hbt - Mathbox for Stefan O'Rear

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Theorem hbt 39155

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 39137	. . 3 ⊢ (𝑅 ∈ LNoeR → 𝑅 ∈ Ring)
2		hbt.p	. . . 4 ⊢ 𝑃 = (Poly₁‘𝑅)
3	2	ply1ring 20131	. . 3 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
4	1, 3	syl 17	. 2 ⊢ (𝑅 ∈ LNoeR → 𝑃 ∈ Ring)
5		eqid 2772	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
6		eqid 2772	. . . . . . . 8 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
7	5, 6	islnr3 39140	. . . . . . 7 ⊢ (𝑅 ∈ LNoeR ↔ (𝑅 ∈ Ring ∧ (LIdeal‘𝑅) ∈ (NoeACS‘(Base‘𝑅))))
8	7	simprbi 489	. . . . . 6 ⊢ (𝑅 ∈ LNoeR → (LIdeal‘𝑅) ∈ (NoeACS‘(Base‘𝑅)))
9	8	adantr 473	. . . . 5 ⊢ ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → (LIdeal‘𝑅) ∈ (NoeACS‘(Base‘𝑅)))
10		eqid 2772	. . . . . . 7 ⊢ (LIdeal‘𝑃) = (LIdeal‘𝑃)
11		eqid 2772	. . . . . . 7 ⊢ (ldgIdlSeq‘𝑅) = (ldgIdlSeq‘𝑅)
12	2, 10, 11, 6	hbtlem7 39150	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ((ldgIdlSeq‘𝑅)‘𝑎):ℕ₀⟶(LIdeal‘𝑅))
13	1, 12	sylan 572	. . . . 5 ⊢ ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ((ldgIdlSeq‘𝑅)‘𝑎):ℕ₀⟶(LIdeal‘𝑅))
14	1	ad2antrr 713	. . . . . . 7 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ₀) → 𝑅 ∈ Ring)
15		simplr 756	. . . . . . 7 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ₀) → 𝑎 ∈ (LIdeal‘𝑃))
16		simpr 477	. . . . . . 7 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ₀) → 𝑏 ∈ ℕ₀)
17		peano2nn0 11747	. . . . . . . 8 ⊢ (𝑏 ∈ ℕ₀ → (𝑏 + 1) ∈ ℕ₀)
18	17	adantl 474	. . . . . . 7 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ₀) → (𝑏 + 1) ∈ ℕ₀)
19		nn0re 11715	. . . . . . . . 9 ⊢ (𝑏 ∈ ℕ₀ → 𝑏 ∈ ℝ)
20	19	lep1d 11370	. . . . . . . 8 ⊢ (𝑏 ∈ ℕ₀ → 𝑏 ≤ (𝑏 + 1))
21	20	adantl 474	. . . . . . 7 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ₀) → 𝑏 ≤ (𝑏 + 1))
22	2, 10, 11, 14, 15, 16, 18, 21	hbtlem4 39151	. . . . . 6 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ₀) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑏) ⊆ (((ldgIdlSeq‘𝑅)‘𝑎)‘(𝑏 + 1)))
23	22	ralrimiva 3126	. . . . 5 ⊢ ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∀𝑏 ∈ ℕ₀ (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑏) ⊆ (((ldgIdlSeq‘𝑅)‘𝑎)‘(𝑏 + 1)))
24		nacsfix 38733	. . . . 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 1351	. . . 4 ⊢ ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∃𝑐 ∈ ℕ₀ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
26		fzfi 13153	. . . . . . 7 ⊢ (0...𝑐) ∈ Fin
27		eqid 2772	. . . . . . . . 9 ⊢ (RSpan‘𝑃) = (RSpan‘𝑃)
28		simpll 754	. . . . . . . . 9 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑒 ∈ (0...𝑐)) → 𝑅 ∈ LNoeR)
29		simplr 756	. . . . . . . . 9 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑒 ∈ (0...𝑐)) → 𝑎 ∈ (LIdeal‘𝑃))
30		elfznn0 12814	. . . . . . . . . 10 ⊢ (𝑒 ∈ (0...𝑐) → 𝑒 ∈ ℕ₀)
31	30	adantl 474	. . . . . . . . 9 ⊢ (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑒 ∈ (0...𝑐)) → 𝑒 ∈ ℕ₀)
32	2, 10, 11, 27, 28, 29, 31	hbtlem6 39154	. . . . . . . 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 3126	. . . . . . 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 6501	. . . . . . . . . 10 ⊢ (𝑏 = (𝑓‘𝑒) → ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏)) = ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒))))
35	34	fveq1d 6498	. . . . . . . . 9 ⊢ (𝑏 = (𝑓‘𝑒) → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏))‘𝑒) = (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))
36	35	sseq2d 3883	. . . . . . . 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 8555	. . . . . . 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 578	. . . . . 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 473	. . . . 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 6347	. . . . . . . . . . . . 13 ⊢ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) → ran 𝑓 ⊆ (𝒫 𝑎 ∩ Fin))
41	40	ad2antrl 715	. . . . . . . . . . . 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 4086	. . . . . . . . . . . 12 ⊢ (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑎
43	41, 42	syl6ss 3864	. . . . . . . . . . 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 4732	. . . . . . . . . 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 5195	. . . . . . . . . 10 ⊢ ∪ 𝒫 𝑎 = 𝑎
46	44, 45	syl6sseq 3901	. . . . . . . . 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 763	. . . . . . . . . 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 2772	. . . . . . . . . . 11 ⊢ (Base‘𝑃) = (Base‘𝑃)
49	48, 10	lidlss 19716	. . . . . . . . . 10 ⊢ (𝑎 ∈ (LIdeal‘𝑃) → 𝑎 ⊆ (Base‘𝑃))
50	47, 49	syl 17	. . . . . . . . 9 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → 𝑎 ⊆ (Base‘𝑃))
51	46, 50	sstrd 3862	. . . . . . . 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 6509	. . . . . . . . 9 ⊢ (Base‘𝑃) ∈ V
53	52	elpw2 5100	. . . . . . . 8 ⊢ (∪ ran 𝑓 ∈ 𝒫 (Base‘𝑃) ↔ ∪ ran 𝑓 ⊆ (Base‘𝑃))
54	51, 53	sylibr 226	. . . . . . 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 758	. . . . . . . . 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 6341	. . . . . . . . 9 ⊢ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) → 𝑓 Fn (0...𝑐))
57		fniunfv 6829	. . . . . . . . 9 ⊢ (𝑓 Fn (0...𝑐) → ∪ 𝑔 ∈ (0...𝑐)(𝑓‘𝑔) = ∪ ran 𝑓)
58	55, 56, 57	3syl 18	. . . . . . . 8 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → ∪ 𝑔 ∈ (0...𝑐)(𝑓‘𝑔) = ∪ ran 𝑓)
59		inss2 4087	. . . . . . . . . . 11 ⊢ (𝒫 𝑎 ∩ Fin) ⊆ Fin
60	55	ffvelrnda 6674	. . . . . . . . . . 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	sseldi 3850	. . . . . . . . . 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 3126	. . . . . . . . 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 8605	. . . . . . . . 9 ⊢ (((0...𝑐) ∈ Fin ∧ ∀𝑔 ∈ (0...𝑐)(𝑓‘𝑔) ∈ Fin) → ∪ 𝑔 ∈ (0...𝑐)(𝑓‘𝑔) ∈ Fin)
64	26, 62, 63	sylancr 578	. . . . . . . 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 4053	. . . . . 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 717	. . . . . . . 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 717	. . . . . . . . 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 19728	. . . . . . . . 9 ⊢ ((𝑃 ∈ Ring ∧ ∪ ran 𝑓 ⊆ (Base‘𝑃)) → ((RSpan‘𝑃)‘∪ ran 𝑓) ∈ (LIdeal‘𝑃))
70	68, 51, 69	syl2anc 576	. . . . . . . 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 19732	. . . . . . . . 9 ⊢ ((𝑃 ∈ Ring ∧ 𝑎 ∈ (LIdeal‘𝑃) ∧ ∪ ran 𝑓 ⊆ 𝑎) → ((RSpan‘𝑃)‘∪ ran 𝑓) ⊆ 𝑎)
72	68, 47, 46, 71	syl3anc 1351	. . . . . . . 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 11715	. . . . . . . . . . 11 ⊢ (𝑔 ∈ ℕ₀ → 𝑔 ∈ ℝ)
74	73	adantl 474	. . . . . . . . . 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 764	. . . . . . . . . . . 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 473	. . . . . . . . . . 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 11766	. . . . . . . . . 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 758	. . . . . . . . . . . . . 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 760	. . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . 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 12813	. . . . . . . . . . . . . . 15 ⊢ (𝑐 ∈ ℕ₀ → (𝑔 ∈ (0...𝑐) ↔ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)))
82	80, 81	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)) → (𝑔 ∈ (0...𝑐) ↔ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)))
83	78, 79, 82	mpbir2and 700	. . . . . . . . . . . . 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 765	. . . . . . . . . . . . 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 6496	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = 𝑔 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔))
86		2fveq3 6501	. . . . . . . . . . . . . . . . 17 ⊢ (𝑒 = 𝑔 → ((RSpan‘𝑃)‘(𝑓‘𝑒)) = ((RSpan‘𝑃)‘(𝑓‘𝑔)))
87	86	fveq2d 6500	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = 𝑔 → ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒))) = ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑔))))
88		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 = 𝑔 → 𝑒 = 𝑔)
89	87, 88	fveq12d 6503	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = 𝑔 → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒) = (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑔)))‘𝑔))
90	85, 89	sseq12d 3884	. . . . . . . . . . . . . 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 3527	. . . . . . . . . . . . 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 576	. . . . . . . . . . . 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 473	. . . . . . . . . . . . 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 6525	. . . . . . . . . . . . . . . 16 ⊢ (𝑓‘𝑔) ⊆ ∪ ran 𝑓
95	94, 51	syl5ss 3863	. . . . . . . . . . . . . . 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 19728	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ∈ Ring ∧ (𝑓‘𝑔) ⊆ (Base‘𝑃)) → ((RSpan‘𝑃)‘(𝑓‘𝑔)) ∈ (LIdeal‘𝑃))
97	68, 95, 96	syl2anc 576	. . . . . . . . . . . . . 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 473	. . . . . . . . . . . . 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 473	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ₀ ∧ 𝑔 ≤ 𝑐)) → ((RSpan‘𝑃)‘∪ ran 𝑓) ∈ (LIdeal‘𝑃))
100	67, 3	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → 𝑃 ∈ Ring)
101	100	adantr 473	. . . . . . . . . . . . . 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 19729	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃 ∈ Ring ∧ ∪ ran 𝑓 ⊆ (Base‘𝑃)) → ∪ ran 𝑓 ⊆ ((RSpan‘𝑃)‘∪ ran 𝑓))
103	68, 51, 102	syl2anc 576	. . . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . 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	syl5ss 3863	. . . . . . . . . . . . . 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 19732	. . . . . . . . . . . . . 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 1351	. . . . . . . . . . . . 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 39152	. . . . . . . . . . . 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 3862	. . . . . . . . . . 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 460	. . . . . . . . . 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 11816	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ ℕ₀ → 𝑐 ∈ ℤ)
112	111	adantr 473	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ ℕ₀ ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → 𝑐 ∈ ℤ)
113		nn0z 11816	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 ∈ ℕ₀ → 𝑔 ∈ ℤ)
114	113	ad2antrl 715	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ ℕ₀ ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → 𝑔 ∈ ℤ)
115		simprr 760	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ ℕ₀ ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → 𝑐 ≤ 𝑔)
116		eluz2 12062	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ (ℤ_≥‘𝑐) ↔ (𝑐 ∈ ℤ ∧ 𝑔 ∈ ℤ ∧ 𝑐 ≤ 𝑔))
117	112, 114, 115, 116	syl3anbrc 1323	. . . . . . . . . . . . . 14 ⊢ ((𝑐 ∈ ℕ₀ ∧ (𝑔 ∈ ℕ₀ ∧ 𝑐 ≤ 𝑔)) → 𝑔 ∈ (ℤ_≥‘𝑐))
118	75, 117	sylan 572	. . . . . . . . . . . . 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 760	. . . . . . . . . . . . . 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 713	. . . . . . . . . . . . 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 6505	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑔 → ((((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐) ↔ (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐)))
122	121	rspcva 3527	. . . . . . . . . . . . 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 576	. . . . . . . . . . . 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 11766	. . . . . . . . . . . . . . . 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 11005	. . . . . . . . . . . . . . 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 449	. . . . . . . . . . . . . . . . 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 3126	. . . . . . . . . . . . . . . 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 4928	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = 𝑐 → (𝑔 ≤ 𝑐 ↔ 𝑐 ≤ 𝑐))
129		fveq2 6496	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑐 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
130		fveq2 6496	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑐 → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑔) = (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑐))
131	129, 130	sseq12d 3884	. . . . . . . . . . . . . . . . . 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 337	. . . . . . . . . . . . . . . . 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 3527	. . . . . . . . . . . . . . . 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 576	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → (𝑐 ≤ 𝑐 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑐)))
135	125, 134	mpd 15	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ₀ ∧ ∀𝑑 ∈ (ℤ_≥‘𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓‘𝑒)))‘𝑒))) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘∪ ran 𝑓))‘𝑐))
136	135	adantr 473	. . . . . . . . . . . . 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 473	. . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . 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 758	. . . . . . . . . . . . . 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 760	. . . . . . . . . . . . . 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 39151	. . . . . . . . . . . . 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 3862	. . . . . . . . . . . 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 3889	. . . . . . . . . . 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 460	. . . . . . . . . 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 10544	. . . . . . . . 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 3126	. . . . . . . 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 39153	. . . . . . 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 2778	. . . . . 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 6496	. . . . . . 7 ⊢ (𝑏 = ∪ ran 𝑓 → ((RSpan‘𝑃)‘𝑏) = ((RSpan‘𝑃)‘∪ ran 𝑓))
151	150	rspceeqv 3547	. . . . . 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 576	. . . . 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 1894	. . . 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 3230	. . 3 ⊢ ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏))
155	154	ralrimiva 3126	. 2 ⊢ (𝑅 ∈ LNoeR → ∀𝑎 ∈ (LIdeal‘𝑃)∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏))
156	48, 10, 27	islnr2 39139	. 2 ⊢ (𝑃 ∈ LNoeR ↔ (𝑃 ∈ Ring ∧ ∀𝑎 ∈ (LIdeal‘𝑃)∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏)))
157	4, 155, 156	sylanbrc 575	1 ⊢ (𝑅 ∈ LNoeR → 𝑃 ∈ LNoeR)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∃wex 1742 ∈ wcel 2050 ∀wral 3082 ∃wrex 3083 ∩ cin 3822 ⊆ wss 3823 𝒫 cpw 4416 ∪ cuni 4708 ∪ ciun 4788 class class class wbr 4925 ran crn 5404 Fn wfn 6180 ⟶wf 6181 ‘cfv 6185 (class class class)co 6974 Fincfn 8304 ℝcr 10332 0cc0 10333 1c1 10334 + caddc 10336 ≤ cle 10473 ℕ₀cn0 11705 ℤcz 11791 ℤ_≥cuz 12056 ...cfz 12706 Basecbs 16337 Ringcrg 19032 LIdealclidl 19676 RSpancrsp 19677 Poly₁cpl1 20060 NoeACScnacs 38723 LNoeRclnr 39134 ldgIdlSeqcldgis 39146

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 ax-addf 10412 ax-mulf 10413

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-of 7225 df-ofr 7226 df-om 7395 df-1st 7499 df-2nd 7500 df-supp 7632 df-tpos 7693 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-2o 7904 df-oadd 7907 df-er 8087 df-map 8206 df-pm 8207 df-ixp 8258 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fsupp 8627 df-sup 8699 df-oi 8767 df-card 9160 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-dec 11910 df-uz 12057 df-fz 12707 df-fzo 12848 df-seq 13183 df-hash 13504 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-starv 16434 df-sca 16435 df-vsca 16436 df-ip 16437 df-tset 16438 df-ple 16439 df-ocomp 16440 df-ds 16441 df-unif 16442 df-0g 16569 df-gsum 16570 df-mre 16727 df-mrc 16728 df-acs 16730 df-proset 17408 df-drs 17409 df-poset 17426 df-ipo 17632 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-mhm 17815 df-submnd 17816 df-grp 17906 df-minusg 17907 df-sbg 17908 df-mulg 18024 df-subg 18072 df-ghm 18139 df-cntz 18230 df-cmn 18680 df-abl 18681 df-mgp 18975 df-ur 18987 df-ring 19034 df-cring 19035 df-oppr 19108 df-dvdsr 19126 df-unit 19127 df-invr 19157 df-subrg 19268 df-lmod 19370 df-lss 19438 df-lsp 19478 df-sra 19678 df-rgmod 19679 df-lidl 19680 df-rsp 19681 df-rlreg 19789 df-ascl 19820 df-psr 19862 df-mvr 19863 df-mpl 19864 df-opsr 19866 df-psr1 20063 df-vr1 20064 df-ply1 20065 df-coe1 20066 df-cnfld 20260 df-mdeg 24364 df-deg1 24365 df-nacs 38724 df-lfig 39093 df-lnm 39101 df-lnr 39135 df-ldgis 39147

This theorem is referenced by: (None)

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