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Theorem unichnidl 36899
Description: The union of a nonempty chain of ideals is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.)
Assertion
Ref Expression
unichnidl ((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) → 𝐶 ∈ (Idl‘𝑅))
Distinct variable groups:   𝑅,𝑖   𝐶,𝑖,𝑗
Allowed substitution hint:   𝑅(𝑗)

Proof of Theorem unichnidl
Dummy variables 𝑘 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfss3 3971 . . . . 5 (𝐶 ⊆ (Idl‘𝑅) ↔ ∀𝑖𝐶 𝑖 ∈ (Idl‘𝑅))
2 eqid 2733 . . . . . . . . 9 (1st𝑅) = (1st𝑅)
3 eqid 2733 . . . . . . . . 9 ran (1st𝑅) = ran (1st𝑅)
42, 3idlss 36884 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑖 ⊆ ran (1st𝑅))
54ex 414 . . . . . . 7 (𝑅 ∈ RingOps → (𝑖 ∈ (Idl‘𝑅) → 𝑖 ⊆ ran (1st𝑅)))
65ralimdv 3170 . . . . . 6 (𝑅 ∈ RingOps → (∀𝑖𝐶 𝑖 ∈ (Idl‘𝑅) → ∀𝑖𝐶 𝑖 ⊆ ran (1st𝑅)))
76imp 408 . . . . 5 ((𝑅 ∈ RingOps ∧ ∀𝑖𝐶 𝑖 ∈ (Idl‘𝑅)) → ∀𝑖𝐶 𝑖 ⊆ ran (1st𝑅))
81, 7sylan2b 595 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖𝐶 𝑖 ⊆ ran (1st𝑅))
9 unissb 4944 . . . 4 ( 𝐶 ⊆ ran (1st𝑅) ↔ ∀𝑖𝐶 𝑖 ⊆ ran (1st𝑅))
108, 9sylibr 233 . . 3 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 ⊆ ran (1st𝑅))
11103ad2antr2 1190 . 2 ((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) → 𝐶 ⊆ ran (1st𝑅))
12 eqid 2733 . . . . . . . . . . 11 (GId‘(1st𝑅)) = (GId‘(1st𝑅))
132, 12idl0cl 36886 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → (GId‘(1st𝑅)) ∈ 𝑖)
1413ex 414 . . . . . . . . 9 (𝑅 ∈ RingOps → (𝑖 ∈ (Idl‘𝑅) → (GId‘(1st𝑅)) ∈ 𝑖))
1514ralimdv 3170 . . . . . . . 8 (𝑅 ∈ RingOps → (∀𝑖𝐶 𝑖 ∈ (Idl‘𝑅) → ∀𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖))
1615imp 408 . . . . . . 7 ((𝑅 ∈ RingOps ∧ ∀𝑖𝐶 𝑖 ∈ (Idl‘𝑅)) → ∀𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖)
171, 16sylan2b 595 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖)
18 r19.2z 4495 . . . . . 6 ((𝐶 ≠ ∅ ∧ ∀𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖) → ∃𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖)
1917, 18sylan2 594 . . . . 5 ((𝐶 ≠ ∅ ∧ (𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅))) → ∃𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖)
2019an12s 648 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅))) → ∃𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖)
21 eluni2 4913 . . . 4 ((GId‘(1st𝑅)) ∈ 𝐶 ↔ ∃𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖)
2220, 21sylibr 233 . . 3 ((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅))) → (GId‘(1st𝑅)) ∈ 𝐶)
23223adantr3 1172 . 2 ((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) → (GId‘(1st𝑅)) ∈ 𝐶)
24 eluni2 4913 . . . 4 (𝑥 𝐶 ↔ ∃𝑘𝐶 𝑥𝑘)
25 sseq1 4008 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑘 → (𝑖𝑗𝑘𝑗))
26 sseq2 4009 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑘 → (𝑗𝑖𝑗𝑘))
2725, 26orbi12d 918 . . . . . . . . . . . . . . 15 (𝑖 = 𝑘 → ((𝑖𝑗𝑗𝑖) ↔ (𝑘𝑗𝑗𝑘)))
2827ralbidv 3178 . . . . . . . . . . . . . 14 (𝑖 = 𝑘 → (∀𝑗𝐶 (𝑖𝑗𝑗𝑖) ↔ ∀𝑗𝐶 (𝑘𝑗𝑗𝑘)))
2928rspcv 3609 . . . . . . . . . . . . 13 (𝑘𝐶 → (∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖) → ∀𝑗𝐶 (𝑘𝑗𝑗𝑘)))
3029adantr 482 . . . . . . . . . . . 12 ((𝑘𝐶𝑥𝑘) → (∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖) → ∀𝑗𝐶 (𝑘𝑗𝑗𝑘)))
3130ad2antlr 726 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) → (∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖) → ∀𝑗𝐶 (𝑘𝑗𝑗𝑘)))
3231imp 408 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖)) → ∀𝑗𝐶 (𝑘𝑗𝑗𝑘))
33 eluni2 4913 . . . . . . . . . . . 12 (𝑦 𝐶 ↔ ∃𝑖𝐶 𝑦𝑖)
34 sseq2 4009 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑖 → (𝑘𝑗𝑘𝑖))
35 sseq1 4008 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑖 → (𝑗𝑘𝑖𝑘))
3634, 35orbi12d 918 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑖 → ((𝑘𝑗𝑗𝑘) ↔ (𝑘𝑖𝑖𝑘)))
3736rspcv 3609 . . . . . . . . . . . . . . . . 17 (𝑖𝐶 → (∀𝑗𝐶 (𝑘𝑗𝑗𝑘) → (𝑘𝑖𝑖𝑘)))
3837ad2antrl 727 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑖𝐶𝑦𝑖)) → (∀𝑗𝐶 (𝑘𝑗𝑗𝑘) → (𝑘𝑖𝑖𝑘)))
3938imp 408 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑖𝐶𝑦𝑖)) ∧ ∀𝑗𝐶 (𝑘𝑗𝑗𝑘)) → (𝑘𝑖𝑖𝑘))
40 ssel2 3978 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘𝑖𝑥𝑘) → 𝑥𝑖)
4140ancoms 460 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝑘𝑘𝑖) → 𝑥𝑖)
4241adantll 713 . . . . . . . . . . . . . . . . . . . . 21 (((𝑘𝐶𝑥𝑘) ∧ 𝑘𝑖) → 𝑥𝑖)
43 ssel2 3978 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶) → 𝑖 ∈ (Idl‘𝑅))
442idladdcl 36887 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑥𝑖𝑦𝑖)) → (𝑥(1st𝑅)𝑦) ∈ 𝑖)
4544ancom2s 649 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑦𝑖𝑥𝑖)) → (𝑥(1st𝑅)𝑦) ∈ 𝑖)
4645expr 458 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑦𝑖) → (𝑥𝑖 → (𝑥(1st𝑅)𝑦) ∈ 𝑖))
4746an32s 651 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑅 ∈ RingOps ∧ 𝑦𝑖) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑥𝑖 → (𝑥(1st𝑅)𝑦) ∈ 𝑖))
4843, 47sylan2 594 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑅 ∈ RingOps ∧ 𝑦𝑖) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶)) → (𝑥𝑖 → (𝑥(1st𝑅)𝑦) ∈ 𝑖))
4948an42s 660 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑖𝐶𝑦𝑖)) → (𝑥𝑖 → (𝑥(1st𝑅)𝑦) ∈ 𝑖))
5049anasss 468 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ (𝑖𝐶𝑦𝑖))) → (𝑥𝑖 → (𝑥(1st𝑅)𝑦) ∈ 𝑖))
5150imp 408 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ (𝑖𝐶𝑦𝑖))) ∧ 𝑥𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝑖)
52 simprl 770 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 ⊆ (Idl‘𝑅) ∧ (𝑖𝐶𝑦𝑖)) → 𝑖𝐶)
5352ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ (𝑖𝐶𝑦𝑖))) ∧ 𝑥𝑖) → 𝑖𝐶)
54 elunii 4914 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥(1st𝑅)𝑦) ∈ 𝑖𝑖𝐶) → (𝑥(1st𝑅)𝑦) ∈ 𝐶)
5551, 53, 54syl2anc 585 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ (𝑖𝐶𝑦𝑖))) ∧ 𝑥𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝐶)
5642, 55sylan2 594 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ (𝑖𝐶𝑦𝑖))) ∧ ((𝑘𝐶𝑥𝑘) ∧ 𝑘𝑖)) → (𝑥(1st𝑅)𝑦) ∈ 𝐶)
5756expr 458 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ (𝑖𝐶𝑦𝑖))) ∧ (𝑘𝐶𝑥𝑘)) → (𝑘𝑖 → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
5857an32s 651 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ (𝑖𝐶𝑦𝑖))) → (𝑘𝑖 → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
5958anassrs 469 . . . . . . . . . . . . . . . . 17 ((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑖𝐶𝑦𝑖)) → (𝑘𝑖 → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
6059imp 408 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑖𝐶𝑦𝑖)) ∧ 𝑘𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝐶)
61 ssel2 3978 . . . . . . . . . . . . . . . . . . . 20 ((𝑖𝑘𝑦𝑖) → 𝑦𝑘)
6261ancoms 460 . . . . . . . . . . . . . . . . . . 19 ((𝑦𝑖𝑖𝑘) → 𝑦𝑘)
6362adantll 713 . . . . . . . . . . . . . . . . . 18 (((𝑖𝐶𝑦𝑖) ∧ 𝑖𝑘) → 𝑦𝑘)
64 ssel2 3978 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 ⊆ (Idl‘𝑅) ∧ 𝑘𝐶) → 𝑘 ∈ (Idl‘𝑅))
652idladdcl 36887 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑅 ∈ RingOps ∧ 𝑘 ∈ (Idl‘𝑅)) ∧ (𝑥𝑘𝑦𝑘)) → (𝑥(1st𝑅)𝑦) ∈ 𝑘)
6665expr 458 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑅 ∈ RingOps ∧ 𝑘 ∈ (Idl‘𝑅)) ∧ 𝑥𝑘) → (𝑦𝑘 → (𝑥(1st𝑅)𝑦) ∈ 𝑘))
6766an32s 651 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ RingOps ∧ 𝑥𝑘) ∧ 𝑘 ∈ (Idl‘𝑅)) → (𝑦𝑘 → (𝑥(1st𝑅)𝑦) ∈ 𝑘))
6864, 67sylan2 594 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ RingOps ∧ 𝑥𝑘) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑘𝐶)) → (𝑦𝑘 → (𝑥(1st𝑅)𝑦) ∈ 𝑘))
6968an42s 660 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑘𝐶𝑥𝑘)) → (𝑦𝑘 → (𝑥(1st𝑅)𝑦) ∈ 𝑘))
7069an32s 651 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) → (𝑦𝑘 → (𝑥(1st𝑅)𝑦) ∈ 𝑘))
7170imp 408 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑦𝑘) → (𝑥(1st𝑅)𝑦) ∈ 𝑘)
72 simprl 770 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) → 𝑘𝐶)
7372ad2antrr 725 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑦𝑘) → 𝑘𝐶)
74 elunii 4914 . . . . . . . . . . . . . . . . . . 19 (((𝑥(1st𝑅)𝑦) ∈ 𝑘𝑘𝐶) → (𝑥(1st𝑅)𝑦) ∈ 𝐶)
7571, 73, 74syl2anc 585 . . . . . . . . . . . . . . . . . 18 ((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑦𝑘) → (𝑥(1st𝑅)𝑦) ∈ 𝐶)
7663, 75sylan2 594 . . . . . . . . . . . . . . . . 17 ((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ((𝑖𝐶𝑦𝑖) ∧ 𝑖𝑘)) → (𝑥(1st𝑅)𝑦) ∈ 𝐶)
7776anassrs 469 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑖𝐶𝑦𝑖)) ∧ 𝑖𝑘) → (𝑥(1st𝑅)𝑦) ∈ 𝐶)
7860, 77jaodan 957 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑖𝐶𝑦𝑖)) ∧ (𝑘𝑖𝑖𝑘)) → (𝑥(1st𝑅)𝑦) ∈ 𝐶)
7939, 78syldan 592 . . . . . . . . . . . . . 14 (((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑖𝐶𝑦𝑖)) ∧ ∀𝑗𝐶 (𝑘𝑗𝑗𝑘)) → (𝑥(1st𝑅)𝑦) ∈ 𝐶)
8079an32s 651 . . . . . . . . . . . . 13 (((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑗𝐶 (𝑘𝑗𝑗𝑘)) ∧ (𝑖𝐶𝑦𝑖)) → (𝑥(1st𝑅)𝑦) ∈ 𝐶)
8180rexlimdvaa 3157 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑗𝐶 (𝑘𝑗𝑗𝑘)) → (∃𝑖𝐶 𝑦𝑖 → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
8233, 81biimtrid 241 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑗𝐶 (𝑘𝑗𝑗𝑘)) → (𝑦 𝐶 → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
8382ralrimiv 3146 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑗𝐶 (𝑘𝑗𝑗𝑘)) → ∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶)
8432, 83syldan 592 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖)) → ∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶)
8584anasss 468 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) → ∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶)
86853adantr1 1170 . . . . . . 7 (((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) → ∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶)
8786an32s 651 . . . . . 6 (((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) ∧ (𝑘𝐶𝑥𝑘)) → ∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶)
88 eqid 2733 . . . . . . . . . . . . . . . . . 18 (2nd𝑅) = (2nd𝑅)
892, 88, 3idllmulcl 36888 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ RingOps ∧ 𝑘 ∈ (Idl‘𝑅)) ∧ (𝑥𝑘𝑧 ∈ ran (1st𝑅))) → (𝑧(2nd𝑅)𝑥) ∈ 𝑘)
9089exp43 438 . . . . . . . . . . . . . . . 16 (𝑅 ∈ RingOps → (𝑘 ∈ (Idl‘𝑅) → (𝑥𝑘 → (𝑧 ∈ ran (1st𝑅) → (𝑧(2nd𝑅)𝑥) ∈ 𝑘))))
9190com23 86 . . . . . . . . . . . . . . 15 (𝑅 ∈ RingOps → (𝑥𝑘 → (𝑘 ∈ (Idl‘𝑅) → (𝑧 ∈ ran (1st𝑅) → (𝑧(2nd𝑅)𝑥) ∈ 𝑘))))
9291imp41 427 . . . . . . . . . . . . . 14 ((((𝑅 ∈ RingOps ∧ 𝑥𝑘) ∧ 𝑘 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑧(2nd𝑅)𝑥) ∈ 𝑘)
9364, 92sylanl2 680 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑥𝑘) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑘𝐶)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑧(2nd𝑅)𝑥) ∈ 𝑘)
94 simplrr 777 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑥𝑘) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑘𝐶)) ∧ 𝑧 ∈ ran (1st𝑅)) → 𝑘𝐶)
95 elunii 4914 . . . . . . . . . . . . 13 (((𝑧(2nd𝑅)𝑥) ∈ 𝑘𝑘𝐶) → (𝑧(2nd𝑅)𝑥) ∈ 𝐶)
9693, 94, 95syl2anc 585 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑥𝑘) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑘𝐶)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑧(2nd𝑅)𝑥) ∈ 𝐶)
972, 88, 3idlrmulcl 36889 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ RingOps ∧ 𝑘 ∈ (Idl‘𝑅)) ∧ (𝑥𝑘𝑧 ∈ ran (1st𝑅))) → (𝑥(2nd𝑅)𝑧) ∈ 𝑘)
9897exp43 438 . . . . . . . . . . . . . . . 16 (𝑅 ∈ RingOps → (𝑘 ∈ (Idl‘𝑅) → (𝑥𝑘 → (𝑧 ∈ ran (1st𝑅) → (𝑥(2nd𝑅)𝑧) ∈ 𝑘))))
9998com23 86 . . . . . . . . . . . . . . 15 (𝑅 ∈ RingOps → (𝑥𝑘 → (𝑘 ∈ (Idl‘𝑅) → (𝑧 ∈ ran (1st𝑅) → (𝑥(2nd𝑅)𝑧) ∈ 𝑘))))
10099imp41 427 . . . . . . . . . . . . . 14 ((((𝑅 ∈ RingOps ∧ 𝑥𝑘) ∧ 𝑘 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥(2nd𝑅)𝑧) ∈ 𝑘)
10164, 100sylanl2 680 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑥𝑘) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑘𝐶)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥(2nd𝑅)𝑧) ∈ 𝑘)
102 elunii 4914 . . . . . . . . . . . . 13 (((𝑥(2nd𝑅)𝑧) ∈ 𝑘𝑘𝐶) → (𝑥(2nd𝑅)𝑧) ∈ 𝐶)
103101, 94, 102syl2anc 585 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑥𝑘) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑘𝐶)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥(2nd𝑅)𝑧) ∈ 𝐶)
10496, 103jca 513 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑥𝑘) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑘𝐶)) ∧ 𝑧 ∈ ran (1st𝑅)) → ((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
105104ralrimiva 3147 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ 𝑥𝑘) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑘𝐶)) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
106105an42s 660 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑘𝐶𝑥𝑘)) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
107106an32s 651 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
1081073ad2antr2 1190 . . . . . . 7 (((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
109108an32s 651 . . . . . 6 (((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) ∧ (𝑘𝐶𝑥𝑘)) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
11087, 109jca 513 . . . . 5 (((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) ∧ (𝑘𝐶𝑥𝑘)) → (∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶 ∧ ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶)))
111110rexlimdvaa 3157 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) → (∃𝑘𝐶 𝑥𝑘 → (∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶 ∧ ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))))
11224, 111biimtrid 241 . . 3 ((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) → (𝑥 𝐶 → (∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶 ∧ ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))))
113112ralrimiv 3146 . 2 ((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) → ∀𝑥 𝐶(∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶 ∧ ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶)))
1142, 88, 3, 12isidl 36882 . . 3 (𝑅 ∈ RingOps → ( 𝐶 ∈ (Idl‘𝑅) ↔ ( 𝐶 ⊆ ran (1st𝑅) ∧ (GId‘(1st𝑅)) ∈ 𝐶 ∧ ∀𝑥 𝐶(∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶 ∧ ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶)))))
115114adantr 482 . 2 ((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) → ( 𝐶 ∈ (Idl‘𝑅) ↔ ( 𝐶 ⊆ ran (1st𝑅) ∧ (GId‘(1st𝑅)) ∈ 𝐶 ∧ ∀𝑥 𝐶(∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶 ∧ ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶)))))
11611, 23, 113, 115mpbir3and 1343 1 ((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) → 𝐶 ∈ (Idl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846  w3a 1088  wcel 2107  wne 2941  wral 3062  wrex 3071  wss 3949  c0 4323   cuni 4909  ran crn 5678  cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  GIdcgi 29743  RingOpscrngo 36762  Idlcidl 36875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-idl 36878
This theorem is referenced by: (None)
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