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Theorem intidl 38016
Description: The intersection of a nonempty collection of ideals is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
Assertion
Ref Expression
intidl ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 ∈ (Idl‘𝑅))

Proof of Theorem intidl
Dummy variables 𝑖 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 intssuni 4975 . . . 4 (𝐶 ≠ ∅ → 𝐶 𝐶)
213ad2ant2 1133 . . 3 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 𝐶)
3 ssel2 3990 . . . . . . . 8 ((𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶) → 𝑖 ∈ (Idl‘𝑅))
4 eqid 2735 . . . . . . . . 9 (1st𝑅) = (1st𝑅)
5 eqid 2735 . . . . . . . . 9 ran (1st𝑅) = ran (1st𝑅)
64, 5idlss 38003 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑖 ⊆ ran (1st𝑅))
73, 6sylan2 593 . . . . . . 7 ((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶)) → 𝑖 ⊆ ran (1st𝑅))
87anassrs 467 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑖𝐶) → 𝑖 ⊆ ran (1st𝑅))
98ralrimiva 3144 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖𝐶 𝑖 ⊆ ran (1st𝑅))
1093adant2 1130 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖𝐶 𝑖 ⊆ ran (1st𝑅))
11 unissb 4944 . . . 4 ( 𝐶 ⊆ ran (1st𝑅) ↔ ∀𝑖𝐶 𝑖 ⊆ ran (1st𝑅))
1210, 11sylibr 234 . . 3 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 ⊆ ran (1st𝑅))
132, 12sstrd 4006 . 2 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 ⊆ ran (1st𝑅))
14 eqid 2735 . . . . . . . 8 (GId‘(1st𝑅)) = (GId‘(1st𝑅))
154, 14idl0cl 38005 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → (GId‘(1st𝑅)) ∈ 𝑖)
163, 15sylan2 593 . . . . . 6 ((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶)) → (GId‘(1st𝑅)) ∈ 𝑖)
1716anassrs 467 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑖𝐶) → (GId‘(1st𝑅)) ∈ 𝑖)
1817ralrimiva 3144 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖)
19 fvex 6920 . . . . 5 (GId‘(1st𝑅)) ∈ V
2019elint2 4958 . . . 4 ((GId‘(1st𝑅)) ∈ 𝐶 ↔ ∀𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖)
2118, 20sylibr 234 . . 3 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (GId‘(1st𝑅)) ∈ 𝐶)
22213adant2 1130 . 2 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → (GId‘(1st𝑅)) ∈ 𝐶)
23 vex 3482 . . . . . 6 𝑥 ∈ V
2423elint2 4958 . . . . 5 (𝑥 𝐶 ↔ ∀𝑖𝐶 𝑥𝑖)
25 vex 3482 . . . . . . . . . 10 𝑦 ∈ V
2625elint2 4958 . . . . . . . . 9 (𝑦 𝐶 ↔ ∀𝑖𝐶 𝑦𝑖)
27 r19.26 3109 . . . . . . . . . . 11 (∀𝑖𝐶 (𝑥𝑖𝑦𝑖) ↔ (∀𝑖𝐶 𝑥𝑖 ∧ ∀𝑖𝐶 𝑦𝑖))
284idladdcl 38006 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑥𝑖𝑦𝑖)) → (𝑥(1st𝑅)𝑦) ∈ 𝑖)
2928ex 412 . . . . . . . . . . . . . . 15 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → ((𝑥𝑖𝑦𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝑖))
303, 29sylan2 593 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶)) → ((𝑥𝑖𝑦𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝑖))
3130anassrs 467 . . . . . . . . . . . . 13 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑖𝐶) → ((𝑥𝑖𝑦𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝑖))
3231ralimdva 3165 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (∀𝑖𝐶 (𝑥𝑖𝑦𝑖) → ∀𝑖𝐶 (𝑥(1st𝑅)𝑦) ∈ 𝑖))
33 ovex 7464 . . . . . . . . . . . . 13 (𝑥(1st𝑅)𝑦) ∈ V
3433elint2 4958 . . . . . . . . . . . 12 ((𝑥(1st𝑅)𝑦) ∈ 𝐶 ↔ ∀𝑖𝐶 (𝑥(1st𝑅)𝑦) ∈ 𝑖)
3532, 34imbitrrdi 252 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (∀𝑖𝐶 (𝑥𝑖𝑦𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
3627, 35biimtrrid 243 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ((∀𝑖𝐶 𝑥𝑖 ∧ ∀𝑖𝐶 𝑦𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
3736expdimp 452 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → (∀𝑖𝐶 𝑦𝑖 → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
3826, 37biimtrid 242 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → (𝑦 𝐶 → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
3938ralrimiv 3143 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → ∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶)
40 eqid 2735 . . . . . . . . . . . . . . . . . . . 20 (2nd𝑅) = (2nd𝑅)
414, 40, 5idllmulcl 38007 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑥𝑖𝑧 ∈ ran (1st𝑅))) → (𝑧(2nd𝑅)𝑥) ∈ 𝑖)
4241anass1rs 655 . . . . . . . . . . . . . . . . . 18 ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑥𝑖) → (𝑧(2nd𝑅)𝑥) ∈ 𝑖)
4342ex 412 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥𝑖 → (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
4443an32s 652 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑥𝑖 → (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
453, 44sylan2 593 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st𝑅)) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶)) → (𝑥𝑖 → (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
4645an4s 660 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑧 ∈ ran (1st𝑅) ∧ 𝑖𝐶)) → (𝑥𝑖 → (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
4746anassrs 467 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑖𝐶) → (𝑥𝑖 → (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
4847ralimdva 3165 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (∀𝑖𝐶 𝑥𝑖 → ∀𝑖𝐶 (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
4948imp 406 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → ∀𝑖𝐶 (𝑧(2nd𝑅)𝑥) ∈ 𝑖)
50 ovex 7464 . . . . . . . . . . . 12 (𝑧(2nd𝑅)𝑥) ∈ V
5150elint2 4958 . . . . . . . . . . 11 ((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ↔ ∀𝑖𝐶 (𝑧(2nd𝑅)𝑥) ∈ 𝑖)
5249, 51sylibr 234 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → (𝑧(2nd𝑅)𝑥) ∈ 𝐶)
534, 40, 5idlrmulcl 38008 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑥𝑖𝑧 ∈ ran (1st𝑅))) → (𝑥(2nd𝑅)𝑧) ∈ 𝑖)
5453anass1rs 655 . . . . . . . . . . . . . . . . . 18 ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑥𝑖) → (𝑥(2nd𝑅)𝑧) ∈ 𝑖)
5554ex 412 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥𝑖 → (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
5655an32s 652 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑥𝑖 → (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
573, 56sylan2 593 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st𝑅)) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶)) → (𝑥𝑖 → (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
5857an4s 660 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑧 ∈ ran (1st𝑅) ∧ 𝑖𝐶)) → (𝑥𝑖 → (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
5958anassrs 467 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑖𝐶) → (𝑥𝑖 → (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
6059ralimdva 3165 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (∀𝑖𝐶 𝑥𝑖 → ∀𝑖𝐶 (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
6160imp 406 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → ∀𝑖𝐶 (𝑥(2nd𝑅)𝑧) ∈ 𝑖)
62 ovex 7464 . . . . . . . . . . . 12 (𝑥(2nd𝑅)𝑧) ∈ V
6362elint2 4958 . . . . . . . . . . 11 ((𝑥(2nd𝑅)𝑧) ∈ 𝐶 ↔ ∀𝑖𝐶 (𝑥(2nd𝑅)𝑧) ∈ 𝑖)
6461, 63sylibr 234 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → (𝑥(2nd𝑅)𝑧) ∈ 𝐶)
6552, 64jca 511 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → ((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
6665an32s 652 . . . . . . . 8 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) ∧ 𝑧 ∈ ran (1st𝑅)) → ((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
6766ralrimiva 3144 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
6839, 67jca 511 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → (∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶 ∧ ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶)))
6968ex 412 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (∀𝑖𝐶 𝑥𝑖 → (∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶 ∧ ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))))
7024, 69biimtrid 242 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (𝑥 𝐶 → (∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶 ∧ ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))))
7170ralrimiv 3143 . . 3 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑥 𝐶(∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶 ∧ ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶)))
72713adant2 1130 . 2 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑥 𝐶(∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶 ∧ ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶)))
734, 40, 5, 14isidl 38001 . . 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)))))
74733ad2ant1 1132 . 2 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → ( 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)))))
7513, 22, 72, 74mpbir3and 1341 1 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 ∈ (Idl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086  wcel 2106  wne 2938  wral 3059  wss 3963  c0 4339   cuni 4912   cint 4951  ran crn 5690  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  GIdcgi 30519  RingOpscrngo 37881  Idlcidl 37994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-idl 37997
This theorem is referenced by:  inidl  38017  igenidl  38050
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