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Theorem intidl 36897
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 1135 . . 3 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 𝐶)
3 ssel2 3978 . . . . . . . 8 ((𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶) → 𝑖 ∈ (Idl‘𝑅))
4 eqid 2733 . . . . . . . . 9 (1st𝑅) = (1st𝑅)
5 eqid 2733 . . . . . . . . 9 ran (1st𝑅) = ran (1st𝑅)
64, 5idlss 36884 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑖 ⊆ ran (1st𝑅))
73, 6sylan2 594 . . . . . . 7 ((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶)) → 𝑖 ⊆ ran (1st𝑅))
87anassrs 469 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑖𝐶) → 𝑖 ⊆ ran (1st𝑅))
98ralrimiva 3147 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖𝐶 𝑖 ⊆ ran (1st𝑅))
1093adant2 1132 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖𝐶 𝑖 ⊆ ran (1st𝑅))
11 unissb 4944 . . . 4 ( 𝐶 ⊆ ran (1st𝑅) ↔ ∀𝑖𝐶 𝑖 ⊆ ran (1st𝑅))
1210, 11sylibr 233 . . 3 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 ⊆ ran (1st𝑅))
132, 12sstrd 3993 . 2 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 ⊆ ran (1st𝑅))
14 eqid 2733 . . . . . . . 8 (GId‘(1st𝑅)) = (GId‘(1st𝑅))
154, 14idl0cl 36886 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → (GId‘(1st𝑅)) ∈ 𝑖)
163, 15sylan2 594 . . . . . 6 ((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶)) → (GId‘(1st𝑅)) ∈ 𝑖)
1716anassrs 469 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑖𝐶) → (GId‘(1st𝑅)) ∈ 𝑖)
1817ralrimiva 3147 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖)
19 fvex 6905 . . . . 5 (GId‘(1st𝑅)) ∈ V
2019elint2 4958 . . . 4 ((GId‘(1st𝑅)) ∈ 𝐶 ↔ ∀𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖)
2118, 20sylibr 233 . . 3 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (GId‘(1st𝑅)) ∈ 𝐶)
22213adant2 1132 . 2 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → (GId‘(1st𝑅)) ∈ 𝐶)
23 vex 3479 . . . . . 6 𝑥 ∈ V
2423elint2 4958 . . . . 5 (𝑥 𝐶 ↔ ∀𝑖𝐶 𝑥𝑖)
25 vex 3479 . . . . . . . . . 10 𝑦 ∈ V
2625elint2 4958 . . . . . . . . 9 (𝑦 𝐶 ↔ ∀𝑖𝐶 𝑦𝑖)
27 r19.26 3112 . . . . . . . . . . 11 (∀𝑖𝐶 (𝑥𝑖𝑦𝑖) ↔ (∀𝑖𝐶 𝑥𝑖 ∧ ∀𝑖𝐶 𝑦𝑖))
284idladdcl 36887 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑥𝑖𝑦𝑖)) → (𝑥(1st𝑅)𝑦) ∈ 𝑖)
2928ex 414 . . . . . . . . . . . . . . 15 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → ((𝑥𝑖𝑦𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝑖))
303, 29sylan2 594 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶)) → ((𝑥𝑖𝑦𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝑖))
3130anassrs 469 . . . . . . . . . . . . 13 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑖𝐶) → ((𝑥𝑖𝑦𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝑖))
3231ralimdva 3168 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (∀𝑖𝐶 (𝑥𝑖𝑦𝑖) → ∀𝑖𝐶 (𝑥(1st𝑅)𝑦) ∈ 𝑖))
33 ovex 7442 . . . . . . . . . . . . 13 (𝑥(1st𝑅)𝑦) ∈ V
3433elint2 4958 . . . . . . . . . . . 12 ((𝑥(1st𝑅)𝑦) ∈ 𝐶 ↔ ∀𝑖𝐶 (𝑥(1st𝑅)𝑦) ∈ 𝑖)
3532, 34syl6ibr 252 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (∀𝑖𝐶 (𝑥𝑖𝑦𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
3627, 35biimtrrid 242 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ((∀𝑖𝐶 𝑥𝑖 ∧ ∀𝑖𝐶 𝑦𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
3736expdimp 454 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → (∀𝑖𝐶 𝑦𝑖 → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
3826, 37biimtrid 241 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → (𝑦 𝐶 → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
3938ralrimiv 3146 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → ∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶)
40 eqid 2733 . . . . . . . . . . . . . . . . . . . 20 (2nd𝑅) = (2nd𝑅)
414, 40, 5idllmulcl 36888 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑥𝑖𝑧 ∈ ran (1st𝑅))) → (𝑧(2nd𝑅)𝑥) ∈ 𝑖)
4241anass1rs 654 . . . . . . . . . . . . . . . . . 18 ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑥𝑖) → (𝑧(2nd𝑅)𝑥) ∈ 𝑖)
4342ex 414 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥𝑖 → (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
4443an32s 651 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑥𝑖 → (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
453, 44sylan2 594 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st𝑅)) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶)) → (𝑥𝑖 → (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
4645an4s 659 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑧 ∈ ran (1st𝑅) ∧ 𝑖𝐶)) → (𝑥𝑖 → (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
4746anassrs 469 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑖𝐶) → (𝑥𝑖 → (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
4847ralimdva 3168 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (∀𝑖𝐶 𝑥𝑖 → ∀𝑖𝐶 (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
4948imp 408 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → ∀𝑖𝐶 (𝑧(2nd𝑅)𝑥) ∈ 𝑖)
50 ovex 7442 . . . . . . . . . . . 12 (𝑧(2nd𝑅)𝑥) ∈ V
5150elint2 4958 . . . . . . . . . . 11 ((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ↔ ∀𝑖𝐶 (𝑧(2nd𝑅)𝑥) ∈ 𝑖)
5249, 51sylibr 233 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → (𝑧(2nd𝑅)𝑥) ∈ 𝐶)
534, 40, 5idlrmulcl 36889 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑥𝑖𝑧 ∈ ran (1st𝑅))) → (𝑥(2nd𝑅)𝑧) ∈ 𝑖)
5453anass1rs 654 . . . . . . . . . . . . . . . . . 18 ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑥𝑖) → (𝑥(2nd𝑅)𝑧) ∈ 𝑖)
5554ex 414 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥𝑖 → (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
5655an32s 651 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑥𝑖 → (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
573, 56sylan2 594 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st𝑅)) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶)) → (𝑥𝑖 → (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
5857an4s 659 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑧 ∈ ran (1st𝑅) ∧ 𝑖𝐶)) → (𝑥𝑖 → (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
5958anassrs 469 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑖𝐶) → (𝑥𝑖 → (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
6059ralimdva 3168 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (∀𝑖𝐶 𝑥𝑖 → ∀𝑖𝐶 (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
6160imp 408 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → ∀𝑖𝐶 (𝑥(2nd𝑅)𝑧) ∈ 𝑖)
62 ovex 7442 . . . . . . . . . . . 12 (𝑥(2nd𝑅)𝑧) ∈ V
6362elint2 4958 . . . . . . . . . . 11 ((𝑥(2nd𝑅)𝑧) ∈ 𝐶 ↔ ∀𝑖𝐶 (𝑥(2nd𝑅)𝑧) ∈ 𝑖)
6461, 63sylibr 233 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → (𝑥(2nd𝑅)𝑧) ∈ 𝐶)
6552, 64jca 513 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → ((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
6665an32s 651 . . . . . . . 8 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) ∧ 𝑧 ∈ ran (1st𝑅)) → ((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
6766ralrimiva 3147 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
6839, 67jca 513 . . . . . 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 414 . . . . 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 241 . . . 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 3146 . . 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 1132 . 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 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)))))
74733ad2ant1 1134 . 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 1343 1 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 ∈ (Idl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088  wcel 2107  wne 2941  wral 3062  wss 3949  c0 4323   cuni 4909   cint 4951  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-int 4952  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:  inidl  36898  igenidl  36931
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