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Theorem hbt 42545
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 = (Poly1β€˜π‘…)
Assertion
Ref Expression
hbt (𝑅 ∈ LNoeR β†’ 𝑃 ∈ LNoeR)

Proof of Theorem hbt
Dummy variables π‘Ž 𝑏 𝑐 𝑒 𝑓 𝑔 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lnrring 42527 . . 3 (𝑅 ∈ LNoeR β†’ 𝑅 ∈ Ring)
2 hbt.p . . . 4 𝑃 = (Poly1β€˜π‘…)
32ply1ring 22160 . . 3 (𝑅 ∈ Ring β†’ 𝑃 ∈ Ring)
41, 3syl 17 . 2 (𝑅 ∈ LNoeR β†’ 𝑃 ∈ Ring)
5 eqid 2728 . . . . . . . 8 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
6 eqid 2728 . . . . . . . 8 (LIdealβ€˜π‘…) = (LIdealβ€˜π‘…)
75, 6islnr3 42530 . . . . . . 7 (𝑅 ∈ LNoeR ↔ (𝑅 ∈ Ring ∧ (LIdealβ€˜π‘…) ∈ (NoeACSβ€˜(Baseβ€˜π‘…))))
87simprbi 496 . . . . . 6 (𝑅 ∈ LNoeR β†’ (LIdealβ€˜π‘…) ∈ (NoeACSβ€˜(Baseβ€˜π‘…)))
98adantr 480 . . . . 5 ((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) β†’ (LIdealβ€˜π‘…) ∈ (NoeACSβ€˜(Baseβ€˜π‘…)))
10 eqid 2728 . . . . . . 7 (LIdealβ€˜π‘ƒ) = (LIdealβ€˜π‘ƒ)
11 eqid 2728 . . . . . . 7 (ldgIdlSeqβ€˜π‘…) = (ldgIdlSeqβ€˜π‘…)
122, 10, 11, 6hbtlem7 42540 . . . . . 6 ((𝑅 ∈ Ring ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) β†’ ((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž):β„•0⟢(LIdealβ€˜π‘…))
131, 12sylan 579 . . . . 5 ((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) β†’ ((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž):β„•0⟢(LIdealβ€˜π‘…))
141ad2antrr 725 . . . . . . 7 (((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ 𝑏 ∈ β„•0) β†’ 𝑅 ∈ Ring)
15 simplr 768 . . . . . . 7 (((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ 𝑏 ∈ β„•0) β†’ π‘Ž ∈ (LIdealβ€˜π‘ƒ))
16 simpr 484 . . . . . . 7 (((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ 𝑏 ∈ β„•0) β†’ 𝑏 ∈ β„•0)
17 peano2nn0 12537 . . . . . . . 8 (𝑏 ∈ β„•0 β†’ (𝑏 + 1) ∈ β„•0)
1817adantl 481 . . . . . . 7 (((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ 𝑏 ∈ β„•0) β†’ (𝑏 + 1) ∈ β„•0)
19 nn0re 12506 . . . . . . . . 9 (𝑏 ∈ β„•0 β†’ 𝑏 ∈ ℝ)
2019lep1d 12170 . . . . . . . 8 (𝑏 ∈ β„•0 β†’ 𝑏 ≀ (𝑏 + 1))
2120adantl 481 . . . . . . 7 (((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ 𝑏 ∈ β„•0) β†’ 𝑏 ≀ (𝑏 + 1))
222, 10, 11, 14, 15, 16, 18, 21hbtlem4 42541 . . . . . 6 (((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ 𝑏 ∈ β„•0) β†’ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜(𝑏 + 1)))
2322ralrimiva 3142 . . . . 5 ((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) β†’ βˆ€π‘ ∈ β„•0 (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜(𝑏 + 1)))
24 nacsfix 42123 . . . . 5 (((LIdealβ€˜π‘…) ∈ (NoeACSβ€˜(Baseβ€˜π‘…)) ∧ ((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž):β„•0⟢(LIdealβ€˜π‘…) ∧ βˆ€π‘ ∈ β„•0 (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜(𝑏 + 1))) β†’ βˆƒπ‘ ∈ β„•0 βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))
259, 13, 23, 24syl3anc 1369 . . . 4 ((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) β†’ βˆƒπ‘ ∈ β„•0 βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))
26 fzfi 13964 . . . . . . 7 (0...𝑐) ∈ Fin
27 eqid 2728 . . . . . . . . 9 (RSpanβ€˜π‘ƒ) = (RSpanβ€˜π‘ƒ)
28 simpll 766 . . . . . . . . 9 (((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ 𝑒 ∈ (0...𝑐)) β†’ 𝑅 ∈ LNoeR)
29 simplr 768 . . . . . . . . 9 (((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ 𝑒 ∈ (0...𝑐)) β†’ π‘Ž ∈ (LIdealβ€˜π‘ƒ))
30 elfznn0 13621 . . . . . . . . . 10 (𝑒 ∈ (0...𝑐) β†’ 𝑒 ∈ β„•0)
3130adantl 481 . . . . . . . . 9 (((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ 𝑒 ∈ (0...𝑐)) β†’ 𝑒 ∈ β„•0)
322, 10, 11, 27, 28, 29, 31hbtlem6 42544 . . . . . . . 8 (((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ 𝑒 ∈ (0...𝑐)) β†’ βˆƒπ‘ ∈ (𝒫 π‘Ž ∩ Fin)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜π‘))β€˜π‘’))
3332ralrimiva 3142 . . . . . . 7 ((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) β†’ βˆ€π‘’ ∈ (0...𝑐)βˆƒπ‘ ∈ (𝒫 π‘Ž ∩ Fin)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜π‘))β€˜π‘’))
34 2fveq3 6897 . . . . . . . . . 10 (𝑏 = (π‘“β€˜π‘’) β†’ ((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜π‘)) = ((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’))))
3534fveq1d 6894 . . . . . . . . 9 (𝑏 = (π‘“β€˜π‘’) β†’ (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜π‘))β€˜π‘’) = (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))
3635sseq2d 4011 . . . . . . . 8 (𝑏 = (π‘“β€˜π‘’) β†’ ((((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜π‘))β€˜π‘’) ↔ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’)))
3736ac6sfi 9306 . . . . . . 7 (((0...𝑐) ∈ Fin ∧ βˆ€π‘’ ∈ (0...𝑐)βˆƒπ‘ ∈ (𝒫 π‘Ž ∩ Fin)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜π‘))β€˜π‘’)) β†’ βˆƒπ‘“(𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’)))
3826, 33, 37sylancr 586 . . . . . 6 ((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) β†’ βˆƒπ‘“(𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’)))
3938adantr 480 . . . . 5 (((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) β†’ βˆƒπ‘“(𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’)))
40 frn 6724 . . . . . . . . . . . . 13 (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) β†’ ran 𝑓 βŠ† (𝒫 π‘Ž ∩ Fin))
4140ad2antrl 727 . . . . . . . . . . . 12 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ ran 𝑓 βŠ† (𝒫 π‘Ž ∩ Fin))
42 inss1 4225 . . . . . . . . . . . 12 (𝒫 π‘Ž ∩ Fin) βŠ† 𝒫 π‘Ž
4341, 42sstrdi 3991 . . . . . . . . . . 11 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ ran 𝑓 βŠ† 𝒫 π‘Ž)
4443unissd 4914 . . . . . . . . . 10 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ βˆͺ ran 𝑓 βŠ† βˆͺ 𝒫 π‘Ž)
45 unipw 5447 . . . . . . . . . 10 βˆͺ 𝒫 π‘Ž = π‘Ž
4644, 45sseqtrdi 4029 . . . . . . . . 9 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ βˆͺ ran 𝑓 βŠ† π‘Ž)
47 simpllr 775 . . . . . . . . . 10 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ π‘Ž ∈ (LIdealβ€˜π‘ƒ))
48 eqid 2728 . . . . . . . . . . 11 (Baseβ€˜π‘ƒ) = (Baseβ€˜π‘ƒ)
4948, 10lidlss 21102 . . . . . . . . . 10 (π‘Ž ∈ (LIdealβ€˜π‘ƒ) β†’ π‘Ž βŠ† (Baseβ€˜π‘ƒ))
5047, 49syl 17 . . . . . . . . 9 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ π‘Ž βŠ† (Baseβ€˜π‘ƒ))
5146, 50sstrd 3989 . . . . . . . 8 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ βˆͺ ran 𝑓 βŠ† (Baseβ€˜π‘ƒ))
52 fvex 6905 . . . . . . . . 9 (Baseβ€˜π‘ƒ) ∈ V
5352elpw2 5342 . . . . . . . 8 (βˆͺ ran 𝑓 ∈ 𝒫 (Baseβ€˜π‘ƒ) ↔ βˆͺ ran 𝑓 βŠ† (Baseβ€˜π‘ƒ))
5451, 53sylibr 233 . . . . . . 7 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ βˆͺ ran 𝑓 ∈ 𝒫 (Baseβ€˜π‘ƒ))
55 simprl 770 . . . . . . . . 9 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ 𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin))
56 ffn 6717 . . . . . . . . 9 (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) β†’ 𝑓 Fn (0...𝑐))
57 fniunfv 7252 . . . . . . . . 9 (𝑓 Fn (0...𝑐) β†’ βˆͺ 𝑔 ∈ (0...𝑐)(π‘“β€˜π‘”) = βˆͺ ran 𝑓)
5855, 56, 573syl 18 . . . . . . . 8 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ βˆͺ 𝑔 ∈ (0...𝑐)(π‘“β€˜π‘”) = βˆͺ ran 𝑓)
59 inss2 4226 . . . . . . . . . . 11 (𝒫 π‘Ž ∩ Fin) βŠ† Fin
6055ffvelcdmda 7089 . . . . . . . . . . 11 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ 𝑔 ∈ (0...𝑐)) β†’ (π‘“β€˜π‘”) ∈ (𝒫 π‘Ž ∩ Fin))
6159, 60sselid 3977 . . . . . . . . . 10 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ 𝑔 ∈ (0...𝑐)) β†’ (π‘“β€˜π‘”) ∈ Fin)
6261ralrimiva 3142 . . . . . . . . 9 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ βˆ€π‘” ∈ (0...𝑐)(π‘“β€˜π‘”) ∈ Fin)
63 iunfi 9359 . . . . . . . . 9 (((0...𝑐) ∈ Fin ∧ βˆ€π‘” ∈ (0...𝑐)(π‘“β€˜π‘”) ∈ Fin) β†’ βˆͺ 𝑔 ∈ (0...𝑐)(π‘“β€˜π‘”) ∈ Fin)
6426, 62, 63sylancr 586 . . . . . . . 8 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ βˆͺ 𝑔 ∈ (0...𝑐)(π‘“β€˜π‘”) ∈ Fin)
6558, 64eqeltrrd 2830 . . . . . . 7 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ βˆͺ ran 𝑓 ∈ Fin)
6654, 65elind 4191 . . . . . 6 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ βˆͺ ran 𝑓 ∈ (𝒫 (Baseβ€˜π‘ƒ) ∩ Fin))
671ad3antrrr 729 . . . . . . . 8 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ 𝑅 ∈ Ring)
684ad3antrrr 729 . . . . . . . . 9 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ 𝑃 ∈ Ring)
6927, 48, 10rspcl 21125 . . . . . . . . 9 ((𝑃 ∈ Ring ∧ βˆͺ ran 𝑓 βŠ† (Baseβ€˜π‘ƒ)) β†’ ((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓) ∈ (LIdealβ€˜π‘ƒ))
7068, 51, 69syl2anc 583 . . . . . . . 8 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ ((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓) ∈ (LIdealβ€˜π‘ƒ))
7127, 10rspssp 21129 . . . . . . . . 9 ((𝑃 ∈ Ring ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ) ∧ βˆͺ ran 𝑓 βŠ† π‘Ž) β†’ ((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓) βŠ† π‘Ž)
7268, 47, 46, 71syl3anc 1369 . . . . . . . 8 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ ((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓) βŠ† π‘Ž)
73 nn0re 12506 . . . . . . . . . . 11 (𝑔 ∈ β„•0 β†’ 𝑔 ∈ ℝ)
7473adantl 481 . . . . . . . . . 10 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ 𝑔 ∈ β„•0) β†’ 𝑔 ∈ ℝ)
75 simplrl 776 . . . . . . . . . . . 12 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ 𝑐 ∈ β„•0)
7675adantr 480 . . . . . . . . . . 11 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ 𝑔 ∈ β„•0) β†’ 𝑐 ∈ β„•0)
7776nn0red 12558 . . . . . . . . . 10 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ 𝑔 ∈ β„•0) β†’ 𝑐 ∈ ℝ)
78 simprl 770 . . . . . . . . . . . . . 14 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑔 ≀ 𝑐)) β†’ 𝑔 ∈ β„•0)
79 simprr 772 . . . . . . . . . . . . . 14 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑔 ≀ 𝑐)) β†’ 𝑔 ≀ 𝑐)
8075adantr 480 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑔 ≀ 𝑐)) β†’ 𝑐 ∈ β„•0)
81 fznn0 13620 . . . . . . . . . . . . . . 15 (𝑐 ∈ β„•0 β†’ (𝑔 ∈ (0...𝑐) ↔ (𝑔 ∈ β„•0 ∧ 𝑔 ≀ 𝑐)))
8280, 81syl 17 . . . . . . . . . . . . . 14 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑔 ≀ 𝑐)) β†’ (𝑔 ∈ (0...𝑐) ↔ (𝑔 ∈ β„•0 ∧ 𝑔 ≀ 𝑐)))
8378, 79, 82mpbir2and 712 . . . . . . . . . . . . 13 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑔 ≀ 𝑐)) β†’ 𝑔 ∈ (0...𝑐))
84 simplrr 777 . . . . . . . . . . . . 13 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑔 ≀ 𝑐)) β†’ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))
85 fveq2 6892 . . . . . . . . . . . . . . 15 (𝑒 = 𝑔 β†’ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘”))
86 2fveq3 6897 . . . . . . . . . . . . . . . . 17 (𝑒 = 𝑔 β†’ ((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)) = ((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘”)))
8786fveq2d 6896 . . . . . . . . . . . . . . . 16 (𝑒 = 𝑔 β†’ ((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’))) = ((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘”))))
88 id 22 . . . . . . . . . . . . . . . 16 (𝑒 = 𝑔 β†’ 𝑒 = 𝑔)
8987, 88fveq12d 6899 . . . . . . . . . . . . . . 15 (𝑒 = 𝑔 β†’ (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’) = (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘”)))β€˜π‘”))
9085, 89sseq12d 4012 . . . . . . . . . . . . . 14 (𝑒 = 𝑔 β†’ ((((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’) ↔ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘”) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘”)))β€˜π‘”)))
9190rspcva 3606 . . . . . . . . . . . . 13 ((𝑔 ∈ (0...𝑐) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’)) β†’ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘”) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘”)))β€˜π‘”))
9283, 84, 91syl2anc 583 . . . . . . . . . . . 12 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑔 ≀ 𝑐)) β†’ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘”) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘”)))β€˜π‘”))
9367adantr 480 . . . . . . . . . . . . 13 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑔 ≀ 𝑐)) β†’ 𝑅 ∈ Ring)
94 fvssunirn 6925 . . . . . . . . . . . . . . . 16 (π‘“β€˜π‘”) βŠ† βˆͺ ran 𝑓
9594, 51sstrid 3990 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ (π‘“β€˜π‘”) βŠ† (Baseβ€˜π‘ƒ))
9627, 48, 10rspcl 21125 . . . . . . . . . . . . . . 15 ((𝑃 ∈ Ring ∧ (π‘“β€˜π‘”) βŠ† (Baseβ€˜π‘ƒ)) β†’ ((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘”)) ∈ (LIdealβ€˜π‘ƒ))
9768, 95, 96syl2anc 583 . . . . . . . . . . . . . 14 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ ((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘”)) ∈ (LIdealβ€˜π‘ƒ))
9897adantr 480 . . . . . . . . . . . . 13 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑔 ≀ 𝑐)) β†’ ((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘”)) ∈ (LIdealβ€˜π‘ƒ))
9970adantr 480 . . . . . . . . . . . . 13 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑔 ≀ 𝑐)) β†’ ((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓) ∈ (LIdealβ€˜π‘ƒ))
10067, 3syl 17 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ 𝑃 ∈ Ring)
101100adantr 480 . . . . . . . . . . . . . 14 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑔 ≀ 𝑐)) β†’ 𝑃 ∈ Ring)
10227, 48rspssid 21126 . . . . . . . . . . . . . . . . 17 ((𝑃 ∈ Ring ∧ βˆͺ ran 𝑓 βŠ† (Baseβ€˜π‘ƒ)) β†’ βˆͺ ran 𝑓 βŠ† ((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))
10368, 51, 102syl2anc 583 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ βˆͺ ran 𝑓 βŠ† ((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))
104103adantr 480 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑔 ≀ 𝑐)) β†’ βˆͺ ran 𝑓 βŠ† ((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))
10594, 104sstrid 3990 . . . . . . . . . . . . . 14 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑔 ≀ 𝑐)) β†’ (π‘“β€˜π‘”) βŠ† ((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))
10627, 10rspssp 21129 . . . . . . . . . . . . . 14 ((𝑃 ∈ Ring ∧ ((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓) ∈ (LIdealβ€˜π‘ƒ) ∧ (π‘“β€˜π‘”) βŠ† ((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓)) β†’ ((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘”)) βŠ† ((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))
107101, 99, 105, 106syl3anc 1369 . . . . . . . . . . . . 13 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑔 ≀ 𝑐)) β†’ ((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘”)) βŠ† ((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))
1082, 10, 11, 93, 98, 99, 107, 78hbtlem3 42542 . . . . . . . . . . . 12 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑔 ≀ 𝑐)) β†’ (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘”)))β€˜π‘”) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘”))
10992, 108sstrd 3989 . . . . . . . . . . 11 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑔 ≀ 𝑐)) β†’ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘”) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘”))
110109anassrs 467 . . . . . . . . . 10 ((((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ 𝑔 ∈ β„•0) ∧ 𝑔 ≀ 𝑐) β†’ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘”) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘”))
111 nn0z 12608 . . . . . . . . . . . . . . . 16 (𝑐 ∈ β„•0 β†’ 𝑐 ∈ β„€)
112111adantr 480 . . . . . . . . . . . . . . 15 ((𝑐 ∈ β„•0 ∧ (𝑔 ∈ β„•0 ∧ 𝑐 ≀ 𝑔)) β†’ 𝑐 ∈ β„€)
113 nn0z 12608 . . . . . . . . . . . . . . . 16 (𝑔 ∈ β„•0 β†’ 𝑔 ∈ β„€)
114113ad2antrl 727 . . . . . . . . . . . . . . 15 ((𝑐 ∈ β„•0 ∧ (𝑔 ∈ β„•0 ∧ 𝑐 ≀ 𝑔)) β†’ 𝑔 ∈ β„€)
115 simprr 772 . . . . . . . . . . . . . . 15 ((𝑐 ∈ β„•0 ∧ (𝑔 ∈ β„•0 ∧ 𝑐 ≀ 𝑔)) β†’ 𝑐 ≀ 𝑔)
116 eluz2 12853 . . . . . . . . . . . . . . 15 (𝑔 ∈ (β„€β‰₯β€˜π‘) ↔ (𝑐 ∈ β„€ ∧ 𝑔 ∈ β„€ ∧ 𝑐 ≀ 𝑔))
117112, 114, 115, 116syl3anbrc 1341 . . . . . . . . . . . . . 14 ((𝑐 ∈ β„•0 ∧ (𝑔 ∈ β„•0 ∧ 𝑐 ≀ 𝑔)) β†’ 𝑔 ∈ (β„€β‰₯β€˜π‘))
11875, 117sylan 579 . . . . . . . . . . . . 13 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑐 ≀ 𝑔)) β†’ 𝑔 ∈ (β„€β‰₯β€˜π‘))
119 simprr 772 . . . . . . . . . . . . . 14 (((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) β†’ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))
120119ad2antrr 725 . . . . . . . . . . . . 13 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑐 ≀ 𝑔)) β†’ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))
121 fveqeq2 6901 . . . . . . . . . . . . . 14 (𝑑 = 𝑔 β†’ ((((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘) ↔ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘”) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘)))
122121rspcva 3606 . . . . . . . . . . . . 13 ((𝑔 ∈ (β„€β‰₯β€˜π‘) ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘)) β†’ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘”) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))
123118, 120, 122syl2anc 583 . . . . . . . . . . . 12 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑐 ≀ 𝑔)) β†’ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘”) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))
12475nn0red 12558 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ 𝑐 ∈ ℝ)
125124leidd 11805 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ 𝑐 ≀ 𝑐)
126109expr 456 . . . . . . . . . . . . . . . . 17 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ 𝑔 ∈ β„•0) β†’ (𝑔 ≀ 𝑐 β†’ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘”) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘”)))
127126ralrimiva 3142 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ βˆ€π‘” ∈ β„•0 (𝑔 ≀ 𝑐 β†’ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘”) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘”)))
128 breq1 5146 . . . . . . . . . . . . . . . . . 18 (𝑔 = 𝑐 β†’ (𝑔 ≀ 𝑐 ↔ 𝑐 ≀ 𝑐))
129 fveq2 6892 . . . . . . . . . . . . . . . . . . 19 (𝑔 = 𝑐 β†’ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘”) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))
130 fveq2 6892 . . . . . . . . . . . . . . . . . . 19 (𝑔 = 𝑐 β†’ (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘”) = (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘))
131129, 130sseq12d 4012 . . . . . . . . . . . . . . . . . 18 (𝑔 = 𝑐 β†’ ((((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘”) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘”) ↔ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘)))
132128, 131imbi12d 344 . . . . . . . . . . . . . . . . 17 (𝑔 = 𝑐 β†’ ((𝑔 ≀ 𝑐 β†’ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘”) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘”)) ↔ (𝑐 ≀ 𝑐 β†’ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘))))
133132rspcva 3606 . . . . . . . . . . . . . . . 16 ((𝑐 ∈ β„•0 ∧ βˆ€π‘” ∈ β„•0 (𝑔 ≀ 𝑐 β†’ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘”) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘”))) β†’ (𝑐 ≀ 𝑐 β†’ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘)))
13475, 127, 133syl2anc 583 . . . . . . . . . . . . . . 15 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ (𝑐 ≀ 𝑐 β†’ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘)))
135125, 134mpd 15 . . . . . . . . . . . . . 14 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘))
136135adantr 480 . . . . . . . . . . . . 13 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑐 ≀ 𝑔)) β†’ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘))
13767adantr 480 . . . . . . . . . . . . . 14 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑐 ≀ 𝑔)) β†’ 𝑅 ∈ Ring)
13870adantr 480 . . . . . . . . . . . . . 14 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑐 ≀ 𝑔)) β†’ ((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓) ∈ (LIdealβ€˜π‘ƒ))
13975adantr 480 . . . . . . . . . . . . . 14 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑐 ≀ 𝑔)) β†’ 𝑐 ∈ β„•0)
140 simprl 770 . . . . . . . . . . . . . 14 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑐 ≀ 𝑔)) β†’ 𝑔 ∈ β„•0)
141 simprr 772 . . . . . . . . . . . . . 14 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑐 ≀ 𝑔)) β†’ 𝑐 ≀ 𝑔)
1422, 10, 11, 137, 138, 139, 140, 141hbtlem4 42541 . . . . . . . . . . . . 13 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑐 ≀ 𝑔)) β†’ (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘”))
143136, 142sstrd 3989 . . . . . . . . . . . 12 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑐 ≀ 𝑔)) β†’ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘”))
144123, 143eqsstrd 4017 . . . . . . . . . . 11 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ (𝑔 ∈ β„•0 ∧ 𝑐 ≀ 𝑔)) β†’ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘”) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘”))
145144anassrs 467 . . . . . . . . . 10 ((((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ 𝑔 ∈ β„•0) ∧ 𝑐 ≀ 𝑔) β†’ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘”) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘”))
14674, 77, 110, 145lecasei 11345 . . . . . . . . 9 (((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) ∧ 𝑔 ∈ β„•0) β†’ (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘”) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘”))
147146ralrimiva 3142 . . . . . . . 8 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ βˆ€π‘” ∈ β„•0 (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘”) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))β€˜π‘”))
1482, 10, 11, 67, 70, 47, 72, 147hbtlem5 42543 . . . . . . 7 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ ((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓) = π‘Ž)
149148eqcomd 2734 . . . . . 6 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ π‘Ž = ((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))
150 fveq2 6892 . . . . . . 7 (𝑏 = βˆͺ ran 𝑓 β†’ ((RSpanβ€˜π‘ƒ)β€˜π‘) = ((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓))
151150rspceeqv 3630 . . . . . 6 ((βˆͺ ran 𝑓 ∈ (𝒫 (Baseβ€˜π‘ƒ) ∩ Fin) ∧ π‘Ž = ((RSpanβ€˜π‘ƒ)β€˜βˆͺ ran 𝑓)) β†’ βˆƒπ‘ ∈ (𝒫 (Baseβ€˜π‘ƒ) ∩ Fin)π‘Ž = ((RSpanβ€˜π‘ƒ)β€˜π‘))
15266, 149, 151syl2anc 583 . . . . 5 ((((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) ∧ (𝑓:(0...𝑐)⟢(𝒫 π‘Ž ∩ Fin) ∧ βˆ€π‘’ ∈ (0...𝑐)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘’) βŠ† (((ldgIdlSeqβ€˜π‘…)β€˜((RSpanβ€˜π‘ƒ)β€˜(π‘“β€˜π‘’)))β€˜π‘’))) β†’ βˆƒπ‘ ∈ (𝒫 (Baseβ€˜π‘ƒ) ∩ Fin)π‘Ž = ((RSpanβ€˜π‘ƒ)β€˜π‘))
15339, 152exlimddv 1931 . . . 4 (((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) ∧ (𝑐 ∈ β„•0 ∧ βˆ€π‘‘ ∈ (β„€β‰₯β€˜π‘)(((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘‘) = (((ldgIdlSeqβ€˜π‘…)β€˜π‘Ž)β€˜π‘))) β†’ βˆƒπ‘ ∈ (𝒫 (Baseβ€˜π‘ƒ) ∩ Fin)π‘Ž = ((RSpanβ€˜π‘ƒ)β€˜π‘))
15425, 153rexlimddv 3157 . . 3 ((𝑅 ∈ LNoeR ∧ π‘Ž ∈ (LIdealβ€˜π‘ƒ)) β†’ βˆƒπ‘ ∈ (𝒫 (Baseβ€˜π‘ƒ) ∩ Fin)π‘Ž = ((RSpanβ€˜π‘ƒ)β€˜π‘))
155154ralrimiva 3142 . 2 (𝑅 ∈ LNoeR β†’ βˆ€π‘Ž ∈ (LIdealβ€˜π‘ƒ)βˆƒπ‘ ∈ (𝒫 (Baseβ€˜π‘ƒ) ∩ Fin)π‘Ž = ((RSpanβ€˜π‘ƒ)β€˜π‘))
15648, 10, 27islnr2 42529 . 2 (𝑃 ∈ LNoeR ↔ (𝑃 ∈ Ring ∧ βˆ€π‘Ž ∈ (LIdealβ€˜π‘ƒ)βˆƒπ‘ ∈ (𝒫 (Baseβ€˜π‘ƒ) ∩ Fin)π‘Ž = ((RSpanβ€˜π‘ƒ)β€˜π‘)))
1574, 155, 156sylanbrc 582 1 (𝑅 ∈ LNoeR β†’ 𝑃 ∈ LNoeR)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534  βˆƒwex 1774   ∈ wcel 2099  βˆ€wral 3057  βˆƒwrex 3066   ∩ cin 3944   βŠ† wss 3945  π’« cpw 4599  βˆͺ cuni 4904  βˆͺ ciun 4992   class class class wbr 5143  ran crn 5674   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7415  Fincfn 8958  β„cr 11132  0cc0 11133  1c1 11134   + caddc 11136   ≀ cle 11274  β„•0cn0 12497  β„€cz 12583  β„€β‰₯cuz 12847  ...cfz 13511  Basecbs 17174  Ringcrg 20167  LIdealclidl 21096  RSpancrsp 21097  Poly1cpl1 22090  NoeACScnacs 42113  LNoeRclnr 42524  ldgIdlSeqcldgis 42536
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 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210  ax-pre-sup 11211  ax-addf 11212
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-iin 4995  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-se 5629  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-of 7680  df-ofr 7681  df-om 7866  df-1st 7988  df-2nd 7989  df-supp 8161  df-tpos 8226  df-frecs 8281  df-wrecs 8312  df-recs 8386  df-rdg 8425  df-1o 8481  df-er 8719  df-map 8841  df-pm 8842  df-ixp 8911  df-en 8959  df-dom 8960  df-sdom 8961  df-fin 8962  df-fsupp 9381  df-sup 9460  df-oi 9528  df-card 9957  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-2 12300  df-3 12301  df-4 12302  df-5 12303  df-6 12304  df-7 12305  df-8 12306  df-9 12307  df-n0 12498  df-z 12584  df-dec 12703  df-uz 12848  df-fz 13512  df-fzo 13655  df-seq 13994  df-hash 14317  df-struct 17110  df-sets 17127  df-slot 17145  df-ndx 17157  df-base 17175  df-ress 17204  df-plusg 17240  df-mulr 17241  df-starv 17242  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ocomp 17248  df-ds 17249  df-unif 17250  df-hom 17251  df-cco 17252  df-0g 17417  df-gsum 17418  df-prds 17423  df-pws 17425  df-mre 17560  df-mrc 17561  df-acs 17563  df-proset 18281  df-drs 18282  df-poset 18299  df-ipo 18514  df-mgm 18594  df-sgrp 18673  df-mnd 18689  df-mhm 18734  df-submnd 18735  df-grp 18887  df-minusg 18888  df-sbg 18889  df-mulg 19018  df-subg 19072  df-ghm 19162  df-cntz 19262  df-cmn 19731  df-abl 19732  df-mgp 20069  df-rng 20087  df-ur 20116  df-ring 20169  df-cring 20170  df-oppr 20267  df-dvdsr 20290  df-unit 20291  df-invr 20321  df-subrng 20477  df-subrg 20502  df-lmod 20739  df-lss 20810  df-lsp 20850  df-sra 21052  df-rgmod 21053  df-lidl 21098  df-rsp 21099  df-rlreg 21224  df-cnfld 21274  df-ascl 21783  df-psr 21836  df-mvr 21837  df-mpl 21838  df-opsr 21840  df-psr1 22093  df-vr1 22094  df-ply1 22095  df-coe1 22096  df-mdeg 25982  df-deg1 25983  df-nacs 42114  df-lfig 42483  df-lnm 42491  df-lnr 42525  df-ldgis 42537
This theorem is referenced by: (None)
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