hbt - Mathbox for Stefan O'Rear

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Theorem hbt 43917

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 43899	. . 3 ⊢ (𝑅 ∈ LNoeR → 𝑅 ∈ Ring)
2		hbt.p	. . . 4 ⊢ 𝑃 = (Poly₁‘𝑅)
3	2	ply1ring 22459	. . 3 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
4	1, 3	syl 18	. 2 ⊢ (𝑅 ∈ LNoeR → 𝑃 ∈ Ring)
5		eqid 2765	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
6		eqid 2765	. . . . . . . 8 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
7	5, 6	islnr3 43902	. . . . . . 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 2765	. . . . . . 7 ⊢ (LIdeal‘𝑃) = (LIdeal‘𝑃)
11		eqid 2765	. . . . . . 7 ⊢ (ldgIdlSeq‘𝑅) = (ldgIdlSeq‘𝑅)
12	2, 10, 11, 6	hbtlem7 43912	. . . . . 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 12561	. . . . . . . 8 ⊢ (𝑏 ∈ ℕ₀ → (𝑏 + 1) ∈ ℕ₀)
18	17	adantl 487	. . . . . . 7 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ₀) → (𝑏 + 1) ∈ ℕ₀)
19		nn0re 12530	. . . . . . . . 9 ⊢ (𝑏 ∈ ℕ₀ → 𝑏 ∈ ℝ)
20	19	lep1d 12163	. . . . . . . 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 43913	. . . . . 6 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ₀) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑏) ⊆ (((ldgIdlSeq‘𝑅)‘𝑎)‘(𝑏 + 1)))
23	22	ralrimiva 3159	. . . . 5 ⊢ ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∀𝑏 ∈ ℕ₀ (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑏) ⊆ (((ldgIdlSeq‘𝑅)‘𝑎)‘(𝑏 + 1)))
24		nacsfix 43503	. . . . 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 14028	. . . . . . 7 ⊢ (0...𝑐) ∈ Fin
27		eqid 2765	. . . . . . . . 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 13667	. . . . . . . . . 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 43916	. . . . . . . 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 3159	. . . . . . 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 6890	. . . . . . . . . 10 ⊢ (𝑏 = (𝑓‘𝑒) → ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏)) = ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒))))
35	34	fveq1d 6887	. . . . . . . . 9 ⊢ (𝑏 = (𝑓‘𝑒) → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏))‘𝑒) = (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))
36	35	sseq2d 3970	. . . . . . . 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 9251	. . . . . . 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 6717	. . . . . . . . . . . . 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 4189	. . . . . . . . . . . 12 ⊢ (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑎
43	41, 42	sstrdi 3950	. . . . . . . . . . 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 4884	. . . . . . . . . 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 5433	. . . . . . . . . 10 ⊢ ∪ 𝒫 𝑎 = 𝑎
46	44, 45	sseqtrdi 3978	. . . . . . . . 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 2765	. . . . . . . . . . 11 ⊢ (Base‘𝑃) = (Base‘𝑃)
49	48, 10	lidlss 21388	. . . . . . . . . 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 3948	. . . . . . . 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 6898	. . . . . . . . 9 ⊢ (Base‘𝑃) ∈ V
53	52	elpw2 5307	. . . . . . . 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 6709	. . . . . . . . 9 ⊢ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) → 𝑓 Fn (0...𝑐))
57		fniunfv 7250	. . . . . . . . 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 4190	. . . . . . . . . . 11 ⊢ (𝒫 𝑎 ∩ Fin) ⊆ Fin
60	55	ffvelcdmda 7083	. . . . . . . . . . 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 3936	. . . . . . . . . 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 3159	. . . . . . . . 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 9307	. . . . . . . . 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 2866	. . . . . . 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 4153	. . . . . 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 21416	. . . . . . . . 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 21420	. . . . . . . . 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 12530	. . . . . . . . . . 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 12583	. . . . . . . . . 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 13666	. . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = 𝑔 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔))
86		2fveq3 6890	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = 𝑔 → ((RSpan‘𝑃)‘(𝑓‘𝑒)) = ((RSpan‘𝑃)‘(𝑓‘𝑔)))
87	86	fveq2d 6889	. . . . . . . . . . . . . . . 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 6892	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = 𝑔 → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒) = (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑔)))‘𝑔))
90	85, 89	sseq12d 3971	. . . . . . . . . . . . . 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 3581	. . . . . . . . . . . . 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 6916	. . . . . . . . . . . . . . . 16 ⊢ (𝑓‘𝑔) ⊆ ∪ ran 𝑓
95	94, 51	sstrid 3949	. . . . . . . . . . . . . . 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 21416	. . . . . . . . . . . . . . 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 21417	. . . . . . . . . . . . . . . . 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 3949	. . . . . . . . . . . . . 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 21420	. . . . . . . . . . . . . 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 43914	. . . . . . . . . . . 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 3948	. . . . . . . . . . 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 12632	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ ℕ₀ → 𝑐 ∈ ℤ)
112	111	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ ℕ₀ ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → 𝑐 ∈ ℤ)
113		nn0z 12632	. . . . . . . . . . . . . . . 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 12886	. . . . . . . . . . . . . . 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 6894	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑔 → ((((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐) ↔ (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐)))
122	121	rspcva 3581	. . . . . . . . . . . . 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 12583	. . . . . . . . . . . . . . . 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 11797	. . . . . . . . . . . . . . 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 3159	. . . . . . . . . . . . . . . 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 5114	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = 𝑐 → (𝑔 ≤ 𝑐 ↔ 𝑐 ≤ 𝑐))
129		fveq2 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑐 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
130		fveq2 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑐 → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑔) = (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑐))
131	129, 130	sseq12d 3971	. . . . . . . . . . . . . . . . . 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 3581	. . . . . . . . . . . . . . . 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 43913	. . . . . . . . . . . . 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 3948	. . . . . . . . . . . 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 3972	. . . . . . . . . . 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 11333	. . . . . . . . 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 3159	. . . . . . . 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 43915	. . . . . . 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 2771	. . . . . 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 6885	. . . . . . 7 ⊢ (𝑏 = ∪ ran 𝑓 → ((RSpan‘𝑃)‘𝑏) = ((RSpan‘𝑃)‘∪ ran 𝑓))
151	150	rspceeqv 3606	. . . . . 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 1968	. . . 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 3174	. . 3 ⊢ ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏))
155	154	ralrimiva 3159	. 2 ⊢ (𝑅 ∈ LNoeR → ∀𝑎 ∈ (LIdeal‘𝑃)∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏))
156	48, 10, 27	islnr2 43901	. 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 1570 ∃wex 1812 ∈ wcel 2146 ∀wral 3081 ∃wrex 3091 ∩ cin 3905 ⊆ wss 3906 𝒫 cpw 4564 ∪ cuni 4874 ∪ ciun 4958 class class class wbr 5111 ran crn 5664 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 Fincfn 8949 ℝcr 11116 0cc0 11117 1c1 11118 + caddc 11120 ≤ cle 11261 ℕ₀cn0 12521 ℤcz 12608 ℤ_≥cuz 12880 ...cfz 13553 Basecbs 17293 Ringcrg 20361 LIdealclidl 21382 RSpancrsp 21383 Poly₁cpl1 22389 NoeACScnacs 43493 LNoeRclnr 43896 ldgIdlSeqcldgis 43908

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 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194 ax-pre-sup 11195 ax-addf 11196

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-ofr 7685 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-tpos 8228 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-sup 9409 df-oi 9479 df-card 9941 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-nn 12251 df-2 12320 df-3 12321 df-4 12322 df-5 12323 df-6 12324 df-7 12325 df-8 12326 df-9 12327 df-n0 12522 df-z 12609 df-dec 12730 df-uz 12881 df-fz 13554 df-fzo 13702 df-seq 14058 df-hash 14387 df-struct 17231 df-sets 17248 df-slot 17266 df-ndx 17278 df-base 17294 df-ress 17315 df-plusg 17347 df-mulr 17348 df-starv 17349 df-sca 17350 df-vsca 17351 df-ip 17352 df-tset 17353 df-ple 17354 df-ocomp 17355 df-ds 17356 df-unif 17357 df-hom 17358 df-cco 17359 df-0g 17518 df-gsum 17519 df-prds 17524 df-pws 17526 df-mre 17662 df-mrc 17663 df-acs 17665 df-proset 18374 df-drs 18375 df-poset 18393 df-ipo 18608 df-mgm 18722 df-sgrp 18811 df-mnd 18827 df-mhm 18880 df-submnd 18881 df-grp 19049 df-minusg 19050 df-sbg 19051 df-mulg 19180 df-subg 19235 df-ghm 19330 df-cntz 19433 df-cmn 19898 df-abl 19899 df-mgp 20263 df-rng 20277 df-ur 20310 df-ring 20363 df-cring 20364 df-oppr 20467 df-dvdsr 20487 df-unit 20488 df-invr 20518 df-subrng 20697 df-subrg 20721 df-rlreg 20845 df-lmod 21035 df-lss 21105 df-lsp 21145 df-sra 21346 df-rgmod 21347 df-lidl 21384 df-rsp 21385 df-cnfld 21575 df-ascl 22057 df-psr 22111 df-mvr 22112 df-mpl 22113 df-opsr 22115 df-psr1 22392 df-vr1 22393 df-ply1 22394 df-coe1 22395 df-mdeg 26265 df-deg1 26266 df-nacs 43494 df-lfig 43855 df-lnm 43863 df-lnr 43897 df-ldgis 43909

This theorem is used by: (None)

W3C validator