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Theorem unichnidl 34273
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 3752 . . . . 5 (𝐶 ⊆ (Idl‘𝑅) ↔ ∀𝑖𝐶 𝑖 ∈ (Idl‘𝑅))
2 eqid 2765 . . . . . . . . 9 (1st𝑅) = (1st𝑅)
3 eqid 2765 . . . . . . . . 9 ran (1st𝑅) = ran (1st𝑅)
42, 3idlss 34258 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑖 ⊆ ran (1st𝑅))
54ex 401 . . . . . . 7 (𝑅 ∈ RingOps → (𝑖 ∈ (Idl‘𝑅) → 𝑖 ⊆ ran (1st𝑅)))
65ralimdv 3110 . . . . . 6 (𝑅 ∈ RingOps → (∀𝑖𝐶 𝑖 ∈ (Idl‘𝑅) → ∀𝑖𝐶 𝑖 ⊆ ran (1st𝑅)))
76imp 395 . . . . 5 ((𝑅 ∈ RingOps ∧ ∀𝑖𝐶 𝑖 ∈ (Idl‘𝑅)) → ∀𝑖𝐶 𝑖 ⊆ ran (1st𝑅))
81, 7sylan2b 587 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖𝐶 𝑖 ⊆ ran (1st𝑅))
9 unissb 4629 . . . 4 ( 𝐶 ⊆ ran (1st𝑅) ↔ ∀𝑖𝐶 𝑖 ⊆ ran (1st𝑅))
108, 9sylibr 225 . . 3 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 ⊆ ran (1st𝑅))
11103ad2antr2 1240 . 2 ((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) → 𝐶 ⊆ ran (1st𝑅))
12 eqid 2765 . . . . . . . . . . 11 (GId‘(1st𝑅)) = (GId‘(1st𝑅))
132, 12idl0cl 34260 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → (GId‘(1st𝑅)) ∈ 𝑖)
1413ex 401 . . . . . . . . 9 (𝑅 ∈ RingOps → (𝑖 ∈ (Idl‘𝑅) → (GId‘(1st𝑅)) ∈ 𝑖))
1514ralimdv 3110 . . . . . . . 8 (𝑅 ∈ RingOps → (∀𝑖𝐶 𝑖 ∈ (Idl‘𝑅) → ∀𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖))
1615imp 395 . . . . . . 7 ((𝑅 ∈ RingOps ∧ ∀𝑖𝐶 𝑖 ∈ (Idl‘𝑅)) → ∀𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖)
171, 16sylan2b 587 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖)
18 r19.2z 4221 . . . . . 6 ((𝐶 ≠ ∅ ∧ ∀𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖) → ∃𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖)
1917, 18sylan2 586 . . . . 5 ((𝐶 ≠ ∅ ∧ (𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅))) → ∃𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖)
2019an12s 639 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅))) → ∃𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖)
21 eluni2 4600 . . . 4 ((GId‘(1st𝑅)) ∈ 𝐶 ↔ ∃𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖)
2220, 21sylibr 225 . . 3 ((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅))) → (GId‘(1st𝑅)) ∈ 𝐶)
23223adantr3 1212 . 2 ((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) → (GId‘(1st𝑅)) ∈ 𝐶)
24 eluni2 4600 . . . 4 (𝑥 𝐶 ↔ ∃𝑘𝐶 𝑥𝑘)
25 sseq1 3788 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑘 → (𝑖𝑗𝑘𝑗))
26 sseq2 3789 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑘 → (𝑗𝑖𝑗𝑘))
2725, 26orbi12d 942 . . . . . . . . . . . . . . 15 (𝑖 = 𝑘 → ((𝑖𝑗𝑗𝑖) ↔ (𝑘𝑗𝑗𝑘)))
2827ralbidv 3133 . . . . . . . . . . . . . 14 (𝑖 = 𝑘 → (∀𝑗𝐶 (𝑖𝑗𝑗𝑖) ↔ ∀𝑗𝐶 (𝑘𝑗𝑗𝑘)))
2928rspcv 3458 . . . . . . . . . . . . 13 (𝑘𝐶 → (∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖) → ∀𝑗𝐶 (𝑘𝑗𝑗𝑘)))
3029adantr 472 . . . . . . . . . . . 12 ((𝑘𝐶𝑥𝑘) → (∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖) → ∀𝑗𝐶 (𝑘𝑗𝑗𝑘)))
3130ad2antlr 718 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) → (∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖) → ∀𝑗𝐶 (𝑘𝑗𝑗𝑘)))
3231imp 395 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖)) → ∀𝑗𝐶 (𝑘𝑗𝑗𝑘))
33 eluni2 4600 . . . . . . . . . . . 12 (𝑦 𝐶 ↔ ∃𝑖𝐶 𝑦𝑖)
34 sseq2 3789 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑖 → (𝑘𝑗𝑘𝑖))
35 sseq1 3788 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑖 → (𝑗𝑘𝑖𝑘))
3634, 35orbi12d 942 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑖 → ((𝑘𝑗𝑗𝑘) ↔ (𝑘𝑖𝑖𝑘)))
3736rspcv 3458 . . . . . . . . . . . . . . . . 17 (𝑖𝐶 → (∀𝑗𝐶 (𝑘𝑗𝑗𝑘) → (𝑘𝑖𝑖𝑘)))
3837ad2antrl 719 . . . . . . . . . . . . . . . 16 ((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑖𝐶𝑦𝑖)) → (∀𝑗𝐶 (𝑘𝑗𝑗𝑘) → (𝑘𝑖𝑖𝑘)))
3938imp 395 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑖𝐶𝑦𝑖)) ∧ ∀𝑗𝐶 (𝑘𝑗𝑗𝑘)) → (𝑘𝑖𝑖𝑘))
40 ssel2 3758 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘𝑖𝑥𝑘) → 𝑥𝑖)
4140ancoms 450 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝑘𝑘𝑖) → 𝑥𝑖)
4241adantll 705 . . . . . . . . . . . . . . . . . . . . 21 (((𝑘𝐶𝑥𝑘) ∧ 𝑘𝑖) → 𝑥𝑖)
43 ssel2 3758 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶) → 𝑖 ∈ (Idl‘𝑅))
442idladdcl 34261 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑥𝑖𝑦𝑖)) → (𝑥(1st𝑅)𝑦) ∈ 𝑖)
4544ancom2s 640 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑦𝑖𝑥𝑖)) → (𝑥(1st𝑅)𝑦) ∈ 𝑖)
4645expr 448 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑦𝑖) → (𝑥𝑖 → (𝑥(1st𝑅)𝑦) ∈ 𝑖))
4746an32s 642 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑅 ∈ RingOps ∧ 𝑦𝑖) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑥𝑖 → (𝑥(1st𝑅)𝑦) ∈ 𝑖))
4843, 47sylan2 586 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑅 ∈ RingOps ∧ 𝑦𝑖) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶)) → (𝑥𝑖 → (𝑥(1st𝑅)𝑦) ∈ 𝑖))
4948an42s 651 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑖𝐶𝑦𝑖)) → (𝑥𝑖 → (𝑥(1st𝑅)𝑦) ∈ 𝑖))
5049anasss 458 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ (𝑖𝐶𝑦𝑖))) → (𝑥𝑖 → (𝑥(1st𝑅)𝑦) ∈ 𝑖))
5150imp 395 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ (𝑖𝐶𝑦𝑖))) ∧ 𝑥𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝑖)
52 simprl 787 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 ⊆ (Idl‘𝑅) ∧ (𝑖𝐶𝑦𝑖)) → 𝑖𝐶)
5352ad2antlr 718 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ (𝑖𝐶𝑦𝑖))) ∧ 𝑥𝑖) → 𝑖𝐶)
54 elunii 4601 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥(1st𝑅)𝑦) ∈ 𝑖𝑖𝐶) → (𝑥(1st𝑅)𝑦) ∈ 𝐶)
5551, 53, 54syl2anc 579 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ (𝑖𝐶𝑦𝑖))) ∧ 𝑥𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝐶)
5642, 55sylan2 586 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ (𝑖𝐶𝑦𝑖))) ∧ ((𝑘𝐶𝑥𝑘) ∧ 𝑘𝑖)) → (𝑥(1st𝑅)𝑦) ∈ 𝐶)
5756expr 448 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ (𝑖𝐶𝑦𝑖))) ∧ (𝑘𝐶𝑥𝑘)) → (𝑘𝑖 → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
5857an32s 642 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ (𝑖𝐶𝑦𝑖))) → (𝑘𝑖 → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
5958anassrs 459 . . . . . . . . . . . . . . . . 17 ((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑖𝐶𝑦𝑖)) → (𝑘𝑖 → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
6059imp 395 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑖𝐶𝑦𝑖)) ∧ 𝑘𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝐶)
61 ssel2 3758 . . . . . . . . . . . . . . . . . . . 20 ((𝑖𝑘𝑦𝑖) → 𝑦𝑘)
6261ancoms 450 . . . . . . . . . . . . . . . . . . 19 ((𝑦𝑖𝑖𝑘) → 𝑦𝑘)
6362adantll 705 . . . . . . . . . . . . . . . . . 18 (((𝑖𝐶𝑦𝑖) ∧ 𝑖𝑘) → 𝑦𝑘)
64 ssel2 3758 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 ⊆ (Idl‘𝑅) ∧ 𝑘𝐶) → 𝑘 ∈ (Idl‘𝑅))
652idladdcl 34261 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑅 ∈ RingOps ∧ 𝑘 ∈ (Idl‘𝑅)) ∧ (𝑥𝑘𝑦𝑘)) → (𝑥(1st𝑅)𝑦) ∈ 𝑘)
6665expr 448 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑅 ∈ RingOps ∧ 𝑘 ∈ (Idl‘𝑅)) ∧ 𝑥𝑘) → (𝑦𝑘 → (𝑥(1st𝑅)𝑦) ∈ 𝑘))
6766an32s 642 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑅 ∈ RingOps ∧ 𝑥𝑘) ∧ 𝑘 ∈ (Idl‘𝑅)) → (𝑦𝑘 → (𝑥(1st𝑅)𝑦) ∈ 𝑘))
6864, 67sylan2 586 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑅 ∈ RingOps ∧ 𝑥𝑘) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑘𝐶)) → (𝑦𝑘 → (𝑥(1st𝑅)𝑦) ∈ 𝑘))
6968an42s 651 . . . . . . . . . . . . . . . . . . . . 21 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑘𝐶𝑥𝑘)) → (𝑦𝑘 → (𝑥(1st𝑅)𝑦) ∈ 𝑘))
7069an32s 642 . . . . . . . . . . . . . . . . . . . 20 (((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) → (𝑦𝑘 → (𝑥(1st𝑅)𝑦) ∈ 𝑘))
7170imp 395 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑦𝑘) → (𝑥(1st𝑅)𝑦) ∈ 𝑘)
72 simprl 787 . . . . . . . . . . . . . . . . . . . 20 ((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) → 𝑘𝐶)
7372ad2antrr 717 . . . . . . . . . . . . . . . . . . 19 ((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑦𝑘) → 𝑘𝐶)
74 elunii 4601 . . . . . . . . . . . . . . . . . . 19 (((𝑥(1st𝑅)𝑦) ∈ 𝑘𝑘𝐶) → (𝑥(1st𝑅)𝑦) ∈ 𝐶)
7571, 73, 74syl2anc 579 . . . . . . . . . . . . . . . . . 18 ((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑦𝑘) → (𝑥(1st𝑅)𝑦) ∈ 𝐶)
7663, 75sylan2 586 . . . . . . . . . . . . . . . . 17 ((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ((𝑖𝐶𝑦𝑖) ∧ 𝑖𝑘)) → (𝑥(1st𝑅)𝑦) ∈ 𝐶)
7776anassrs 459 . . . . . . . . . . . . . . . 16 (((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑖𝐶𝑦𝑖)) ∧ 𝑖𝑘) → (𝑥(1st𝑅)𝑦) ∈ 𝐶)
7860, 77jaodan 980 . . . . . . . . . . . . . . 15 (((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑖𝐶𝑦𝑖)) ∧ (𝑘𝑖𝑖𝑘)) → (𝑥(1st𝑅)𝑦) ∈ 𝐶)
7939, 78syldan 585 . . . . . . . . . . . . . 14 (((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑖𝐶𝑦𝑖)) ∧ ∀𝑗𝐶 (𝑘𝑗𝑗𝑘)) → (𝑥(1st𝑅)𝑦) ∈ 𝐶)
8079an32s 642 . . . . . . . . . . . . 13 (((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑗𝐶 (𝑘𝑗𝑗𝑘)) ∧ (𝑖𝐶𝑦𝑖)) → (𝑥(1st𝑅)𝑦) ∈ 𝐶)
8180rexlimdvaa 3179 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑗𝐶 (𝑘𝑗𝑗𝑘)) → (∃𝑖𝐶 𝑦𝑖 → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
8233, 81syl5bi 233 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑗𝐶 (𝑘𝑗𝑗𝑘)) → (𝑦 𝐶 → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
8382ralrimiv 3112 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑗𝐶 (𝑘𝑗𝑗𝑘)) → ∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶)
8432, 83syldan 585 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖)) → ∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶)
8584anasss 458 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) → ∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶)
86853adantr1 1210 . . . . . . 7 (((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) → ∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶)
8786an32s 642 . . . . . 6 (((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) ∧ (𝑘𝐶𝑥𝑘)) → ∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶)
88 eqid 2765 . . . . . . . . . . . . . . . . . 18 (2nd𝑅) = (2nd𝑅)
892, 88, 3idllmulcl 34262 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ RingOps ∧ 𝑘 ∈ (Idl‘𝑅)) ∧ (𝑥𝑘𝑧 ∈ ran (1st𝑅))) → (𝑧(2nd𝑅)𝑥) ∈ 𝑘)
9089exp43 427 . . . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . . 14 ((((𝑅 ∈ RingOps ∧ 𝑥𝑘) ∧ 𝑘 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑧(2nd𝑅)𝑥) ∈ 𝑘)
9364, 92sylanl2 671 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑥𝑘) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑘𝐶)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑧(2nd𝑅)𝑥) ∈ 𝑘)
94 simplrr 796 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑥𝑘) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑘𝐶)) ∧ 𝑧 ∈ ran (1st𝑅)) → 𝑘𝐶)
95 elunii 4601 . . . . . . . . . . . . 13 (((𝑧(2nd𝑅)𝑥) ∈ 𝑘𝑘𝐶) → (𝑧(2nd𝑅)𝑥) ∈ 𝐶)
9693, 94, 95syl2anc 579 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑥𝑘) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑘𝐶)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑧(2nd𝑅)𝑥) ∈ 𝐶)
972, 88, 3idlrmulcl 34263 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ RingOps ∧ 𝑘 ∈ (Idl‘𝑅)) ∧ (𝑥𝑘𝑧 ∈ ran (1st𝑅))) → (𝑥(2nd𝑅)𝑧) ∈ 𝑘)
9897exp43 427 . . . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . . 14 ((((𝑅 ∈ RingOps ∧ 𝑥𝑘) ∧ 𝑘 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥(2nd𝑅)𝑧) ∈ 𝑘)
10164, 100sylanl2 671 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝑥𝑘) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑘𝐶)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥(2nd𝑅)𝑧) ∈ 𝑘)
102 elunii 4601 . . . . . . . . . . . . 13 (((𝑥(2nd𝑅)𝑧) ∈ 𝑘𝑘𝐶) → (𝑥(2nd𝑅)𝑧) ∈ 𝐶)
103101, 94, 102syl2anc 579 . . . . . . . . . . . 12 ((((𝑅 ∈ RingOps ∧ 𝑥𝑘) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑘𝐶)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥(2nd𝑅)𝑧) ∈ 𝐶)
10496, 103jca 507 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑥𝑘) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑘𝐶)) ∧ 𝑧 ∈ ran (1st𝑅)) → ((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
105104ralrimiva 3113 . . . . . . . . . 10 (((𝑅 ∈ RingOps ∧ 𝑥𝑘) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑘𝐶)) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
106105an42s 651 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑘𝐶𝑥𝑘)) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
107106an32s 642 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
1081073ad2antr2 1240 . . . . . . 7 (((𝑅 ∈ RingOps ∧ (𝑘𝐶𝑥𝑘)) ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
109108an32s 642 . . . . . 6 (((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) ∧ (𝑘𝐶𝑥𝑘)) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
11087, 109jca 507 . . . . 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 3179 . . . 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, 111syl5bi 233 . . 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 3112 . 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 34256 . . 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 472 . 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 1442 1 ((𝑅 ∈ RingOps ∧ (𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅) ∧ ∀𝑖𝐶𝑗𝐶 (𝑖𝑗𝑗𝑖))) → 𝐶 ∈ (Idl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 873  w3a 1107  wcel 2155  wne 2937  wral 3055  wrex 3056  wss 3734  c0 4081   cuni 4596  ran crn 5280  cfv 6070  (class class class)co 6846  1st c1st 7368  2nd c2nd 7369  GIdcgi 27822  RingOpscrngo 34136  Idlcidl 34249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-iota 6033  df-fun 6072  df-fv 6078  df-ov 6849  df-idl 34252
This theorem is referenced by: (None)
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