ)
20	19	lep1d 12078	. . . . . . . 8 ⊢ (𝑏 ∈ ℕ₀ → 𝑏 ≤ (𝑏 + 1))
21	20	adantl 482	. . . . . . 7 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ₀) → 𝑏 ≤ (𝑏 + 1))
22	2, 10, 11, 14, 15, 16, 18, 21	hbtlem4 43571	. . . . . 6 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ₀) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑏) ⊆ (((ldgIdlSeq‘𝑅)‘𝑎)‘(𝑏 + 1)))
23	22	ralrimiva 3131	. . . . 5 ⊢ ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∀𝑏 ∈ ℕ₀ (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑏) ⊆ (((ldgIdlSeq‘𝑅)‘𝑎)‘(𝑏 + 1)))
24		nacsfix 43161	. . . . 5 ⊢ (((LIdeal‘𝑅) ∈ (NoeACS‘(Base‘𝑅)) ∧ ((ldgIdlSeq‘𝑅)‘𝑎):ℕ₀⟶(LIdeal‘𝑅) ∧ ∀𝑏 ∈ ℕ₀ (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑏) ⊆ (((ldgIdlSeq‘𝑅)‘𝑎)‘(𝑏 + 1))) → ∃𝑐 ∈ ℕ₀ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
25	9, 13, 23, 24	syl3anc 1379	. . . 4 ⊢ ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∃𝑐 ∈ ℕ₀ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
26		fzfi 13925	. . . . . . 7 ⊢ (0...𝑐) ∈ Fin
27		eqid 2739	. . . . . . . . 9 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
28		simpll 772	. . . . . . . . 9 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑒 ∈ (0...𝑐)) → 𝑅 ∈ LNoeR)
29		simplr 774	. . . . . . . . 9 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑒 ∈ (0...𝑐)) → 𝑎 ∈ (LIdeal‘𝑃))
30		elfznn0 13565	. . . . . . . . . 10 ⊢ (𝑒 ∈ (0...𝑐) → 𝑒 ∈ ℕ₀)
31	30	adantl 482	. . . . . . . . 9 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑒 ∈ (0...𝑐)) → 𝑒 ∈ ℕ₀)
32	2, 10, 11, 27, 28, 29, 31	hbtlem6 43574	. . . . . . . 8 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑒 ∈ (0...𝑐)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏))‘𝑒))
33	32	ralrimiva 3131	. . . . . . 7 ⊢ ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∀𝑒 ∈ (0...𝑐)∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏))‘𝑒))
34		2fveq3 6832	. . . . . . . . . 10 ⊢ (𝑏 = (𝑓‘𝑒) → ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏)) = ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒))))
35	34	fveq1d 6829	. . . . . . . . 9 ⊢ (𝑏 = (𝑓‘𝑒) → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏))‘𝑒) = (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))
36	35	sseq2d 3947	. . . . . . . 8 ⊢ (𝑏 = (𝑓‘𝑒) → ((((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏))‘𝑒) ↔ (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒)))
37	36	ac6sfi 9184	. . . . . . 7 ⊢ (((0...𝑐) ∈ Fin ∧ ∀𝑒 ∈ (0...𝑐)∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏))‘𝑒)) → ∃𝑓(𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒)))
38	26, 33, 37	sylancr 593	. . . . . 6 ⊢ ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∃𝑓(𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒)))
39	38	adantr 481	. . . . 5 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) → ∃𝑓(𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒)))
40		frn 6662	. . . . . . . . . . . . 13 ⊢ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) → ran 𝑓 ⊆ (𝒫 𝑎 ∩ Fin))
41	40	ad2antrl 734	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → ran 𝑓 ⊆ (𝒫 𝑎 ∩ Fin))
42		inss1 4165	. . . . . . . . . . . 12 ⊢ (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑎
43	41, 42	sstrdi 3927	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → ran 𝑓 ⊆ 𝒫 𝑎)
44	43	unissd 4848	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → ∪ ran 𝑓 ⊆ ∪ 𝒫 𝑎)
45		unipw 5389	. . . . . . . . . 10 ⊢ ∪ 𝒫 𝑎 = 𝑎
46	44, 45	sseqtrdi 3955	. . . . . . . . 9 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → ∪ ran 𝑓 ⊆ 𝑎)
47		simpllr 781	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → 𝑎 ∈ (LIdeal‘𝑃))
48		eqid 2739	. . . . . . . . . . 11 ⊢ (Base‘𝑃) = (Base‘𝑃)
49	48, 10	lidlss 21205	. . . . . . . . . 10 ⊢ (𝑎 ∈ (LIdeal‘𝑃) → 𝑎 ⊆ (Base‘𝑃))
50	47, 49	syl 17	. . . . . . . . 9 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → 𝑎 ⊆ (Base‘𝑃))
51	46, 50	sstrd 3925	. . . . . . . 8 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → ∪ ran 𝑓 ⊆ (Base‘𝑃))
52		fvex 6840	. . . . . . . . 9 ⊢ (Base‘𝑃) ∈ V
53	52	elpw2 5262	. . . . . . . 8 ⊢ (∪ ran 𝑓 ∈ 𝒫 (Base‘𝑃) ↔ ∪ ran 𝑓 ⊆ (Base‘𝑃))
54	51, 53	sylibr 235	. . . . . . 7 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → ∪ ran 𝑓 ∈ 𝒫 (Base‘𝑃))
55		simprl 776	. . . . . . . . 9 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → 𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin))
56		ffn 6655	. . . . . . . . 9 ⊢ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) → 𝑓 Fn (0...𝑐))
57		fniunfv 7191	. . . . . . . . 9 ⊢ (𝑓 Fn (0...𝑐) → ∪ 𝑔 ∈ (0...𝑐)(𝑓‘𝑔) = ∪ ran 𝑓)
58	55, 56, 57	3syl 18	. . . . . . . 8 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → ∪ 𝑔 ∈ (0...𝑐)(𝑓‘𝑔) = ∪ ran 𝑓)
59		inss2 4166	. . . . . . . . . . 11 ⊢ (𝒫 𝑎 ∩ Fin) ⊆ Fin
60	55	ffvelcdmda 7025	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ 𝑔 ∈ (0...𝑐)) → (𝑓‘𝑔) ∈ (𝒫 𝑎 ∩ Fin))
61	59, 60	sselid 3913	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ 𝑔 ∈ (0...𝑐)) → (𝑓‘𝑔) ∈ Fin)
62	61	ralrimiva 3131	. . . . . . . . 9 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → ∀𝑔 ∈ (0...𝑐)(𝑓‘𝑔) ∈ Fin)
63		iunfi 9243	. . . . . . . . 9 ⊢ (((0...𝑐) ∈ Fin ∧ ∀𝑔 ∈ (0...𝑐)(𝑓‘𝑔) ∈ Fin) → ∪ 𝑔 ∈ (0...𝑐)(𝑓‘𝑔) ∈ Fin)
64	26, 62, 63	sylancr 593	. . . . . . . 8 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → ∪ 𝑔 ∈ (0...𝑐)(𝑓‘𝑔) ∈ Fin)
65	58, 64	eqeltrrd 2840	. . . . . . 7 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → ∪ ran 𝑓 ∈ Fin)
66	54, 65	elind 4129	. . . . . 6 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → ∪ ran 𝑓 ∈ (𝒫 (Base‘𝑃) ∩ Fin))
67	1	ad3antrrr 736	. . . . . . . 8 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → 𝑅 ∈ Ring)
68	4	ad3antrrr 736	. . . . . . . . 9 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → 𝑃 ∈ Ring)
69	27, 48, 10	rspcl 21228	. . . . . . . . 9 ⊢ ((𝑃 ∈ Ring ∧ ∪ ran 𝑓 ⊆ (Base‘𝑃)) → ((RSpan‘𝑃)‘∪ ran 𝑓) ∈ (LIdeal‘𝑃))
70	68, 51, 69	syl2anc 590	. . . . . . . 8 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → ((RSpan‘𝑃)‘∪ ran 𝑓) ∈ (LIdeal‘𝑃))
71	27, 10	rspssp 21232	. . . . . . . . 9 ⊢ ((𝑃 ∈ Ring ∧ 𝑎 ∈ (LIdeal‘𝑃) ∧ ∪ ran 𝑓 ⊆ 𝑎) → ((RSpan‘𝑃)‘∪ ran 𝑓) ⊆ 𝑎)
72	68, 47, 46, 71	syl3anc 1379	. . . . . . . 8 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → ((RSpan‘𝑃)‘∪ ran 𝑓) ⊆ 𝑎)
73		nn0re 12437	. . . . . . . . . . 11 ⊢ (𝑔 ∈ ℕ₀ → 𝑔 ∈ ℝ)
74	73	adantl 482	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ 𝑔 ∈ ℕ₀) → 𝑔 ∈ ℝ)
75		simplrl 782	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → 𝑐 ∈ ℕ₀)
76	75	adantr 481	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ 𝑔 ∈ ℕ₀) → 𝑐 ∈ ℕ₀)
77	76	nn0red 12490	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ 𝑔 ∈ ℕ₀) → 𝑐 ∈ ℝ)
78		simprl 776	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)) → 𝑔 ∈ ℕ₀)
79		simprr 778	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)) → 𝑔 ≤ 𝑐)
80	75	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)) → 𝑐 ∈ ℕ₀)
81		fznn0 13564	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ℕ₀ → (𝑔 ∈ (0...𝑐) ↔ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)))
82	80, 81	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)) → (𝑔 ∈ (0...𝑐) ↔ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)))
83	78, 79, 82	mpbir2and 719	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)) → 𝑔 ∈ (0...𝑐))
84		simplrr 783	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)) → ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))
85		fveq2 6827	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = 𝑔 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔))
86		2fveq3 6832	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = 𝑔 → ((RSpan‘𝑃)‘(𝑓‘𝑒)) = ((RSpan‘𝑃)‘(𝑓‘𝑔)))
87	86	fveq2d 6831	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = 𝑔 → ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒))) = ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑔))))
88		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = 𝑔 → 𝑒 = 𝑔)
89	87, 88	fveq12d 6834	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = 𝑔 → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒) = (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑔)))‘𝑔))
90	85, 89	sseq12d 3948	. . . . . . . . . . . . . 14 ⊢ (𝑒 = 𝑔 → ((((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒) ↔ (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑔)))‘𝑔)))
91	90	rspcva 3558	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∈ (0...𝑐) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒)) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑔)))‘𝑔))
92	83, 84, 91	syl2anc 590	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑔)))‘𝑔))
93	67	adantr 481	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)) → 𝑅 ∈ Ring)
94		fvssunirn 6858	. . . . . . . . . . . . . . . 16 ⊢ (𝑓‘𝑔) ⊆ ∪ ran 𝑓
95	94, 51	sstrid 3926	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → (𝑓‘𝑔) ⊆ (Base‘𝑃))
96	27, 48, 10	rspcl 21228	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ Ring ∧ (𝑓‘𝑔) ⊆ (Base‘𝑃)) → ((RSpan‘𝑃)‘(𝑓‘𝑔)) ∈ (LIdeal‘𝑃))
97	68, 95, 96	syl2anc 590	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → ((RSpan‘𝑃)‘(𝑓‘𝑔)) ∈ (LIdeal‘𝑃))
98	97	adantr 481	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)) → ((RSpan‘𝑃)‘(𝑓‘𝑔)) ∈ (LIdeal‘𝑃))
99	70	adantr 481	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)) → ((RSpan‘𝑃)‘∪ ran 𝑓) ∈ (LIdeal‘𝑃))
100	67, 3	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → 𝑃 ∈ Ring)
101	100	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)) → 𝑃 ∈ Ring)
102	27, 48	rspssid 21229	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ Ring ∧ ∪ ran 𝑓 ⊆ (Base‘𝑃)) → ∪ ran 𝑓 ⊆ ((RSpan‘𝑃)‘∪ ran 𝑓))
103	68, 51, 102	syl2anc 590	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → ∪ ran 𝑓 ⊆ ((RSpan‘𝑃)‘∪ ran 𝑓))
104	103	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)) → ∪ ran 𝑓 ⊆ ((RSpan‘𝑃)‘∪ ran 𝑓))
105	94, 104	sstrid 3926	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)) → (𝑓‘𝑔) ⊆ ((RSpan‘𝑃)‘∪ ran 𝑓))
106	27, 10	rspssp 21232	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ Ring ∧ ((RSpan‘𝑃)‘∪ ran 𝑓) ∈ (LIdeal‘𝑃) ∧ (𝑓‘𝑔) ⊆ ((RSpan‘𝑃)‘∪ ran 𝑓)) → ((RSpan‘𝑃)‘(𝑓‘𝑔)) ⊆ ((RSpan‘𝑃)‘∪ ran 𝑓))
107	101, 99, 105, 106	syl3anc 1379	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)) → ((RSpan‘𝑃)‘(𝑓‘𝑔)) ⊆ ((RSpan‘𝑃)‘∪ ran 𝑓))
108	2, 10, 11, 93, 98, 99, 107, 78	hbtlem3 43572	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)) → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑔)))‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑔))
109	92, 108	sstrd 3925	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑔))
110	109	anassrs 468	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ 𝑔 ∈ ℕ₀) ∧ 𝑔 ≤ 𝑐) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑔))
111		nn0z 12539	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ ℕ₀ → 𝑐 ∈ ℤ)
112	111	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ ℕ₀ ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → 𝑐 ∈ ℤ)
113		nn0z 12539	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 ∈ ℕ₀ → 𝑔 ∈ ℤ)
114	113	ad2antrl 734	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ ℕ₀ ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → 𝑔 ∈ ℤ)
115		simprr 778	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ ℕ₀ ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → 𝑐 ≤ 𝑔)
116		eluz2 12785	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (ℤ_≥‘𝑐) ↔ (𝑐 ∈ ℤ ∧ 𝑔 ∈ ℤ ∧ 𝑐 ≤ 𝑔))
117	112, 114, 115, 116	syl3anbrc 1350	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ ℕ₀ ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → 𝑔 ∈ (ℤ_≥‘𝑐))
118	75, 117	sylan 586	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → 𝑔 ∈ (ℤ_≥‘𝑐))
119		simprr 778	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) → ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
120	119	ad2antrr 732	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
121		fveqeq2 6836	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑔 → ((((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐) ↔ (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐)))
122	121	rspcva 3558	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∈ (ℤ_≥‘𝑐) ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐)) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
123	118, 120, 122	syl2anc 590	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
124	75	nn0red 12490	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → 𝑐 ∈ ℝ)
125	124	leidd 11707	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → 𝑐 ≤ 𝑐)
126	109	expr 457	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ 𝑔 ∈ ℕ₀) → (𝑔 ≤ 𝑐 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑔)))
127	126	ralrimiva 3131	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → ∀𝑔 ∈ ℕ₀ (𝑔 ≤ 𝑐 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑔)))
128		breq1 5075	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = 𝑐 → (𝑔 ≤ 𝑐 ↔ 𝑐 ≤ 𝑐))
129		fveq2 6827	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑐 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
130		fveq2 6827	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑐 → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑔) = (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑐))
131	129, 130	sseq12d 3948	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = 𝑐 → ((((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑔) ↔ (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑐)))
132	128, 131	imbi12d 345	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝑐 → ((𝑔 ≤ 𝑐 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑔)) ↔ (𝑐 ≤ 𝑐 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑐))))
133	132	rspcva 3558	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 ∈ ℕ₀ ∧ ∀𝑔 ∈ ℕ₀ (𝑔 ≤ 𝑐 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑔))) → (𝑐 ≤ 𝑐 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑐)))
134	75, 127, 133	syl2anc 590	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → (𝑐 ≤ 𝑐 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑐)))
135	125, 134	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑐))
136	135	adantr 481	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑐))
137	67	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → 𝑅 ∈ Ring)
138	70	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → ((RSpan‘𝑃)‘∪ ran 𝑓) ∈ (LIdeal‘𝑃))
139	75	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → 𝑐 ∈ ℕ₀)
140		simprl 776	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → 𝑔 ∈ ℕ₀)
141		simprr 778	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → 𝑐 ≤ 𝑔)
142	2, 10, 11, 137, 138, 139, 140, 141	hbtlem4 43571	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑐) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑔))
143	136, 142	sstrd 3925	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑔))
144	123, 143	eqsstrd 3949	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑔))
145	144	anassrs 468	. . . . . . . . . 10 ⊢ ((((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ 𝑔 ∈ ℕ₀) ∧ 𝑐 ≤ 𝑔) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑔))
146	74, 77, 110, 145	lecasei 11243	. . . . . . . . 9 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ 𝑔 ∈ ℕ₀) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑔))
147	146	ralrimiva 3131	. . . . . . . 8 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → ∀𝑔 ∈ ℕ₀ (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑔))
148	2, 10, 11, 67, 70, 47, 72, 147	hbtlem5 43573	. . . . . . 7 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → ((RSpan‘𝑃)‘∪ ran 𝑓) = 𝑎)
149	148	eqcomd 2745	. . . . . 6 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → 𝑎 = ((RSpan‘𝑃)‘∪ ran 𝑓))
150		fveq2 6827	. . . . . . 7 ⊢ (𝑏 = ∪ ran 𝑓 → ((RSpan‘𝑃)‘𝑏) = ((RSpan‘𝑃)‘∪ ran 𝑓))
151	150	rspceeqv 3583	. . . . . 6 ⊢ ((∪ ran 𝑓 ∈ (𝒫 (Base‘𝑃) ∩ Fin) ∧ 𝑎 = ((RSpan‘𝑃)‘∪ ran 𝑓)) → ∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏))
152	66, 149, 151	syl2anc 590	. . . . 5 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → ∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏))
153	39, 152	exlimddv 1942	. . . 4 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) → ∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏))
154	25, 153	rexlimddv 3146	. . 3 ⊢ ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏))
155	154	ralrimiva 3131	. 2 ⊢ (𝑅 ∈ LNoeR → ∀𝑎 ∈ (LIdeal‘𝑃)∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏))
156	48, 10, 27	islnr2 43559	. 2 ⊢ (𝑃 ∈ LNoeR ↔ (𝑃 ∈ Ring ∧ ∀𝑎 ∈ (LIdeal‘𝑃)∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏)))
157	4, 155, 156	sylanbrc 589	1 ⊢ (𝑅 ∈ LNoeR → 𝑃 ∈ LNoeR)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∃wex 1786 ∈ wcel 2119 ∀wral 3053 ∃wrex 3063 ∩ cin 3882 ⊆ wss 3883 𝒫 cpw 4529 ∪ cuni 4838 ∪ ciun 4921 class class class wbr 5072 ran crn 5619 Fn wfn 6480 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 Fincfn 8883 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 ≤ cle 11171 ℕ₀cn0 12428 ℤcz 12515 ℤ_≥cuz 12779 ...cfz 13452 Basecbs 17170 Ringcrg 20205 LIdealclidl 21199 RSpancrsp 21200 Poly₁cpl1 22162 NoeACScnacs 43151 LNoeRclnr 43554 ldgIdlSeqcldgis 43566

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 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 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-ofr 7621 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-sup 9345 df-oi 9415 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-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-ocomp 17232 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-proset 18251 df-drs 18252 df-poset 18270 df-ipo 18485 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-ring 20207 df-cring 20208 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-subrng 20518 df-subrg 20542 df-rlreg 20666 df-lmod 20852 df-lss 20922 df-lsp 20962 df-sra 21163 df-rgmod 21164 df-lidl 21201 df-rsp 21202 df-cnfld 21348 df-ascl 21830 df-psr 21884 df-mvr 21885 df-mpl 21886 df-opsr 21888 df-psr1 22165 df-vr1 22166 df-ply1 22167 df-coe1 22168 df-mdeg 26038 df-deg1 26039 df-nacs 43152 df-lfig 43513 df-lnm 43521 df-lnr 43555 df-ldgis 43567

This theorem is referenced by: (None)

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