hbt - Mathbox for Stefan O'Rear

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Theorem hbt 44272

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 44254	. . 3 ⊢ (𝑅 ∈ LNoeR → 𝑅 ∈ Ring)
2		hbt.p	. . . 4 ⊢ 𝑃 = (Poly₁‘𝑅)
3	2	ply1ring 22577	. . 3 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
4	1, 3	syl 18	. 2 ⊢ (𝑅 ∈ LNoeR → 𝑃 ∈ Ring)
5		eqid 2763	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
6		eqid 2763	. . . . . . . 8 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
7	5, 6	islnr3 44257	. . . . . . 7 ⊢ (𝑅 ∈ LNoeR ↔ (𝑅 ∈ Ring ∧ (LIdeal‘𝑅) ∈ (NoeACS‘(Base‘𝑅))))
8	7	simprbi 503	. . . . . 6 ⊢ (𝑅 ∈ LNoeR → (LIdeal‘𝑅) ∈ (NoeACS‘(Base‘𝑅)))
9	8	adantr 486	. . . . 5 ⊢ ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → (LIdeal‘𝑅) ∈ (NoeACS‘(Base‘𝑅)))
10		eqid 2763	. . . . . . 7 ⊢ (LIdeal‘𝑃) = (LIdeal‘𝑃)
11		eqid 2763	. . . . . . 7 ⊢ (ldgIdlSeq‘𝑅) = (ldgIdlSeq‘𝑅)
12	2, 10, 11, 6	hbtlem7 44267	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ((ldgIdlSeq‘𝑅)‘𝑎):ℕ₀⟶(LIdeal‘𝑅))
13	1, 12	sylan 592	. . . . 5 ⊢ ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ((ldgIdlSeq‘𝑅)‘𝑎):ℕ₀⟶(LIdeal‘𝑅))
14	1	ad2antrr 739	. . . . . . 7 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ₀) → 𝑅 ∈ Ring)
15		simplr 781	. . . . . . 7 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ₀) → 𝑎 ∈ (LIdeal‘𝑃))
16		simpr 490	. . . . . . 7 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ₀) → 𝑏 ∈ ℕ₀)
17		peano2nn0 12656	. . . . . . . 8 ⊢ (𝑏 ∈ ℕ₀ → (𝑏 + 1) ∈ ℕ₀)
18	17	adantl 487	. . . . . . 7 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ₀) → (𝑏 + 1) ∈ ℕ₀)
19		nn0re 12625	. . . . . . . . 9 ⊢ (𝑏 ∈ ℕ₀ → 𝑏 ∈ ℝ)
20	19	lep1d 12258	. . . . . . . 8 ⊢ (𝑏 ∈ ℕ₀ → 𝑏 ≤ (𝑏 + 1))
21	20	adantl 487	. . . . . . 7 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ₀) → 𝑏 ≤ (𝑏 + 1))
22	2, 10, 11, 14, 15, 16, 18, 21	hbtlem4 44268	. . . . . 6 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ₀) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑏) ⊆ (((ldgIdlSeq‘𝑅)‘𝑎)‘(𝑏 + 1)))
23	22	ralrimiva 3157	. . . . 5 ⊢ ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∀𝑏 ∈ ℕ₀ (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑏) ⊆ (((ldgIdlSeq‘𝑅)‘𝑎)‘(𝑏 + 1)))
24		nacsfix 43863	. . . . 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 1398	. . . 4 ⊢ ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∃𝑐 ∈ ℕ₀ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
26		fzfi 14125	. . . . . . 7 ⊢ (0...𝑐) ∈ Fin
27		eqid 2763	. . . . . . . . 9 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
28		simpll 779	. . . . . . . . 9 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑒 ∈ (0...𝑐)) → 𝑅 ∈ LNoeR)
29		simplr 781	. . . . . . . . 9 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑒 ∈ (0...𝑐)) → 𝑎 ∈ (LIdeal‘𝑃))
30		elfznn0 13764	. . . . . . . . . 10 ⊢ (𝑒 ∈ (0...𝑐) → 𝑒 ∈ ℕ₀)
31	30	adantl 487	. . . . . . . . 9 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑒 ∈ (0...𝑐)) → 𝑒 ∈ ℕ₀)
32	2, 10, 11, 27, 28, 29, 31	hbtlem6 44271	. . . . . . . 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 3157	. . . . . . 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 6898	. . . . . . . . . 10 ⊢ (𝑏 = (𝑓‘𝑒) → ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏)) = ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒))))
35	34	fveq1d 6895	. . . . . . . . 9 ⊢ (𝑏 = (𝑓‘𝑒) → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏))‘𝑒) = (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))
36	35	sseq2d 3944	. . . . . . . 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 9286	. . . . . . 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 599	. . . . . 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 486	. . . . 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 6725	. . . . . . . . . . . . 13 ⊢ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) → ran 𝑓 ⊆ (𝒫 𝑎 ∩ Fin))
41	40	ad2antrl 741	. . . . . . . . . . . 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 4135	. . . . . . . . . . . 12 ⊢ (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑎
43	41, 42	sstrdi 3924	. . . . . . . . . . 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 4879	. . . . . . . . . 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 5421	. . . . . . . . . 10 ⊢ ∪ 𝒫 𝑎 = 𝑎
46	44, 45	sseqtrdi 3952	. . . . . . . . 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 788	. . . . . . . . . 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 2763	. . . . . . . . . . 11 ⊢ (Base‘𝑃) = (Base‘𝑃)
49	48, 10	lidlss 21502	. . . . . . . . . 10 ⊢ (𝑎 ∈ (LIdeal‘𝑃) → 𝑎 ⊆ (Base‘𝑃))
50	47, 49	syl 18	. . . . . . . . 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 3922	. . . . . . . 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 6906	. . . . . . . . 9 ⊢ (Base‘𝑃) ∈ V
53	52	elpw2 5299	. . . . . . . 8 ⊢ (∪ ran 𝑓 ∈ 𝒫 (Base‘𝑃) ↔ ∪ ran 𝑓 ⊆ (Base‘𝑃))
54	51, 53	sylibr 237	. . . . . . 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 783	. . . . . . . . 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 6717	. . . . . . . . 9 ⊢ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) → 𝑓 Fn (0...𝑐))
57		fniunfv 7258	. . . . . . . . 9 ⊢ (𝑓 Fn (0...𝑐) → ∪ 𝑔 ∈ (0...𝑐)(𝑓‘𝑔) = ∪ ran 𝑓)
58	55, 56, 57	3syl 19	. . . . . . . 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 4136	. . . . . . . . . . 11 ⊢ (𝒫 𝑎 ∩ Fin) ⊆ Fin
60	55	ffvelcdmda 7092	. . . . . . . . . . 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 3910	. . . . . . . . . 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 3157	. . . . . . . . 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 9343	. . . . . . . . 9 ⊢ (((0...𝑐) ∈ Fin ∧ ∀𝑔 ∈ (0...𝑐)(𝑓‘𝑔) ∈ Fin) → ∪ 𝑔 ∈ (0...𝑐)(𝑓‘𝑔) ∈ Fin)
64	26, 62, 63	sylancr 599	. . . . . . . 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 2864	. . . . . . 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 4099	. . . . . 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 743	. . . . . . . 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 743	. . . . . . . . 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 21530	. . . . . . . . 9 ⊢ ((𝑃 ∈ Ring ∧ ∪ ran 𝑓 ⊆ (Base‘𝑃)) → ((RSpan‘𝑃)‘∪ ran 𝑓) ∈ (LIdeal‘𝑃))
70	68, 51, 69	syl2anc 596	. . . . . . . 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 21534	. . . . . . . . 9 ⊢ ((𝑃 ∈ Ring ∧ 𝑎 ∈ (LIdeal‘𝑃) ∧ ∪ ran 𝑓 ⊆ 𝑎) → ((RSpan‘𝑃)‘∪ ran 𝑓) ⊆ 𝑎)
72	68, 47, 46, 71	syl3anc 1398	. . . . . . . 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 12625	. . . . . . . . . . 11 ⊢ (𝑔 ∈ ℕ₀ → 𝑔 ∈ ℝ)
74	73	adantl 487	. . . . . . . . . 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 789	. . . . . . . . . . . 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 486	. . . . . . . . . . 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 12678	. . . . . . . . . 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 783	. . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . 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 13763	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ℕ₀ → (𝑔 ∈ (0...𝑐) ↔ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)))
82	80, 81	syl 18	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . . 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 790	. . . . . . . . . . . . 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 6893	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = 𝑔 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔))
86		2fveq3 6898	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = 𝑔 → ((RSpan‘𝑃)‘(𝑓‘𝑒)) = ((RSpan‘𝑃)‘(𝑓‘𝑔)))
87	86	fveq2d 6897	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = 𝑔 → ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒))) = ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑔))))
88		id 23	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = 𝑔 → 𝑒 = 𝑔)
89	87, 88	fveq12d 6900	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = 𝑔 → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒) = (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑔)))‘𝑔))
90	85, 89	sseq12d 3945	. . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 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 486	. . . . . . . . . . . . 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 6924	. . . . . . . . . . . . . . . 16 ⊢ (𝑓‘𝑔) ⊆ ∪ ran 𝑓
95	94, 51	sstrid 3923	. . . . . . . . . . . . . . 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 21530	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ Ring ∧ (𝑓‘𝑔) ⊆ (Base‘𝑃)) → ((RSpan‘𝑃)‘(𝑓‘𝑔)) ∈ (LIdeal‘𝑃))
97	68, 95, 96	syl2anc 596	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . 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 486	. . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 21531	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ Ring ∧ ∪ ran 𝑓 ⊆ (Base‘𝑃)) → ∪ ran 𝑓 ⊆ ((RSpan‘𝑃)‘∪ ran 𝑓))
103	68, 51, 102	syl2anc 596	. . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . 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 3923	. . . . . . . . . . . . . 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 21534	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . 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 44269	. . . . . . . . . . . 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 3922	. . . . . . . . . . 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 473	. . . . . . . . . 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 12727	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ ℕ₀ → 𝑐 ∈ ℤ)
112	111	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ ℕ₀ ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → 𝑐 ∈ ℤ)
113		nn0z 12727	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 ∈ ℕ₀ → 𝑔 ∈ ℤ)
114	113	ad2antrl 741	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ ℕ₀ ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → 𝑔 ∈ ℤ)
115		simprr 785	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ ℕ₀ ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → 𝑐 ≤ 𝑔)
116		eluz2 12981	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (ℤ_≥‘𝑐) ↔ (𝑐 ∈ ℤ ∧ 𝑔 ∈ ℤ ∧ 𝑐 ≤ 𝑔))
117	112, 114, 115, 116	syl3anbrc 1362	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ ℕ₀ ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → 𝑔 ∈ (ℤ_≥‘𝑐))
118	75, 117	sylan 592	. . . . . . . . . . . . 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 785	. . . . . . . . . . . . . 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 739	. . . . . . . . . . . . 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 6902	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑔 → ((((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐) ↔ (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐)))
122	121	rspcva 3577	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 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 12678	. . . . . . . . . . . . . . . 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 11892	. . . . . . . . . . . . . . 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 462	. . . . . . . . . . . . . . . . 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 3157	. . . . . . . . . . . . . . . 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 5108	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = 𝑐 → (𝑔 ≤ 𝑐 ↔ 𝑐 ≤ 𝑐))
129		fveq2 6893	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑐 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
130		fveq2 6893	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑐 → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑔) = (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑐))
131	129, 130	sseq12d 3945	. . . . . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . 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 44268	. . . . . . . . . . . . 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 3922	. . . . . . . . . . . 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 3946	. . . . . . . . . . 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 473	. . . . . . . . . 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 11426	. . . . . . . . 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 3157	. . . . . . . 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 44270	. . . . . . 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 2769	. . . . . 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 6893	. . . . . . 7 ⊢ (𝑏 = ∪ ran 𝑓 → ((RSpan‘𝑃)‘𝑏) = ((RSpan‘𝑃)‘∪ ran 𝑓))
151	150	rspceeqv 3601	. . . . . 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 596	. . . . 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 1969	. . . 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 3172	. . 3 ⊢ ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏))
155	154	ralrimiva 3157	. 2 ⊢ (𝑅 ∈ LNoeR → ∀𝑎 ∈ (LIdeal‘𝑃)∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏))
156	48, 10, 27	islnr2 44256	. 2 ⊢ (𝑃 ∈ LNoeR ↔ (𝑃 ∈ Ring ∧ ∀𝑎 ∈ (LIdeal‘𝑃)∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏)))
157	4, 155, 156	sylanbrc 595	1 ⊢ (𝑅 ∈ LNoeR → 𝑃 ∈ LNoeR)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∃wex 1811 ∈ wcel 2146 ∀wral 3079 ∃wrex 3089 ⊆ wss 3899 ∩ cin 4091 𝒫 cpw 4559 ∪ cuni 4869 ∪ ciun 4953 class class class wbr 5105 ran crn 5655 Fn wfn 6541 ⟶wf 6542 ‘cfv 6546 (class class class)co 7427 Fincfn 8984 ℝcr 11209 0cc0 11210 1c1 11211 + caddc 11213 ≤ cle 11354 ℕ₀cn0 12616 ℤcz 12703 ℤ_≥cuz 12975 ...cfz 13649 Basecbs 17399 Ringcrg 20471 LIdealclidl 21496 RSpancrsp 21497 Poly₁cpl1 22507 NoeACScnacs 43853 LNoeRclnr 44251 ldgIdlSeqcldgis 44263

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5234 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758 ax-cnex 11266 ax-resscn 11267 ax-1cn 11268 ax-icn 11269 ax-addcl 11270 ax-addrcl 11271 ax-mulcl 11272 ax-mulrcl 11273 ax-mulcom 11274 ax-addass 11275 ax-mulass 11276 ax-distr 11277 ax-i2m1 11278 ax-1ne0 11279 ax-1rid 11280 ax-rnegex 11281 ax-rrecex 11282 ax-cnre 11283 ax-pre-lttri 11284 ax-pre-lttrn 11285 ax-pre-ltadd 11286 ax-pre-mulgt0 11287 ax-pre-sup 11288 ax-addf 11289

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4838 df-uni 4870 df-int 4910 df-iun 4955 df-iin 4956 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-se 5608 df-we 5609 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6310 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7384 df-ov 7430 df-oprab 7431 df-mpo 7432 df-of 7700 df-ofr 7701 df-om 7885 df-1st 8008 df-2nd 8009 df-supp 8188 df-tpos 8254 df-frecs 8310 df-wrecs 8341 df-recs 8390 df-rdg 8429 df-1o 8487 df-2o 8488 df-er 8728 df-map 8860 df-pm 8861 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9365 df-sup 9444 df-oi 9514 df-card 10030 df-pnf 11355 df-mnf 11356 df-xr 11357 df-ltxr 11358 df-le 11359 df-sub 11553 df-neg 11554 df-nn 12346 df-2 12415 df-3 12416 df-4 12417 df-5 12418 df-6 12419 df-7 12420 df-8 12421 df-9 12422 df-n0 12617 df-z 12704 df-dec 12825 df-uz 12976 df-fz 13650 df-fzo 13799 df-seq 14155 df-hash 14485 df-struct 17337 df-sets 17354 df-slot 17372 df-ndx 17384 df-base 17400 df-ress 17421 df-plusg 17453 df-mulr 17454 df-starv 17455 df-sca 17456 df-vsca 17457 df-ip 17458 df-tset 17459 df-ple 17460 df-ocomp 17461 df-ds 17462 df-unif 17463 df-hom 17464 df-cco 17465 df-0g 17624 df-gsum 17625 df-prds 17630 df-pws 17632 df-mre 17768 df-mrc 17769 df-acs 17771 df-proset 18480 df-drs 18481 df-poset 18499 df-ipo 18714 df-mgm 18828 df-sgrp 18920 df-mnd 18936 df-mhm 18990 df-submnd 18991 df-grp 19159 df-minusg 19160 df-sbg 19161 df-mulg 19290 df-subg 19345 df-ghm 19440 df-cntz 19543 df-cmn 20008 df-abl 20009 df-mgp 20373 df-rng 20387 df-ur 20420 df-ring 20473 df-cring 20474 df-oppr 20579 df-dvdsr 20599 df-unit 20600 df-invr 20630 df-subrng 20810 df-subrg 20834 df-rlreg 20958 df-lmod 21149 df-lss 21219 df-lsp 21259 df-sra 21460 df-rgmod 21461 df-lidl 21498 df-rsp 21499 df-cnfld 21691 df-ascl 22175 df-psr 22229 df-mvr 22230 df-mpl 22231 df-opsr 22233 df-psr1 22510 df-vr1 22511 df-ply1 22512 df-coe1 22513 df-mdeg 26385 df-deg1 26386 df-nacs 43854 df-lfig 44210 df-lnm 44218 df-lnr 44252 df-ldgis 44264

This theorem is used by: (None)

W3C validator