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Theorem intidl 36491
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 4932 . . . 4 (𝐶 ≠ ∅ → 𝐶 𝐶)
213ad2ant2 1135 . . 3 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 𝐶)
3 ssel2 3940 . . . . . . . 8 ((𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶) → 𝑖 ∈ (Idl‘𝑅))
4 eqid 2737 . . . . . . . . 9 (1st𝑅) = (1st𝑅)
5 eqid 2737 . . . . . . . . 9 ran (1st𝑅) = ran (1st𝑅)
64, 5idlss 36478 . . . . . . . 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 3144 . . . . 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 4901 . . . 4 ( 𝐶 ⊆ ran (1st𝑅) ↔ ∀𝑖𝐶 𝑖 ⊆ ran (1st𝑅))
1210, 11sylibr 233 . . 3 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 ⊆ ran (1st𝑅))
132, 12sstrd 3955 . 2 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 ⊆ ran (1st𝑅))
14 eqid 2737 . . . . . . . 8 (GId‘(1st𝑅)) = (GId‘(1st𝑅))
154, 14idl0cl 36480 . . . . . . 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 3144 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖)
19 fvex 6856 . . . . 5 (GId‘(1st𝑅)) ∈ V
2019elint2 4915 . . . 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 3450 . . . . . 6 𝑥 ∈ V
2423elint2 4915 . . . . 5 (𝑥 𝐶 ↔ ∀𝑖𝐶 𝑥𝑖)
25 vex 3450 . . . . . . . . . 10 𝑦 ∈ V
2625elint2 4915 . . . . . . . . 9 (𝑦 𝐶 ↔ ∀𝑖𝐶 𝑦𝑖)
27 r19.26 3115 . . . . . . . . . . 11 (∀𝑖𝐶 (𝑥𝑖𝑦𝑖) ↔ (∀𝑖𝐶 𝑥𝑖 ∧ ∀𝑖𝐶 𝑦𝑖))
284idladdcl 36481 . . . . . . . . . . . . . . . 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 3165 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (∀𝑖𝐶 (𝑥𝑖𝑦𝑖) → ∀𝑖𝐶 (𝑥(1st𝑅)𝑦) ∈ 𝑖))
33 ovex 7391 . . . . . . . . . . . . 13 (𝑥(1st𝑅)𝑦) ∈ V
3433elint2 4915 . . . . . . . . . . . 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 3143 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → ∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶)
40 eqid 2737 . . . . . . . . . . . . . . . . . . . 20 (2nd𝑅) = (2nd𝑅)
414, 40, 5idllmulcl 36482 . . . . . . . . . . . . . . . . . . 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 3165 . . . . . . . . . . . 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 7391 . . . . . . . . . . . 12 (𝑧(2nd𝑅)𝑥) ∈ V
5150elint2 4915 . . . . . . . . . . 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 36483 . . . . . . . . . . . . . . . . . . 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 3165 . . . . . . . . . . . 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 7391 . . . . . . . . . . . 12 (𝑥(2nd𝑅)𝑧) ∈ V
6362elint2 4915 . . . . . . . . . . 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 3144 . . . . . . 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 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 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 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)))))
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 2944  wral 3065  wss 3911  c0 4283   cuni 4866   cint 4908  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-int 4909  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:  inidl  36492  igenidl  36525
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