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Theorem unichnidl 36493
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 3933 . . . . 5 (𝐶 ⊆ (Idl‘𝑅) ↔ ∀𝑖𝐶 𝑖 ∈ (Idl‘𝑅))
2 eqid 2737 . . . . . . . . 9 (1st𝑅) = (1st𝑅)
3 eqid 2737 . . . . . . . . 9 ran (1st𝑅) = ran (1st𝑅)
42, 3idlss 36478 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑖 ⊆ ran (1st𝑅))
54ex 414 . . . . . . 7 (𝑅 ∈ RingOps → (𝑖 ∈ (Idl‘𝑅) → 𝑖 ⊆ ran (1st𝑅)))
65ralimdv 3167 . . . . . 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 4901 . . . 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 2737 . . . . . . . . . . 11 (GId‘(1st𝑅)) = (GId‘(1st𝑅))
132, 12idl0cl 36480 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → (GId‘(1st𝑅)) ∈ 𝑖)
1413ex 414 . . . . . . . . 9 (𝑅 ∈ RingOps → (𝑖 ∈ (Idl‘𝑅) → (GId‘(1st𝑅)) ∈ 𝑖))
1514ralimdv 3167 . . . . . . . 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 4453 . . . . . 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 4870 . . . 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 4870 . . . 4 (𝑥 𝐶 ↔ ∃𝑘𝐶 𝑥𝑘)
25 sseq1 3970 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑘 → (𝑖𝑗𝑘𝑗))
26 sseq2 3971 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑘 → (𝑗𝑖𝑗𝑘))
2725, 26orbi12d 918 . . . . . . . . . . . . . . 15 (𝑖 = 𝑘 → ((𝑖𝑗𝑗𝑖) ↔ (𝑘𝑗𝑗𝑘)))
2827ralbidv 3175 . . . . . . . . . . . . . 14 (𝑖 = 𝑘 → (∀𝑗𝐶 (𝑖𝑗𝑗𝑖) ↔ ∀𝑗𝐶 (𝑘𝑗𝑗𝑘)))
2928rspcv 3578 . . . . . . . . . . . . 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 4870 . . . . . . . . . . . 12 (𝑦 𝐶 ↔ ∃𝑖𝐶 𝑦𝑖)
34 sseq2 3971 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑖 → (𝑘𝑗𝑘𝑖))
35 sseq1 3970 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑖 → (𝑗𝑘𝑖𝑘))
3634, 35orbi12d 918 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑖 → ((𝑘𝑗𝑗𝑘) ↔ (𝑘𝑖𝑖𝑘)))
3736rspcv 3578 . . . . . . . . . . . . . . . . 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 3940 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘𝑖𝑥𝑘) → 𝑥𝑖)
4140ancoms 460 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝑘𝑘𝑖) → 𝑥𝑖)
4241adantll 713 . . . . . . . . . . . . . . . . . . . . 21 (((𝑘𝐶𝑥𝑘) ∧ 𝑘𝑖) → 𝑥𝑖)
43 ssel2 3940 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶) → 𝑖 ∈ (Idl‘𝑅))
442idladdcl 36481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4871 . . . . . . . . . . . . . . . . . . . . . 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 3940 . . . . . . . . . . . . . . . . . . . 20 ((𝑖𝑘𝑦𝑖) → 𝑦𝑘)
6261ancoms 460 . . . . . . . . . . . . . . . . . . 19 ((𝑦𝑖𝑖𝑘) → 𝑦𝑘)
6362adantll 713 . . . . . . . . . . . . . . . . . 18 (((𝑖𝐶𝑦𝑖) ∧ 𝑖𝑘) → 𝑦𝑘)
64 ssel2 3940 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 ⊆ (Idl‘𝑅) ∧ 𝑘𝐶) → 𝑘 ∈ (Idl‘𝑅))
652idladdcl 36481 . . . . . . . . . . . . . . . . . . . . . . . . 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 4871 . . . . . . . . . . . . . . . . . . 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 3154 . . . . . . . . . . . 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 3143 . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . . 18 (2nd𝑅) = (2nd𝑅)
892, 88, 3idllmulcl 36482 . . . . . . . . . . . . . . . . 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 4871 . . . . . . . . . . . . 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 36483 . . . . . . . . . . . . . . . . 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 4871 . . . . . . . . . . . . 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 3144 . . . . . . . . . 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 3154 . . . 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 3143 . 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 36476 . . 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 2944  wral 3065  wrex 3074  wss 3911  c0 4283   cuni 4866  ran crn 5635  cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  GIdcgi 29435  RingOpscrngo 36356  Idlcidl 36469
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 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-idl 36472
This theorem is referenced by: (None)
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