hbt - Mathbox for Stefan O'Rear

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Theorem hbt 43974

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 43956	. . 3 ⊢ (𝑅 ∈ LNoeR → 𝑅 ∈ Ring)
2		hbt.p	. . . 4 ⊢ 𝑃 = (Poly₁‘𝑅)
3	2	ply1ring 22475	. . 3 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
4	1, 3	syl 18	. 2 ⊢ (𝑅 ∈ LNoeR → 𝑃 ∈ Ring)
5		eqid 2760	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
6		eqid 2760	. . . . . . . 8 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
7	5, 6	islnr3 43959	. . . . . . 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 2760	. . . . . . 7 ⊢ (LIdeal‘𝑃) = (LIdeal‘𝑃)
11		eqid 2760	. . . . . . 7 ⊢ (ldgIdlSeq‘𝑅) = (ldgIdlSeq‘𝑅)
12	2, 10, 11, 6	hbtlem7 43969	. . . . . 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 12571	. . . . . . . 8 ⊢ (𝑏 ∈ ℕ₀ → (𝑏 + 1) ∈ ℕ₀)
18	17	adantl 487	. . . . . . 7 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ₀) → (𝑏 + 1) ∈ ℕ₀)
19		nn0re 12540	. . . . . . . . 9 ⊢ (𝑏 ∈ ℕ₀ → 𝑏 ∈ ℝ)
20	19	lep1d 12173	. . . . . . . 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 43970	. . . . . 6 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ₀) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑏) ⊆ (((ldgIdlSeq‘𝑅)‘𝑎)‘(𝑏 + 1)))
23	22	ralrimiva 3154	. . . . 5 ⊢ ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∀𝑏 ∈ ℕ₀ (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑏) ⊆ (((ldgIdlSeq‘𝑅)‘𝑎)‘(𝑏 + 1)))
24		nacsfix 43560	. . . . 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 14039	. . . . . . 7 ⊢ (0...𝑐) ∈ Fin
27		eqid 2760	. . . . . . . . 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 13678	. . . . . . . . . 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 43973	. . . . . . . 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 3154	. . . . . . 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 6884	. . . . . . . . . 10 ⊢ (𝑏 = (𝑓‘𝑒) → ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏)) = ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒))))
35	34	fveq1d 6881	. . . . . . . . 9 ⊢ (𝑏 = (𝑓‘𝑒) → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏))‘𝑒) = (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))
36	35	sseq2d 3963	. . . . . . . 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 9257	. . . . . . 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 6711	. . . . . . . . . . . . 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 4182	. . . . . . . . . . . 12 ⊢ (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑎
43	41, 42	sstrdi 3943	. . . . . . . . . . 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 4877	. . . . . . . . . 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 5425	. . . . . . . . . 10 ⊢ ∪ 𝒫 𝑎 = 𝑎
46	44, 45	sseqtrdi 3971	. . . . . . . . 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 2760	. . . . . . . . . . 11 ⊢ (Base‘𝑃) = (Base‘𝑃)
49	48, 10	lidlss 21402	. . . . . . . . . 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 3941	. . . . . . . 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 6892	. . . . . . . . 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 6703	. . . . . . . . 9 ⊢ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) → 𝑓 Fn (0...𝑐))
57		fniunfv 7245	. . . . . . . . 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 4183	. . . . . . . . . . 11 ⊢ (𝒫 𝑎 ∩ Fin) ⊆ Fin
60	55	ffvelcdmda 7078	. . . . . . . . . . 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 3929	. . . . . . . . . 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 3154	. . . . . . . . 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 9313	. . . . . . . . 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 2861	. . . . . . 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 4146	. . . . . 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 21430	. . . . . . . . 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 21434	. . . . . . . . 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 12540	. . . . . . . . . . 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 12593	. . . . . . . . . 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 13677	. . . . . . . . . . . . . . 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 6879	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = 𝑔 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔))
86		2fveq3 6884	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = 𝑔 → ((RSpan‘𝑃)‘(𝑓‘𝑒)) = ((RSpan‘𝑃)‘(𝑓‘𝑔)))
87	86	fveq2d 6883	. . . . . . . . . . . . . . . 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 6886	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = 𝑔 → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒) = (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑔)))‘𝑔))
90	85, 89	sseq12d 3964	. . . . . . . . . . . . . 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 3574	. . . . . . . . . . . . 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 6910	. . . . . . . . . . . . . . . 16 ⊢ (𝑓‘𝑔) ⊆ ∪ ran 𝑓
95	94, 51	sstrid 3942	. . . . . . . . . . . . . . 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 21430	. . . . . . . . . . . . . . 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 21431	. . . . . . . . . . . . . . . . 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 3942	. . . . . . . . . . . . . 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 21434	. . . . . . . . . . . . . 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 43971	. . . . . . . . . . . 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 3941	. . . . . . . . . . 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 12642	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ ℕ₀ → 𝑐 ∈ ℤ)
112	111	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ ℕ₀ ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → 𝑐 ∈ ℤ)
113		nn0z 12642	. . . . . . . . . . . . . . . 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 12896	. . . . . . . . . . . . . . 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 6888	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑔 → ((((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐) ↔ (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐)))
122	121	rspcva 3574	. . . . . . . . . . . . 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 12593	. . . . . . . . . . . . . . . 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 11807	. . . . . . . . . . . . . . 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 3154	. . . . . . . . . . . . . . . 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 5106	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = 𝑐 → (𝑔 ≤ 𝑐 ↔ 𝑐 ≤ 𝑐))
129		fveq2 6879	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑐 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
130		fveq2 6879	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑐 → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑔) = (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑐))
131	129, 130	sseq12d 3964	. . . . . . . . . . . . . . . . . 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 3574	. . . . . . . . . . . . . . . 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 43970	. . . . . . . . . . . . 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 3941	. . . . . . . . . . . 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 3965	. . . . . . . . . . 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 11343	. . . . . . . . 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 3154	. . . . . . . 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 43972	. . . . . . 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 2766	. . . . . 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 6879	. . . . . . 7 ⊢ (𝑏 = ∪ ran 𝑓 → ((RSpan‘𝑃)‘𝑏) = ((RSpan‘𝑃)‘∪ ran 𝑓))
151	150	rspceeqv 3599	. . . . . 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 3169	. . 3 ⊢ ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏))
155	154	ralrimiva 3154	. 2 ⊢ (𝑅 ∈ LNoeR → ∀𝑎 ∈ (LIdeal‘𝑃)∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏))
156	48, 10, 27	islnr2 43958	. 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 2145 ∀wral 3076 ∃wrex 3086 ∩ cin 3898 ⊆ wss 3899 𝒫 cpw 4557 ∪ cuni 4867 ∪ ciun 4951 class class class wbr 5103 ran crn 5656 Fn wfn 6528 ⟶wf 6529 ‘cfv 6533 (class class class)co 7414 Fincfn 8955 ℝcr 11126 0cc0 11127 1c1 11128 + caddc 11130 ≤ cle 11271 ℕ₀cn0 12531 ℤcz 12618 ℤ_≥cuz 12890 ...cfz 13564 Basecbs 17304 Ringcrg 20375 LIdealclidl 21396 RSpancrsp 21397 Poly₁cpl1 22405 NoeACScnacs 43550 LNoeRclnr 43953 ldgIdlSeqcldgis 43965

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-ofr 7680 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8160 df-tpos 8225 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-pm 8832 df-ixp 8908 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-sup 9415 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-fz 13565 df-fzo 13713 df-seq 14069 df-hash 14398 df-struct 17242 df-sets 17259 df-slot 17277 df-ndx 17289 df-base 17305 df-ress 17326 df-plusg 17358 df-mulr 17359 df-starv 17360 df-sca 17361 df-vsca 17362 df-ip 17363 df-tset 17364 df-ple 17365 df-ocomp 17366 df-ds 17367 df-unif 17368 df-hom 17369 df-cco 17370 df-0g 17529 df-gsum 17530 df-prds 17535 df-pws 17537 df-mre 17673 df-mrc 17674 df-acs 17676 df-proset 18385 df-drs 18386 df-poset 18404 df-ipo 18619 df-mgm 18733 df-sgrp 18824 df-mnd 18840 df-mhm 18894 df-submnd 18895 df-grp 19063 df-minusg 19064 df-sbg 19065 df-mulg 19194 df-subg 19249 df-ghm 19344 df-cntz 19447 df-cmn 19912 df-abl 19913 df-mgp 20277 df-rng 20291 df-ur 20324 df-ring 20377 df-cring 20378 df-oppr 20481 df-dvdsr 20501 df-unit 20502 df-invr 20532 df-subrng 20711 df-subrg 20735 df-rlreg 20859 df-lmod 21049 df-lss 21119 df-lsp 21159 df-sra 21360 df-rgmod 21361 df-lidl 21398 df-rsp 21399 df-cnfld 21589 df-ascl 22073 df-psr 22127 df-mvr 22128 df-mpl 22129 df-opsr 22131 df-psr1 22408 df-vr1 22409 df-ply1 22410 df-coe1 22411 df-mdeg 26283 df-deg1 26284 df-nacs 43551 df-lfig 43912 df-lnm 43920 df-lnr 43954 df-ldgis 43966

This theorem is used by: (None)

W3C validator