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Theorem hbt 38226
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 = (Poly1𝑅)
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.
StepHypRef Expression
1 lnrring 38208 . . 3 (𝑅 ∈ LNoeR → 𝑅 ∈ Ring)
2 hbt.p . . . 4 𝑃 = (Poly1𝑅)
32ply1ring 19832 . . 3 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
41, 3syl 17 . 2 (𝑅 ∈ LNoeR → 𝑃 ∈ Ring)
5 eqid 2771 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
6 eqid 2771 . . . . . . . 8 (LIdeal‘𝑅) = (LIdeal‘𝑅)
75, 6islnr3 38211 . . . . . . 7 (𝑅 ∈ LNoeR ↔ (𝑅 ∈ Ring ∧ (LIdeal‘𝑅) ∈ (NoeACS‘(Base‘𝑅))))
87simprbi 484 . . . . . 6 (𝑅 ∈ LNoeR → (LIdeal‘𝑅) ∈ (NoeACS‘(Base‘𝑅)))
98adantr 466 . . . . 5 ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → (LIdeal‘𝑅) ∈ (NoeACS‘(Base‘𝑅)))
10 eqid 2771 . . . . . . 7 (LIdeal‘𝑃) = (LIdeal‘𝑃)
11 eqid 2771 . . . . . . 7 (ldgIdlSeq‘𝑅) = (ldgIdlSeq‘𝑅)
122, 10, 11, 6hbtlem7 38221 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ((ldgIdlSeq‘𝑅)‘𝑎):ℕ0⟶(LIdeal‘𝑅))
131, 12sylan 569 . . . . 5 ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ((ldgIdlSeq‘𝑅)‘𝑎):ℕ0⟶(LIdeal‘𝑅))
141ad2antrr 705 . . . . . . 7 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ0) → 𝑅 ∈ Ring)
15 simplr 752 . . . . . . 7 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ0) → 𝑎 ∈ (LIdeal‘𝑃))
16 simpr 471 . . . . . . 7 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ0) → 𝑏 ∈ ℕ0)
17 peano2nn0 11539 . . . . . . . 8 (𝑏 ∈ ℕ0 → (𝑏 + 1) ∈ ℕ0)
1817adantl 467 . . . . . . 7 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ0) → (𝑏 + 1) ∈ ℕ0)
19 nn0re 11507 . . . . . . . . 9 (𝑏 ∈ ℕ0𝑏 ∈ ℝ)
2019lep1d 11160 . . . . . . . 8 (𝑏 ∈ ℕ0𝑏 ≤ (𝑏 + 1))
2120adantl 467 . . . . . . 7 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ0) → 𝑏 ≤ (𝑏 + 1))
222, 10, 11, 14, 15, 16, 18, 21hbtlem4 38222 . . . . . 6 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ0) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑏) ⊆ (((ldgIdlSeq‘𝑅)‘𝑎)‘(𝑏 + 1)))
2322ralrimiva 3115 . . . . 5 ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∀𝑏 ∈ ℕ0 (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑏) ⊆ (((ldgIdlSeq‘𝑅)‘𝑎)‘(𝑏 + 1)))
24 nacsfix 37801 . . . . 5 (((LIdeal‘𝑅) ∈ (NoeACS‘(Base‘𝑅)) ∧ ((ldgIdlSeq‘𝑅)‘𝑎):ℕ0⟶(LIdeal‘𝑅) ∧ ∀𝑏 ∈ ℕ0 (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑏) ⊆ (((ldgIdlSeq‘𝑅)‘𝑎)‘(𝑏 + 1))) → ∃𝑐 ∈ ℕ0𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
259, 13, 23, 24syl3anc 1476 . . . 4 ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∃𝑐 ∈ ℕ0𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
26 fzfi 12978 . . . . . . 7 (0...𝑐) ∈ Fin
27 eqid 2771 . . . . . . . . 9 (RSpan‘𝑃) = (RSpan‘𝑃)
28 simpll 750 . . . . . . . . 9 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑒 ∈ (0...𝑐)) → 𝑅 ∈ LNoeR)
29 simplr 752 . . . . . . . . 9 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑒 ∈ (0...𝑐)) → 𝑎 ∈ (LIdeal‘𝑃))
30 elfznn0 12639 . . . . . . . . . 10 (𝑒 ∈ (0...𝑐) → 𝑒 ∈ ℕ0)
3130adantl 467 . . . . . . . . 9 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑒 ∈ (0...𝑐)) → 𝑒 ∈ ℕ0)
322, 10, 11, 27, 28, 29, 31hbtlem6 38225 . . . . . . . 8 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑒 ∈ (0...𝑐)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏))‘𝑒))
3332ralrimiva 3115 . . . . . . 7 ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∀𝑒 ∈ (0...𝑐)∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏))‘𝑒))
34 fveq2 6333 . . . . . . . . . . 11 (𝑏 = (𝑓𝑒) → ((RSpan‘𝑃)‘𝑏) = ((RSpan‘𝑃)‘(𝑓𝑒)))
3534fveq2d 6337 . . . . . . . . . 10 (𝑏 = (𝑓𝑒) → ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏)) = ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒))))
3635fveq1d 6335 . . . . . . . . 9 (𝑏 = (𝑓𝑒) → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏))‘𝑒) = (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))
3736sseq2d 3782 . . . . . . . 8 (𝑏 = (𝑓𝑒) → ((((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏))‘𝑒) ↔ (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒)))
3837ac6sfi 8363 . . . . . . 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)‘(𝑓𝑒)))‘𝑒)))
3926, 33, 38sylancr 575 . . . . . 6 ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∃𝑓(𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒)))
4039adantr 466 . . . . 5 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((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)‘(𝑓𝑒)))‘𝑒)))
41 frn 6190 . . . . . . . . . . . . 13 (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) → ran 𝑓 ⊆ (𝒫 𝑎 ∩ Fin))
4241ad2antrl 707 . . . . . . . . . . . 12 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → ran 𝑓 ⊆ (𝒫 𝑎 ∩ Fin))
43 inss1 3981 . . . . . . . . . . . 12 (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑎
4442, 43syl6ss 3764 . . . . . . . . . . 11 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → ran 𝑓 ⊆ 𝒫 𝑎)
4544unissd 4599 . . . . . . . . . 10 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → ran 𝑓 𝒫 𝑎)
46 unipw 5047 . . . . . . . . . 10 𝒫 𝑎 = 𝑎
4745, 46syl6sseq 3800 . . . . . . . . 9 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → ran 𝑓𝑎)
48 simpllr 760 . . . . . . . . . 10 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → 𝑎 ∈ (LIdeal‘𝑃))
49 eqid 2771 . . . . . . . . . . 11 (Base‘𝑃) = (Base‘𝑃)
5049, 10lidlss 19424 . . . . . . . . . 10 (𝑎 ∈ (LIdeal‘𝑃) → 𝑎 ⊆ (Base‘𝑃))
5148, 50syl 17 . . . . . . . . 9 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → 𝑎 ⊆ (Base‘𝑃))
5247, 51sstrd 3762 . . . . . . . 8 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → ran 𝑓 ⊆ (Base‘𝑃))
53 fvex 6344 . . . . . . . . 9 (Base‘𝑃) ∈ V
5453elpw2 4960 . . . . . . . 8 ( ran 𝑓 ∈ 𝒫 (Base‘𝑃) ↔ ran 𝑓 ⊆ (Base‘𝑃))
5552, 54sylibr 224 . . . . . . 7 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → ran 𝑓 ∈ 𝒫 (Base‘𝑃))
56 simprl 754 . . . . . . . . 9 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → 𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin))
57 ffn 6184 . . . . . . . . 9 (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) → 𝑓 Fn (0...𝑐))
58 fniunfv 6650 . . . . . . . . 9 (𝑓 Fn (0...𝑐) → 𝑔 ∈ (0...𝑐)(𝑓𝑔) = ran 𝑓)
5956, 57, 583syl 18 . . . . . . . 8 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → 𝑔 ∈ (0...𝑐)(𝑓𝑔) = ran 𝑓)
60 inss2 3982 . . . . . . . . . . 11 (𝒫 𝑎 ∩ Fin) ⊆ Fin
6156ffvelrnda 6504 . . . . . . . . . . 11 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ 𝑔 ∈ (0...𝑐)) → (𝑓𝑔) ∈ (𝒫 𝑎 ∩ Fin))
6260, 61sseldi 3750 . . . . . . . . . 10 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ 𝑔 ∈ (0...𝑐)) → (𝑓𝑔) ∈ Fin)
6362ralrimiva 3115 . . . . . . . . 9 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → ∀𝑔 ∈ (0...𝑐)(𝑓𝑔) ∈ Fin)
64 iunfi 8413 . . . . . . . . 9 (((0...𝑐) ∈ Fin ∧ ∀𝑔 ∈ (0...𝑐)(𝑓𝑔) ∈ Fin) → 𝑔 ∈ (0...𝑐)(𝑓𝑔) ∈ Fin)
6526, 63, 64sylancr 575 . . . . . . . 8 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → 𝑔 ∈ (0...𝑐)(𝑓𝑔) ∈ Fin)
6659, 65eqeltrrd 2851 . . . . . . 7 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → ran 𝑓 ∈ Fin)
6755, 66elind 3949 . . . . . 6 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → ran 𝑓 ∈ (𝒫 (Base‘𝑃) ∩ Fin))
681ad3antrrr 709 . . . . . . . 8 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → 𝑅 ∈ Ring)
694ad3antrrr 709 . . . . . . . . 9 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → 𝑃 ∈ Ring)
7027, 49, 10rspcl 19436 . . . . . . . . 9 ((𝑃 ∈ Ring ∧ ran 𝑓 ⊆ (Base‘𝑃)) → ((RSpan‘𝑃)‘ ran 𝑓) ∈ (LIdeal‘𝑃))
7169, 52, 70syl2anc 573 . . . . . . . 8 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → ((RSpan‘𝑃)‘ ran 𝑓) ∈ (LIdeal‘𝑃))
7227, 10rspssp 19440 . . . . . . . . 9 ((𝑃 ∈ Ring ∧ 𝑎 ∈ (LIdeal‘𝑃) ∧ ran 𝑓𝑎) → ((RSpan‘𝑃)‘ ran 𝑓) ⊆ 𝑎)
7369, 48, 47, 72syl3anc 1476 . . . . . . . 8 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → ((RSpan‘𝑃)‘ ran 𝑓) ⊆ 𝑎)
74 nn0re 11507 . . . . . . . . . . 11 (𝑔 ∈ ℕ0𝑔 ∈ ℝ)
7574adantl 467 . . . . . . . . . 10 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ 𝑔 ∈ ℕ0) → 𝑔 ∈ ℝ)
76 simplrl 762 . . . . . . . . . . . 12 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → 𝑐 ∈ ℕ0)
7776adantr 466 . . . . . . . . . . 11 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ 𝑔 ∈ ℕ0) → 𝑐 ∈ ℕ0)
7877nn0red 11558 . . . . . . . . . 10 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ 𝑔 ∈ ℕ0) → 𝑐 ∈ ℝ)
79 simprl 754 . . . . . . . . . . . . . 14 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑔𝑐)) → 𝑔 ∈ ℕ0)
80 simprr 756 . . . . . . . . . . . . . 14 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑔𝑐)) → 𝑔𝑐)
8176adantr 466 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑔𝑐)) → 𝑐 ∈ ℕ0)
82 fznn0 12638 . . . . . . . . . . . . . . 15 (𝑐 ∈ ℕ0 → (𝑔 ∈ (0...𝑐) ↔ (𝑔 ∈ ℕ0𝑔𝑐)))
8381, 82syl 17 . . . . . . . . . . . . . 14 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑔𝑐)) → (𝑔 ∈ (0...𝑐) ↔ (𝑔 ∈ ℕ0𝑔𝑐)))
8479, 80, 83mpbir2and 692 . . . . . . . . . . . . 13 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑔𝑐)) → 𝑔 ∈ (0...𝑐))
85 simplrr 763 . . . . . . . . . . . . 13 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑔𝑐)) → ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))
86 fveq2 6333 . . . . . . . . . . . . . . 15 (𝑒 = 𝑔 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔))
87 fveq2 6333 . . . . . . . . . . . . . . . . . 18 (𝑒 = 𝑔 → (𝑓𝑒) = (𝑓𝑔))
8887fveq2d 6337 . . . . . . . . . . . . . . . . 17 (𝑒 = 𝑔 → ((RSpan‘𝑃)‘(𝑓𝑒)) = ((RSpan‘𝑃)‘(𝑓𝑔)))
8988fveq2d 6337 . . . . . . . . . . . . . . . 16 (𝑒 = 𝑔 → ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒))) = ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑔))))
90 id 22 . . . . . . . . . . . . . . . 16 (𝑒 = 𝑔𝑒 = 𝑔)
9189, 90fveq12d 6340 . . . . . . . . . . . . . . 15 (𝑒 = 𝑔 → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒) = (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑔)))‘𝑔))
9286, 91sseq12d 3783 . . . . . . . . . . . . . 14 (𝑒 = 𝑔 → ((((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒) ↔ (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑔)))‘𝑔)))
9392rspcva 3458 . . . . . . . . . . . . 13 ((𝑔 ∈ (0...𝑐) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒)) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑔)))‘𝑔))
9484, 85, 93syl2anc 573 . . . . . . . . . . . 12 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑔𝑐)) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑔)))‘𝑔))
9568adantr 466 . . . . . . . . . . . . 13 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑔𝑐)) → 𝑅 ∈ Ring)
96 fvssunirn 6360 . . . . . . . . . . . . . . . 16 (𝑓𝑔) ⊆ ran 𝑓
9796, 52syl5ss 3763 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → (𝑓𝑔) ⊆ (Base‘𝑃))
9827, 49, 10rspcl 19436 . . . . . . . . . . . . . . 15 ((𝑃 ∈ Ring ∧ (𝑓𝑔) ⊆ (Base‘𝑃)) → ((RSpan‘𝑃)‘(𝑓𝑔)) ∈ (LIdeal‘𝑃))
9969, 97, 98syl2anc 573 . . . . . . . . . . . . . 14 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → ((RSpan‘𝑃)‘(𝑓𝑔)) ∈ (LIdeal‘𝑃))
10099adantr 466 . . . . . . . . . . . . 13 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑔𝑐)) → ((RSpan‘𝑃)‘(𝑓𝑔)) ∈ (LIdeal‘𝑃))
10171adantr 466 . . . . . . . . . . . . 13 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑔𝑐)) → ((RSpan‘𝑃)‘ ran 𝑓) ∈ (LIdeal‘𝑃))
10268, 3syl 17 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → 𝑃 ∈ Ring)
103102adantr 466 . . . . . . . . . . . . . 14 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑔𝑐)) → 𝑃 ∈ Ring)
10427, 49rspssid 19437 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ Ring ∧ ran 𝑓 ⊆ (Base‘𝑃)) → ran 𝑓 ⊆ ((RSpan‘𝑃)‘ ran 𝑓))
10569, 52, 104syl2anc 573 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → ran 𝑓 ⊆ ((RSpan‘𝑃)‘ ran 𝑓))
106105adantr 466 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑔𝑐)) → ran 𝑓 ⊆ ((RSpan‘𝑃)‘ ran 𝑓))
10796, 106syl5ss 3763 . . . . . . . . . . . . . 14 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑔𝑐)) → (𝑓𝑔) ⊆ ((RSpan‘𝑃)‘ ran 𝑓))
10827, 10rspssp 19440 . . . . . . . . . . . . . 14 ((𝑃 ∈ Ring ∧ ((RSpan‘𝑃)‘ ran 𝑓) ∈ (LIdeal‘𝑃) ∧ (𝑓𝑔) ⊆ ((RSpan‘𝑃)‘ ran 𝑓)) → ((RSpan‘𝑃)‘(𝑓𝑔)) ⊆ ((RSpan‘𝑃)‘ ran 𝑓))
109103, 101, 107, 108syl3anc 1476 . . . . . . . . . . . . 13 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑔𝑐)) → ((RSpan‘𝑃)‘(𝑓𝑔)) ⊆ ((RSpan‘𝑃)‘ ran 𝑓))
1102, 10, 11, 95, 100, 101, 109, 79hbtlem3 38223 . . . . . . . . . . . 12 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑔𝑐)) → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑔)))‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑔))
11194, 110sstrd 3762 . . . . . . . . . . 11 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑔𝑐)) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑔))
112111anassrs 453 . . . . . . . . . 10 ((((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ 𝑔 ∈ ℕ0) ∧ 𝑔𝑐) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑔))
113 nn0z 11606 . . . . . . . . . . . . . . . 16 (𝑐 ∈ ℕ0𝑐 ∈ ℤ)
114113adantr 466 . . . . . . . . . . . . . . 15 ((𝑐 ∈ ℕ0 ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → 𝑐 ∈ ℤ)
115 nn0z 11606 . . . . . . . . . . . . . . . 16 (𝑔 ∈ ℕ0𝑔 ∈ ℤ)
116115ad2antrl 707 . . . . . . . . . . . . . . 15 ((𝑐 ∈ ℕ0 ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → 𝑔 ∈ ℤ)
117 simprr 756 . . . . . . . . . . . . . . 15 ((𝑐 ∈ ℕ0 ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → 𝑐𝑔)
118 eluz2 11898 . . . . . . . . . . . . . . 15 (𝑔 ∈ (ℤ𝑐) ↔ (𝑐 ∈ ℤ ∧ 𝑔 ∈ ℤ ∧ 𝑐𝑔))
119114, 116, 117, 118syl3anbrc 1428 . . . . . . . . . . . . . 14 ((𝑐 ∈ ℕ0 ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → 𝑔 ∈ (ℤ𝑐))
12076, 119sylan 569 . . . . . . . . . . . . 13 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → 𝑔 ∈ (ℤ𝑐))
121 simprr 756 . . . . . . . . . . . . . 14 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) → ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
122121ad2antrr 705 . . . . . . . . . . . . 13 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
123 fveq2 6333 . . . . . . . . . . . . . . 15 (𝑑 = 𝑔 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔))
124123eqeq1d 2773 . . . . . . . . . . . . . 14 (𝑑 = 𝑔 → ((((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐) ↔ (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐)))
125124rspcva 3458 . . . . . . . . . . . . 13 ((𝑔 ∈ (ℤ𝑐) ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐)) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
126120, 122, 125syl2anc 573 . . . . . . . . . . . 12 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
12776nn0red 11558 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → 𝑐 ∈ ℝ)
128127leidd 10799 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → 𝑐𝑐)
129111expr 444 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ 𝑔 ∈ ℕ0) → (𝑔𝑐 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑔)))
130129ralrimiva 3115 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → ∀𝑔 ∈ ℕ0 (𝑔𝑐 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑔)))
131 breq1 4790 . . . . . . . . . . . . . . . . . 18 (𝑔 = 𝑐 → (𝑔𝑐𝑐𝑐))
132 fveq2 6333 . . . . . . . . . . . . . . . . . . 19 (𝑔 = 𝑐 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
133 fveq2 6333 . . . . . . . . . . . . . . . . . . 19 (𝑔 = 𝑐 → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑔) = (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑐))
134132, 133sseq12d 3783 . . . . . . . . . . . . . . . . . 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)))
135131, 134imbi12d 333 . . . . . . . . . . . . . . . . 17 (𝑔 = 𝑐 → ((𝑔𝑐 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑔)) ↔ (𝑐𝑐 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑐))))
136135rspcva 3458 . . . . . . . . . . . . . . . 16 ((𝑐 ∈ ℕ0 ∧ ∀𝑔 ∈ ℕ0 (𝑔𝑐 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑔))) → (𝑐𝑐 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑐)))
13776, 130, 136syl2anc 573 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → (𝑐𝑐 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑐)))
138128, 137mpd 15 . . . . . . . . . . . . . 14 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑐))
139138adantr 466 . . . . . . . . . . . . 13 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑐))
14068adantr 466 . . . . . . . . . . . . . 14 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → 𝑅 ∈ Ring)
14171adantr 466 . . . . . . . . . . . . . 14 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → ((RSpan‘𝑃)‘ ran 𝑓) ∈ (LIdeal‘𝑃))
14276adantr 466 . . . . . . . . . . . . . 14 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → 𝑐 ∈ ℕ0)
143 simprl 754 . . . . . . . . . . . . . 14 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → 𝑔 ∈ ℕ0)
144 simprr 756 . . . . . . . . . . . . . 14 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → 𝑐𝑔)
1452, 10, 11, 140, 141, 142, 143, 144hbtlem4 38222 . . . . . . . . . . . . 13 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑐) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑔))
146139, 145sstrd 3762 . . . . . . . . . . . 12 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑔))
147126, 146eqsstrd 3788 . . . . . . . . . . 11 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑔))
148147anassrs 453 . . . . . . . . . 10 ((((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ 𝑔 ∈ ℕ0) ∧ 𝑐𝑔) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑔))
14975, 78, 112, 148lecasei 10348 . . . . . . . . 9 (((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) ∧ 𝑔 ∈ ℕ0) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑔))
150149ralrimiva 3115 . . . . . . . 8 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → ∀𝑔 ∈ ℕ0 (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑔))
1512, 10, 11, 68, 71, 48, 73, 150hbtlem5 38224 . . . . . . 7 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → ((RSpan‘𝑃)‘ ran 𝑓) = 𝑎)
152151eqcomd 2777 . . . . . 6 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → 𝑎 = ((RSpan‘𝑃)‘ ran 𝑓))
153 fveq2 6333 . . . . . . . 8 (𝑏 = ran 𝑓 → ((RSpan‘𝑃)‘𝑏) = ((RSpan‘𝑃)‘ ran 𝑓))
154153eqeq2d 2781 . . . . . . 7 (𝑏 = ran 𝑓 → (𝑎 = ((RSpan‘𝑃)‘𝑏) ↔ 𝑎 = ((RSpan‘𝑃)‘ ran 𝑓)))
155154rspcev 3460 . . . . . 6 (( ran 𝑓 ∈ (𝒫 (Base‘𝑃) ∩ Fin) ∧ 𝑎 = ((RSpan‘𝑃)‘ ran 𝑓)) → ∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏))
15667, 152, 155syl2anc 573 . . . . 5 ((((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) ∧ (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) ∧ ∀𝑒 ∈ (0...𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))) → ∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏))
15740, 156exlimddv 2015 . . . 4 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ (𝑐 ∈ ℕ0 ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))) → ∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏))
15825, 157rexlimddv 3183 . . 3 ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏))
159158ralrimiva 3115 . 2 (𝑅 ∈ LNoeR → ∀𝑎 ∈ (LIdeal‘𝑃)∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏))
16049, 10, 27islnr2 38210 . 2 (𝑃 ∈ LNoeR ↔ (𝑃 ∈ Ring ∧ ∀𝑎 ∈ (LIdeal‘𝑃)∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏)))
1614, 159, 160sylanbrc 572 1 (𝑅 ∈ LNoeR → 𝑃 ∈ LNoeR)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wex 1852  wcel 2145  wral 3061  wrex 3062  cin 3722  wss 3723  𝒫 cpw 4298   cuni 4575   ciun 4655   class class class wbr 4787  ran crn 5251   Fn wfn 6025  wf 6026  cfv 6030  (class class class)co 6795  Fincfn 8112  cr 10140  0cc0 10141  1c1 10142   + caddc 10144  cle 10280  0cn0 11498  cz 11583  cuz 11892  ...cfz 12532  Basecbs 16063  Ringcrg 18754  LIdealclidl 19384  RSpancrsp 19385  Poly1cpl1 19761  NoeACScnacs 37791  LNoeRclnr 38205  ldgIdlSeqcldgis 38217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099  ax-inf2 8705  ax-cnex 10197  ax-resscn 10198  ax-1cn 10199  ax-icn 10200  ax-addcl 10201  ax-addrcl 10202  ax-mulcl 10203  ax-mulrcl 10204  ax-mulcom 10205  ax-addass 10206  ax-mulass 10207  ax-distr 10208  ax-i2m1 10209  ax-1ne0 10210  ax-1rid 10211  ax-rnegex 10212  ax-rrecex 10213  ax-cnre 10214  ax-pre-lttri 10215  ax-pre-lttrn 10216  ax-pre-ltadd 10217  ax-pre-mulgt0 10218  ax-pre-sup 10219  ax-addf 10220  ax-mulf 10221
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-iin 4658  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-se 5210  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-isom 6039  df-riota 6756  df-ov 6798  df-oprab 6799  df-mpt2 6800  df-of 7047  df-ofr 7048  df-om 7216  df-1st 7318  df-2nd 7319  df-supp 7450  df-tpos 7507  df-wrecs 7562  df-recs 7624  df-rdg 7662  df-1o 7716  df-2o 7717  df-oadd 7720  df-er 7899  df-map 8014  df-pm 8015  df-ixp 8066  df-en 8113  df-dom 8114  df-sdom 8115  df-fin 8116  df-fsupp 8435  df-sup 8507  df-oi 8574  df-card 8968  df-pnf 10281  df-mnf 10282  df-xr 10283  df-ltxr 10284  df-le 10285  df-sub 10473  df-neg 10474  df-nn 11226  df-2 11284  df-3 11285  df-4 11286  df-5 11287  df-6 11288  df-7 11289  df-8 11290  df-9 11291  df-n0 11499  df-z 11584  df-dec 11700  df-uz 11893  df-fz 12533  df-fzo 12673  df-seq 13008  df-hash 13321  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-ress 16071  df-plusg 16161  df-mulr 16162  df-starv 16163  df-sca 16164  df-vsca 16165  df-ip 16166  df-tset 16167  df-ple 16168  df-ocomp 16170  df-ds 16171  df-unif 16172  df-0g 16309  df-gsum 16310  df-mre 16453  df-mrc 16454  df-acs 16456  df-preset 17135  df-drs 17136  df-poset 17153  df-ipo 17359  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-mhm 17542  df-submnd 17543  df-grp 17632  df-minusg 17633  df-sbg 17634  df-mulg 17748  df-subg 17798  df-ghm 17865  df-cntz 17956  df-cmn 18401  df-abl 18402  df-mgp 18697  df-ur 18709  df-ring 18756  df-cring 18757  df-oppr 18830  df-dvdsr 18848  df-unit 18849  df-invr 18879  df-subrg 18987  df-lmod 19074  df-lss 19142  df-lsp 19184  df-sra 19386  df-rgmod 19387  df-lidl 19388  df-rsp 19389  df-rlreg 19497  df-ascl 19528  df-psr 19570  df-mvr 19571  df-mpl 19572  df-opsr 19574  df-psr1 19764  df-vr1 19765  df-ply1 19766  df-coe1 19767  df-cnfld 19961  df-mdeg 24034  df-deg1 24035  df-nacs 37792  df-lfig 38164  df-lnm 38172  df-lnr 38206  df-ldgis 38218
This theorem is referenced by: (None)
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