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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 19833 . . 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 11540 . . . . . . . 8 (𝑏 ∈ ℕ0 → (𝑏 + 1) ∈ ℕ0)
1817adantl 467 . . . . . . 7 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ0) → (𝑏 + 1) ∈ ℕ0)
19 nn0re 11508 . . . . . . . . 9 (𝑏 ∈ ℕ0𝑏 ∈ ℝ)
2019lep1d 11161 . . . . . . . 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 12979 . . . . . . 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 12640 . . . . . . . . . 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 8364 . . . . . . 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 19425 . . . . . . . . . 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 6651 . . . . . . . . 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 8414 . . . . . . . . 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 19437 . . . . . . . . 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 19441 . . . . . . . . 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 11508 . . . . . . . . . . 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 11559 . . . . . . . . . 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 12639 . . . . . . . . . . . . . . 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 19437 . . . . . . . . . . . . . . 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 19438 . . . . . . . . . . . . . . . . 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 19441 . . . . . . . . . . . . . 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 11607 . . . . . . . . . . . . . . . 16 (𝑐 ∈ ℕ0𝑐 ∈ ℤ)
114113adantr 466 . . . . . . . . . . . . . . 15 ((𝑐 ∈ ℕ0 ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → 𝑐 ∈ ℤ)
115 nn0z 11607 . . . . . . . . . . . . . . . 16 (𝑔 ∈ ℕ0𝑔 ∈ ℤ)
116115ad2antrl 707 . . . . . . . . . . . . . . 15 ((𝑐 ∈ ℕ0 ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → 𝑔 ∈ ℤ)
117 simprr 756 . . . . . . . . . . . . . . 15 ((𝑐 ∈ ℕ0 ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → 𝑐𝑔)
118 eluz2 11899 . . . . . . . . . . . . . . 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 11559 . . . . . . . . . . . . . . . 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 10800 . . . . . . . . . . . . . . 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 10349 . . . . . . . . 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 6796  Fincfn 8113  cr 10141  0cc0 10142  1c1 10143   + caddc 10145  cle 10281  0cn0 11499  cz 11584  cuz 11893  ...cfz 12533  Basecbs 16064  Ringcrg 18755  LIdealclidl 19385  RSpancrsp 19386  Poly1cpl1 19762  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 7100  ax-inf2 8706  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219  ax-pre-sup 10220  ax-addf 10221  ax-mulf 10222
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 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-of 7048  df-ofr 7049  df-om 7217  df-1st 7319  df-2nd 7320  df-supp 7451  df-tpos 7508  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-2o 7718  df-oadd 7721  df-er 7900  df-map 8015  df-pm 8016  df-ixp 8067  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-fsupp 8436  df-sup 8508  df-oi 8575  df-card 8969  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-2 11285  df-3 11286  df-4 11287  df-5 11288  df-6 11289  df-7 11290  df-8 11291  df-9 11292  df-n0 11500  df-z 11585  df-dec 11701  df-uz 11894  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-mulr 16163  df-starv 16164  df-sca 16165  df-vsca 16166  df-ip 16167  df-tset 16168  df-ple 16169  df-ocomp 16171  df-ds 16172  df-unif 16173  df-0g 16310  df-gsum 16311  df-mre 16454  df-mrc 16455  df-acs 16457  df-preset 17136  df-drs 17137  df-poset 17154  df-ipo 17360  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-mhm 17543  df-submnd 17544  df-grp 17633  df-minusg 17634  df-sbg 17635  df-mulg 17749  df-subg 17799  df-ghm 17866  df-cntz 17957  df-cmn 18402  df-abl 18403  df-mgp 18698  df-ur 18710  df-ring 18757  df-cring 18758  df-oppr 18831  df-dvdsr 18849  df-unit 18850  df-invr 18880  df-subrg 18988  df-lmod 19075  df-lss 19143  df-lsp 19185  df-sra 19387  df-rgmod 19388  df-lidl 19389  df-rsp 19390  df-rlreg 19498  df-ascl 19529  df-psr 19571  df-mvr 19572  df-mpl 19573  df-opsr 19575  df-psr1 19765  df-vr1 19766  df-ply1 19767  df-coe1 19768  df-cnfld 19962  df-mdeg 24035  df-deg1 24036  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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