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Theorem intidl 33487
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 4469 . . . 4 (𝐶 ≠ ∅ → 𝐶 𝐶)
213ad2ant2 1081 . . 3 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 𝐶)
3 ssel2 3582 . . . . . . . 8 ((𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶) → 𝑖 ∈ (Idl‘𝑅))
4 eqid 2621 . . . . . . . . 9 (1st𝑅) = (1st𝑅)
5 eqid 2621 . . . . . . . . 9 ran (1st𝑅) = ran (1st𝑅)
64, 5idlss 33474 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑖 ⊆ ran (1st𝑅))
73, 6sylan2 491 . . . . . . 7 ((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶)) → 𝑖 ⊆ ran (1st𝑅))
87anassrs 679 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑖𝐶) → 𝑖 ⊆ ran (1st𝑅))
98ralrimiva 2961 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖𝐶 𝑖 ⊆ ran (1st𝑅))
1093adant2 1078 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖𝐶 𝑖 ⊆ ran (1st𝑅))
11 unissb 4440 . . . 4 ( 𝐶 ⊆ ran (1st𝑅) ↔ ∀𝑖𝐶 𝑖 ⊆ ran (1st𝑅))
1210, 11sylibr 224 . . 3 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 ⊆ ran (1st𝑅))
132, 12sstrd 3597 . 2 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 ⊆ ran (1st𝑅))
14 eqid 2621 . . . . . . . 8 (GId‘(1st𝑅)) = (GId‘(1st𝑅))
154, 14idl0cl 33476 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → (GId‘(1st𝑅)) ∈ 𝑖)
163, 15sylan2 491 . . . . . 6 ((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶)) → (GId‘(1st𝑅)) ∈ 𝑖)
1716anassrs 679 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑖𝐶) → (GId‘(1st𝑅)) ∈ 𝑖)
1817ralrimiva 2961 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖)
19 fvex 6163 . . . . 5 (GId‘(1st𝑅)) ∈ V
2019elint2 4452 . . . 4 ((GId‘(1st𝑅)) ∈ 𝐶 ↔ ∀𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖)
2118, 20sylibr 224 . . 3 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (GId‘(1st𝑅)) ∈ 𝐶)
22213adant2 1078 . 2 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → (GId‘(1st𝑅)) ∈ 𝐶)
23 vex 3192 . . . . . 6 𝑥 ∈ V
2423elint2 4452 . . . . 5 (𝑥 𝐶 ↔ ∀𝑖𝐶 𝑥𝑖)
25 vex 3192 . . . . . . . . . 10 𝑦 ∈ V
2625elint2 4452 . . . . . . . . 9 (𝑦 𝐶 ↔ ∀𝑖𝐶 𝑦𝑖)
27 r19.26 3058 . . . . . . . . . . 11 (∀𝑖𝐶 (𝑥𝑖𝑦𝑖) ↔ (∀𝑖𝐶 𝑥𝑖 ∧ ∀𝑖𝐶 𝑦𝑖))
284idladdcl 33477 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑥𝑖𝑦𝑖)) → (𝑥(1st𝑅)𝑦) ∈ 𝑖)
2928ex 450 . . . . . . . . . . . . . . 15 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → ((𝑥𝑖𝑦𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝑖))
303, 29sylan2 491 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶)) → ((𝑥𝑖𝑦𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝑖))
3130anassrs 679 . . . . . . . . . . . . 13 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑖𝐶) → ((𝑥𝑖𝑦𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝑖))
3231ralimdva 2957 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (∀𝑖𝐶 (𝑥𝑖𝑦𝑖) → ∀𝑖𝐶 (𝑥(1st𝑅)𝑦) ∈ 𝑖))
33 ovex 6638 . . . . . . . . . . . . 13 (𝑥(1st𝑅)𝑦) ∈ V
3433elint2 4452 . . . . . . . . . . . 12 ((𝑥(1st𝑅)𝑦) ∈ 𝐶 ↔ ∀𝑖𝐶 (𝑥(1st𝑅)𝑦) ∈ 𝑖)
3532, 34syl6ibr 242 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (∀𝑖𝐶 (𝑥𝑖𝑦𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
3627, 35syl5bir 233 . . . . . . . . . 10 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ((∀𝑖𝐶 𝑥𝑖 ∧ ∀𝑖𝐶 𝑦𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
3736expdimp 453 . . . . . . . . 9 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → (∀𝑖𝐶 𝑦𝑖 → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
3826, 37syl5bi 232 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → (𝑦 𝐶 → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
3938ralrimiv 2960 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → ∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶)
40 eqid 2621 . . . . . . . . . . . . . . . . . . . 20 (2nd𝑅) = (2nd𝑅)
414, 40, 5idllmulcl 33478 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑥𝑖𝑧 ∈ ran (1st𝑅))) → (𝑧(2nd𝑅)𝑥) ∈ 𝑖)
4241anass1rs 848 . . . . . . . . . . . . . . . . . 18 ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑥𝑖) → (𝑧(2nd𝑅)𝑥) ∈ 𝑖)
4342ex 450 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥𝑖 → (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
4443an32s 845 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑥𝑖 → (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
453, 44sylan2 491 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st𝑅)) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶)) → (𝑥𝑖 → (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
4645an4s 868 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑧 ∈ ran (1st𝑅) ∧ 𝑖𝐶)) → (𝑥𝑖 → (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
4746anassrs 679 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑖𝐶) → (𝑥𝑖 → (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
4847ralimdva 2957 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (∀𝑖𝐶 𝑥𝑖 → ∀𝑖𝐶 (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
4948imp 445 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → ∀𝑖𝐶 (𝑧(2nd𝑅)𝑥) ∈ 𝑖)
50 ovex 6638 . . . . . . . . . . . 12 (𝑧(2nd𝑅)𝑥) ∈ V
5150elint2 4452 . . . . . . . . . . 11 ((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ↔ ∀𝑖𝐶 (𝑧(2nd𝑅)𝑥) ∈ 𝑖)
5249, 51sylibr 224 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → (𝑧(2nd𝑅)𝑥) ∈ 𝐶)
534, 40, 5idlrmulcl 33479 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑥𝑖𝑧 ∈ ran (1st𝑅))) → (𝑥(2nd𝑅)𝑧) ∈ 𝑖)
5453anass1rs 848 . . . . . . . . . . . . . . . . . 18 ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑥𝑖) → (𝑥(2nd𝑅)𝑧) ∈ 𝑖)
5554ex 450 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥𝑖 → (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
5655an32s 845 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑥𝑖 → (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
573, 56sylan2 491 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st𝑅)) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶)) → (𝑥𝑖 → (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
5857an4s 868 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑧 ∈ ran (1st𝑅) ∧ 𝑖𝐶)) → (𝑥𝑖 → (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
5958anassrs 679 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑖𝐶) → (𝑥𝑖 → (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
6059ralimdva 2957 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (∀𝑖𝐶 𝑥𝑖 → ∀𝑖𝐶 (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
6160imp 445 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → ∀𝑖𝐶 (𝑥(2nd𝑅)𝑧) ∈ 𝑖)
62 ovex 6638 . . . . . . . . . . . 12 (𝑥(2nd𝑅)𝑧) ∈ V
6362elint2 4452 . . . . . . . . . . 11 ((𝑥(2nd𝑅)𝑧) ∈ 𝐶 ↔ ∀𝑖𝐶 (𝑥(2nd𝑅)𝑧) ∈ 𝑖)
6461, 63sylibr 224 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → (𝑥(2nd𝑅)𝑧) ∈ 𝐶)
6552, 64jca 554 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → ((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
6665an32s 845 . . . . . . . 8 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) ∧ 𝑧 ∈ ran (1st𝑅)) → ((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
6766ralrimiva 2961 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
6839, 67jca 554 . . . . . 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 450 . . . . 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, 69syl5bi 232 . . . 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 2960 . . 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 1078 . 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 33472 . . 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 1080 . 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 1243 1 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 ∈ (Idl‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wcel 1987  wne 2790  wral 2907  wss 3559  c0 3896   cuni 4407   cint 4445  ran crn 5080  cfv 5852  (class class class)co 6610  1st c1st 7118  2nd c2nd 7119  GIdcgi 27211  RingOpscrngo 33352  Idlcidl 33465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5815  df-fun 5854  df-fv 5860  df-ov 6613  df-idl 33468
This theorem is referenced by:  inidl  33488  igenidl  33521
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