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Theorem hbt 42791
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 42773 . . 3 (𝑅 ∈ LNoeR → 𝑅 ∈ Ring)
2 hbt.p . . . 4 𝑃 = (Poly1𝑅)
32ply1ring 22237 . . 3 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
41, 3syl 17 . 2 (𝑅 ∈ LNoeR → 𝑃 ∈ Ring)
5 eqid 2726 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
6 eqid 2726 . . . . . . . 8 (LIdeal‘𝑅) = (LIdeal‘𝑅)
75, 6islnr3 42776 . . . . . . 7 (𝑅 ∈ LNoeR ↔ (𝑅 ∈ Ring ∧ (LIdeal‘𝑅) ∈ (NoeACS‘(Base‘𝑅))))
87simprbi 495 . . . . . 6 (𝑅 ∈ LNoeR → (LIdeal‘𝑅) ∈ (NoeACS‘(Base‘𝑅)))
98adantr 479 . . . . 5 ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → (LIdeal‘𝑅) ∈ (NoeACS‘(Base‘𝑅)))
10 eqid 2726 . . . . . . 7 (LIdeal‘𝑃) = (LIdeal‘𝑃)
11 eqid 2726 . . . . . . 7 (ldgIdlSeq‘𝑅) = (ldgIdlSeq‘𝑅)
122, 10, 11, 6hbtlem7 42786 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ((ldgIdlSeq‘𝑅)‘𝑎):ℕ0⟶(LIdeal‘𝑅))
131, 12sylan 578 . . . . 5 ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ((ldgIdlSeq‘𝑅)‘𝑎):ℕ0⟶(LIdeal‘𝑅))
141ad2antrr 724 . . . . . . 7 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ0) → 𝑅 ∈ Ring)
15 simplr 767 . . . . . . 7 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ0) → 𝑎 ∈ (LIdeal‘𝑃))
16 simpr 483 . . . . . . 7 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ0) → 𝑏 ∈ ℕ0)
17 peano2nn0 12564 . . . . . . . 8 (𝑏 ∈ ℕ0 → (𝑏 + 1) ∈ ℕ0)
1817adantl 480 . . . . . . 7 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ0) → (𝑏 + 1) ∈ ℕ0)
19 nn0re 12533 . . . . . . . . 9 (𝑏 ∈ ℕ0𝑏 ∈ ℝ)
2019lep1d 12197 . . . . . . . 8 (𝑏 ∈ ℕ0𝑏 ≤ (𝑏 + 1))
2120adantl 480 . . . . . . 7 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ0) → 𝑏 ≤ (𝑏 + 1))
222, 10, 11, 14, 15, 16, 18, 21hbtlem4 42787 . . . . . 6 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑏 ∈ ℕ0) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑏) ⊆ (((ldgIdlSeq‘𝑅)‘𝑎)‘(𝑏 + 1)))
2322ralrimiva 3136 . . . . 5 ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∀𝑏 ∈ ℕ0 (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑏) ⊆ (((ldgIdlSeq‘𝑅)‘𝑎)‘(𝑏 + 1)))
24 nacsfix 42369 . . . . 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 1368 . . . 4 ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∃𝑐 ∈ ℕ0𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
26 fzfi 13992 . . . . . . 7 (0...𝑐) ∈ Fin
27 eqid 2726 . . . . . . . . 9 (RSpan‘𝑃) = (RSpan‘𝑃)
28 simpll 765 . . . . . . . . 9 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑒 ∈ (0...𝑐)) → 𝑅 ∈ LNoeR)
29 simplr 767 . . . . . . . . 9 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑒 ∈ (0...𝑐)) → 𝑎 ∈ (LIdeal‘𝑃))
30 elfznn0 13648 . . . . . . . . . 10 (𝑒 ∈ (0...𝑐) → 𝑒 ∈ ℕ0)
3130adantl 480 . . . . . . . . 9 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑒 ∈ (0...𝑐)) → 𝑒 ∈ ℕ0)
322, 10, 11, 27, 28, 29, 31hbtlem6 42790 . . . . . . . 8 (((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) ∧ 𝑒 ∈ (0...𝑐)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏))‘𝑒))
3332ralrimiva 3136 . . . . . . 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 6906 . . . . . . . . . 10 (𝑏 = (𝑓𝑒) → ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏)) = ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒))))
3534fveq1d 6903 . . . . . . . . 9 (𝑏 = (𝑓𝑒) → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏))‘𝑒) = (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒))
3635sseq2d 4012 . . . . . . . 8 (𝑏 = (𝑓𝑒) → ((((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘𝑏))‘𝑒) ↔ (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒)))
3736ac6sfi 9321 . . . . . . 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)))
3826, 33, 37sylancr 585 . . . . . 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)))
3938adantr 479 . . . . 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)))
40 frn 6735 . . . . . . . . . . . . 13 (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) → ran 𝑓 ⊆ (𝒫 𝑎 ∩ Fin))
4140ad2antrl 726 . . . . . . . . . . . 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))
42 inss1 4230 . . . . . . . . . . . 12 (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑎
4341, 42sstrdi 3992 . . . . . . . . . . 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)
4443unissd 4923 . . . . . . . . . 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)
45 unipw 5456 . . . . . . . . . 10 𝒫 𝑎 = 𝑎
4644, 45sseqtrdi 4030 . . . . . . . . 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)
47 simpllr 774 . . . . . . . . . 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))
48 eqid 2726 . . . . . . . . . . 11 (Base‘𝑃) = (Base‘𝑃)
4948, 10lidlss 21201 . . . . . . . . . 10 (𝑎 ∈ (LIdeal‘𝑃) → 𝑎 ⊆ (Base‘𝑃))
5047, 49syl 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))
5146, 50sstrd 3990 . . . . . . . 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))
52 fvex 6914 . . . . . . . . 9 (Base‘𝑃) ∈ V
5352elpw2 5352 . . . . . . . 8 ( ran 𝑓 ∈ 𝒫 (Base‘𝑃) ↔ ran 𝑓 ⊆ (Base‘𝑃))
5451, 53sylibr 233 . . . . . . 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))
55 simprl 769 . . . . . . . . 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))
56 ffn 6728 . . . . . . . . 9 (𝑓:(0...𝑐)⟶(𝒫 𝑎 ∩ Fin) → 𝑓 Fn (0...𝑐))
57 fniunfv 7262 . . . . . . . . 9 (𝑓 Fn (0...𝑐) → 𝑔 ∈ (0...𝑐)(𝑓𝑔) = ran 𝑓)
5855, 56, 573syl 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)
59 inss2 4231 . . . . . . . . . . 11 (𝒫 𝑎 ∩ Fin) ⊆ Fin
6055ffvelcdmda 7098 . . . . . . . . . . 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))
6159, 60sselid 3977 . . . . . . . . . 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)
6261ralrimiva 3136 . . . . . . . . 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)
63 iunfi 9385 . . . . . . . . 9 (((0...𝑐) ∈ Fin ∧ ∀𝑔 ∈ (0...𝑐)(𝑓𝑔) ∈ Fin) → 𝑔 ∈ (0...𝑐)(𝑓𝑔) ∈ Fin)
6426, 62, 63sylancr 585 . . . . . . . 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)
6558, 64eqeltrrd 2827 . . . . . . 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)
6654, 65elind 4195 . . . . . 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))
671ad3antrrr 728 . . . . . . . 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)
684ad3antrrr 728 . . . . . . . . 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)
6927, 48, 10rspcl 21224 . . . . . . . . 9 ((𝑃 ∈ Ring ∧ ran 𝑓 ⊆ (Base‘𝑃)) → ((RSpan‘𝑃)‘ ran 𝑓) ∈ (LIdeal‘𝑃))
7068, 51, 69syl2anc 582 . . . . . . . 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))
7127, 10rspssp 21228 . . . . . . . . 9 ((𝑃 ∈ Ring ∧ 𝑎 ∈ (LIdeal‘𝑃) ∧ ran 𝑓𝑎) → ((RSpan‘𝑃)‘ ran 𝑓) ⊆ 𝑎)
7268, 47, 46, 71syl3anc 1368 . . . . . . . 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)
73 nn0re 12533 . . . . . . . . . . 11 (𝑔 ∈ ℕ0𝑔 ∈ ℝ)
7473adantl 480 . . . . . . . . . 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)
75 simplrl 775 . . . . . . . . . . . 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)
7675adantr 479 . . . . . . . . . . 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)
7776nn0red 12585 . . . . . . . . . 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)
78 simprl 769 . . . . . . . . . . . . . 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)
79 simprr 771 . . . . . . . . . . . . . 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)
8075adantr 479 . . . . . . . . . . . . . . 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)
81 fznn0 13647 . . . . . . . . . . . . . . 15 (𝑐 ∈ ℕ0 → (𝑔 ∈ (0...𝑐) ↔ (𝑔 ∈ ℕ0𝑔𝑐)))
8280, 81syl 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)))
8378, 79, 82mpbir2and 711 . . . . . . . . . . . . 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))
84 simplrr 776 . . . . . . . . . . . . 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))
85 fveq2 6901 . . . . . . . . . . . . . . 15 (𝑒 = 𝑔 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔))
86 2fveq3 6906 . . . . . . . . . . . . . . . . 17 (𝑒 = 𝑔 → ((RSpan‘𝑃)‘(𝑓𝑒)) = ((RSpan‘𝑃)‘(𝑓𝑔)))
8786fveq2d 6905 . . . . . . . . . . . . . . . 16 (𝑒 = 𝑔 → ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒))) = ((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑔))))
88 id 22 . . . . . . . . . . . . . . . 16 (𝑒 = 𝑔𝑒 = 𝑔)
8987, 88fveq12d 6908 . . . . . . . . . . . . . . 15 (𝑒 = 𝑔 → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒) = (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑔)))‘𝑔))
9085, 89sseq12d 4013 . . . . . . . . . . . . . 14 (𝑒 = 𝑔 → ((((ldgIdlSeq‘𝑅)‘𝑎)‘𝑒) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑒)))‘𝑒) ↔ (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) ⊆ (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘(𝑓𝑔)))‘𝑔)))
9190rspcva 3606 . . . . . . . . . . . . 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))
9283, 84, 91syl2anc 582 . . . . . . . . . . . 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))
9367adantr 479 . . . . . . . . . . . . 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)
94 fvssunirn 6934 . . . . . . . . . . . . . . . 16 (𝑓𝑔) ⊆ ran 𝑓
9594, 51sstrid 3991 . . . . . . . . . . . . . . 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))
9627, 48, 10rspcl 21224 . . . . . . . . . . . . . . 15 ((𝑃 ∈ Ring ∧ (𝑓𝑔) ⊆ (Base‘𝑃)) → ((RSpan‘𝑃)‘(𝑓𝑔)) ∈ (LIdeal‘𝑃))
9768, 95, 96syl2anc 582 . . . . . . . . . . . . . 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))
9897adantr 479 . . . . . . . . . . . . 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))
9970adantr 479 . . . . . . . . . . . . 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))
10067, 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)
101100adantr 479 . . . . . . . . . . . . . 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)
10227, 48rspssid 21225 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ Ring ∧ ran 𝑓 ⊆ (Base‘𝑃)) → ran 𝑓 ⊆ ((RSpan‘𝑃)‘ ran 𝑓))
10368, 51, 102syl2anc 582 . . . . . . . . . . . . . . . 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))
104103adantr 479 . . . . . . . . . . . . . . 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))
10594, 104sstrid 3991 . . . . . . . . . . . . . 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))
10627, 10rspssp 21228 . . . . . . . . . . . . . 14 ((𝑃 ∈ Ring ∧ ((RSpan‘𝑃)‘ ran 𝑓) ∈ (LIdeal‘𝑃) ∧ (𝑓𝑔) ⊆ ((RSpan‘𝑃)‘ ran 𝑓)) → ((RSpan‘𝑃)‘(𝑓𝑔)) ⊆ ((RSpan‘𝑃)‘ ran 𝑓))
107101, 99, 105, 106syl3anc 1368 . . . . . . . . . . . . 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))
1082, 10, 11, 93, 98, 99, 107, 78hbtlem3 42788 . . . . . . . . . . . 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))
10992, 108sstrd 3990 . . . . . . . . . . 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))
110109anassrs 466 . . . . . . . . . 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))
111 nn0z 12635 . . . . . . . . . . . . . . . 16 (𝑐 ∈ ℕ0𝑐 ∈ ℤ)
112111adantr 479 . . . . . . . . . . . . . . 15 ((𝑐 ∈ ℕ0 ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → 𝑐 ∈ ℤ)
113 nn0z 12635 . . . . . . . . . . . . . . . 16 (𝑔 ∈ ℕ0𝑔 ∈ ℤ)
114113ad2antrl 726 . . . . . . . . . . . . . . 15 ((𝑐 ∈ ℕ0 ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → 𝑔 ∈ ℤ)
115 simprr 771 . . . . . . . . . . . . . . 15 ((𝑐 ∈ ℕ0 ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → 𝑐𝑔)
116 eluz2 12880 . . . . . . . . . . . . . . 15 (𝑔 ∈ (ℤ𝑐) ↔ (𝑐 ∈ ℤ ∧ 𝑔 ∈ ℤ ∧ 𝑐𝑔))
117112, 114, 115, 116syl3anbrc 1340 . . . . . . . . . . . . . 14 ((𝑐 ∈ ℕ0 ∧ (𝑔 ∈ ℕ0𝑐𝑔)) → 𝑔 ∈ (ℤ𝑐))
11875, 117sylan 578 . . . . . . . . . . . . 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))
119 simprr 771 . . . . . . . . . . . . . 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))
120119ad2antrr 724 . . . . . . . . . . . . 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))
121 fveqeq2 6910 . . . . . . . . . . . . . 14 (𝑑 = 𝑔 → ((((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐) ↔ (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐)))
122121rspcva 3606 . . . . . . . . . . . . 13 ((𝑔 ∈ (ℤ𝑐) ∧ ∀𝑑 ∈ (ℤ𝑐)(((ldgIdlSeq‘𝑅)‘𝑎)‘𝑑) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐)) → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
123118, 120, 122syl2anc 582 . . . . . . . . . . . 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))
12475nn0red 12585 . . . . . . . . . . . . . . . 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)
125124leidd 11830 . . . . . . . . . . . . . . 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)
126109expr 455 . . . . . . . . . . . . . . . . 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)))
127126ralrimiva 3136 . . . . . . . . . . . . . . . 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)))
128 breq1 5156 . . . . . . . . . . . . . . . . . 18 (𝑔 = 𝑐 → (𝑔𝑐𝑐𝑐))
129 fveq2 6901 . . . . . . . . . . . . . . . . . . 19 (𝑔 = 𝑐 → (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑔) = (((ldgIdlSeq‘𝑅)‘𝑎)‘𝑐))
130 fveq2 6901 . . . . . . . . . . . . . . . . . . 19 (𝑔 = 𝑐 → (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑔) = (((ldgIdlSeq‘𝑅)‘((RSpan‘𝑃)‘ ran 𝑓))‘𝑐))
131129, 130sseq12d 4013 . . . . . . . . . . . . . . . . . 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)))
132128, 131imbi12d 343 . . . . . . . . . . . . . . . . 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))))
133132rspcva 3606 . . . . . . . . . . . . . . . 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)))
13475, 127, 133syl2anc 582 . . . . . . . . . . . . . . 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)))
135125, 134mpd 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))
136135adantr 479 . . . . . . . . . . . . 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))
13767adantr 479 . . . . . . . . . . . . . 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)
13870adantr 479 . . . . . . . . . . . . . 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))
13975adantr 479 . . . . . . . . . . . . . 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)
140 simprl 769 . . . . . . . . . . . . . 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)
141 simprr 771 . . . . . . . . . . . . . 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)
1422, 10, 11, 137, 138, 139, 140, 141hbtlem4 42787 . . . . . . . . . . . . 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))
143136, 142sstrd 3990 . . . . . . . . . . . 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))
144123, 143eqsstrd 4018 . . . . . . . . . . 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))
145144anassrs 466 . . . . . . . . . 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))
14674, 77, 110, 145lecasei 11370 . . . . . . . . 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))
147146ralrimiva 3136 . . . . . . . 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))
1482, 10, 11, 67, 70, 47, 72, 147hbtlem5 42789 . . . . . . 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)
149148eqcomd 2732 . . . . . 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))
150 fveq2 6901 . . . . . . 7 (𝑏 = ran 𝑓 → ((RSpan‘𝑃)‘𝑏) = ((RSpan‘𝑃)‘ ran 𝑓))
151150rspceeqv 3630 . . . . . 6 (( ran 𝑓 ∈ (𝒫 (Base‘𝑃) ∩ Fin) ∧ 𝑎 = ((RSpan‘𝑃)‘ ran 𝑓)) → ∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏))
15266, 149, 151syl2anc 582 . . . . 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))
15339, 152exlimddv 1931 . . . 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))
15425, 153rexlimddv 3151 . . 3 ((𝑅 ∈ LNoeR ∧ 𝑎 ∈ (LIdeal‘𝑃)) → ∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏))
155154ralrimiva 3136 . 2 (𝑅 ∈ LNoeR → ∀𝑎 ∈ (LIdeal‘𝑃)∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏))
15648, 10, 27islnr2 42775 . 2 (𝑃 ∈ LNoeR ↔ (𝑃 ∈ Ring ∧ ∀𝑎 ∈ (LIdeal‘𝑃)∃𝑏 ∈ (𝒫 (Base‘𝑃) ∩ Fin)𝑎 = ((RSpan‘𝑃)‘𝑏)))
1574, 155, 156sylanbrc 581 1 (𝑅 ∈ LNoeR → 𝑃 ∈ LNoeR)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wex 1774  wcel 2099  wral 3051  wrex 3060  cin 3946  wss 3947  𝒫 cpw 4607   cuni 4913   ciun 5001   class class class wbr 5153  ran crn 5683   Fn wfn 6549  wf 6550  cfv 6554  (class class class)co 7424  Fincfn 8974  cr 11157  0cc0 11158  1c1 11159   + caddc 11161  cle 11299  0cn0 12524  cz 12610  cuz 12874  ...cfz 13538  Basecbs 17213  Ringcrg 20216  LIdealclidl 21195  RSpancrsp 21196  Poly1cpl1 22166  NoeACScnacs 42359  LNoeRclnr 42770  ldgIdlSeqcldgis 42782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-pre-sup 11236  ax-addf 11237
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-of 7690  df-ofr 7691  df-om 7877  df-1st 8003  df-2nd 8004  df-supp 8175  df-tpos 8241  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-er 8734  df-map 8857  df-pm 8858  df-ixp 8927  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-fsupp 9406  df-sup 9485  df-oi 9553  df-card 9982  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12611  df-dec 12730  df-uz 12875  df-fz 13539  df-fzo 13682  df-seq 14022  df-hash 14348  df-struct 17149  df-sets 17166  df-slot 17184  df-ndx 17196  df-base 17214  df-ress 17243  df-plusg 17279  df-mulr 17280  df-starv 17281  df-sca 17282  df-vsca 17283  df-ip 17284  df-tset 17285  df-ple 17286  df-ocomp 17287  df-ds 17288  df-unif 17289  df-hom 17290  df-cco 17291  df-0g 17456  df-gsum 17457  df-prds 17462  df-pws 17464  df-mre 17599  df-mrc 17600  df-acs 17602  df-proset 18320  df-drs 18321  df-poset 18338  df-ipo 18553  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-mhm 18773  df-submnd 18774  df-grp 18931  df-minusg 18932  df-sbg 18933  df-mulg 19062  df-subg 19117  df-ghm 19207  df-cntz 19311  df-cmn 19780  df-abl 19781  df-mgp 20118  df-rng 20136  df-ur 20165  df-ring 20218  df-cring 20219  df-oppr 20316  df-dvdsr 20339  df-unit 20340  df-invr 20370  df-subrng 20528  df-subrg 20553  df-rlreg 20672  df-lmod 20838  df-lss 20909  df-lsp 20949  df-sra 21151  df-rgmod 21152  df-lidl 21197  df-rsp 21198  df-cnfld 21344  df-ascl 21853  df-psr 21906  df-mvr 21907  df-mpl 21908  df-opsr 21910  df-psr1 22169  df-vr1 22170  df-ply1 22171  df-coe1 22172  df-mdeg 26079  df-deg1 26080  df-nacs 42360  df-lfig 42729  df-lnm 42737  df-lnr 42771  df-ldgis 42783
This theorem is referenced by: (None)
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