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Theorem intidl 32998
 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 4434 . . . 4 (𝐶 ≠ ∅ → 𝐶 𝐶)
213ad2ant2 1076 . . 3 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 𝐶)
3 ssel2 3563 . . . . . . . 8 ((𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶) → 𝑖 ∈ (Idl‘𝑅))
4 eqid 2610 . . . . . . . . 9 (1st𝑅) = (1st𝑅)
5 eqid 2610 . . . . . . . . 9 ran (1st𝑅) = ran (1st𝑅)
64, 5idlss 32985 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → 𝑖 ⊆ ran (1st𝑅))
73, 6sylan2 490 . . . . . . 7 ((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶)) → 𝑖 ⊆ ran (1st𝑅))
87anassrs 678 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑖𝐶) → 𝑖 ⊆ ran (1st𝑅))
98ralrimiva 2949 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖𝐶 𝑖 ⊆ ran (1st𝑅))
1093adant2 1073 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖𝐶 𝑖 ⊆ ran (1st𝑅))
11 unissb 4405 . . . 4 ( 𝐶 ⊆ ran (1st𝑅) ↔ ∀𝑖𝐶 𝑖 ⊆ ran (1st𝑅))
1210, 11sylibr 223 . . 3 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 ⊆ ran (1st𝑅))
132, 12sstrd 3578 . 2 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 ⊆ ran (1st𝑅))
14 eqid 2610 . . . . . . . 8 (GId‘(1st𝑅)) = (GId‘(1st𝑅))
154, 14idl0cl 32987 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → (GId‘(1st𝑅)) ∈ 𝑖)
163, 15sylan2 490 . . . . . 6 ((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶)) → (GId‘(1st𝑅)) ∈ 𝑖)
1716anassrs 678 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑖𝐶) → (GId‘(1st𝑅)) ∈ 𝑖)
1817ralrimiva 2949 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → ∀𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖)
19 fvex 6113 . . . . 5 (GId‘(1st𝑅)) ∈ V
2019elint2 4417 . . . 4 ((GId‘(1st𝑅)) ∈ 𝐶 ↔ ∀𝑖𝐶 (GId‘(1st𝑅)) ∈ 𝑖)
2118, 20sylibr 223 . . 3 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (GId‘(1st𝑅)) ∈ 𝐶)
22213adant2 1073 . 2 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → (GId‘(1st𝑅)) ∈ 𝐶)
23 vex 3176 . . . . . 6 𝑥 ∈ V
2423elint2 4417 . . . . 5 (𝑥 𝐶 ↔ ∀𝑖𝐶 𝑥𝑖)
25 vex 3176 . . . . . . . . . 10 𝑦 ∈ V
2625elint2 4417 . . . . . . . . 9 (𝑦 𝐶 ↔ ∀𝑖𝐶 𝑦𝑖)
27 r19.26 3046 . . . . . . . . . . 11 (∀𝑖𝐶 (𝑥𝑖𝑦𝑖) ↔ (∀𝑖𝐶 𝑥𝑖 ∧ ∀𝑖𝐶 𝑦𝑖))
284idladdcl 32988 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑥𝑖𝑦𝑖)) → (𝑥(1st𝑅)𝑦) ∈ 𝑖)
2928ex 449 . . . . . . . . . . . . . . 15 ((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) → ((𝑥𝑖𝑦𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝑖))
303, 29sylan2 490 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶)) → ((𝑥𝑖𝑦𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝑖))
3130anassrs 678 . . . . . . . . . . . . 13 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑖𝐶) → ((𝑥𝑖𝑦𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝑖))
3231ralimdva 2945 . . . . . . . . . . . 12 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (∀𝑖𝐶 (𝑥𝑖𝑦𝑖) → ∀𝑖𝐶 (𝑥(1st𝑅)𝑦) ∈ 𝑖))
33 ovex 6577 . . . . . . . . . . . . 13 (𝑥(1st𝑅)𝑦) ∈ V
3433elint2 4417 . . . . . . . . . . . 12 ((𝑥(1st𝑅)𝑦) ∈ 𝐶 ↔ ∀𝑖𝐶 (𝑥(1st𝑅)𝑦) ∈ 𝑖)
3532, 34syl6ibr 241 . . . . . . . . . . 11 ((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) → (∀𝑖𝐶 (𝑥𝑖𝑦𝑖) → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
3627, 35syl5bir 232 . . . . . . . . . 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, 37syl5bi 231 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → (𝑦 𝐶 → (𝑥(1st𝑅)𝑦) ∈ 𝐶))
3938ralrimiv 2948 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → ∀𝑦 𝐶(𝑥(1st𝑅)𝑦) ∈ 𝐶)
40 eqid 2610 . . . . . . . . . . . . . . . . . . . 20 (2nd𝑅) = (2nd𝑅)
414, 40, 5idllmulcl 32989 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑥𝑖𝑧 ∈ ran (1st𝑅))) → (𝑧(2nd𝑅)𝑥) ∈ 𝑖)
4241anass1rs 845 . . . . . . . . . . . . . . . . . 18 ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑥𝑖) → (𝑧(2nd𝑅)𝑥) ∈ 𝑖)
4342ex 449 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥𝑖 → (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
4443an32s 842 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑥𝑖 → (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
453, 44sylan2 490 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st𝑅)) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶)) → (𝑥𝑖 → (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
4645an4s 865 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑧 ∈ ran (1st𝑅) ∧ 𝑖𝐶)) → (𝑥𝑖 → (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
4746anassrs 678 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑖𝐶) → (𝑥𝑖 → (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
4847ralimdva 2945 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (∀𝑖𝐶 𝑥𝑖 → ∀𝑖𝐶 (𝑧(2nd𝑅)𝑥) ∈ 𝑖))
4948imp 444 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → ∀𝑖𝐶 (𝑧(2nd𝑅)𝑥) ∈ 𝑖)
50 ovex 6577 . . . . . . . . . . . 12 (𝑧(2nd𝑅)𝑥) ∈ V
5150elint2 4417 . . . . . . . . . . 11 ((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ↔ ∀𝑖𝐶 (𝑧(2nd𝑅)𝑥) ∈ 𝑖)
5249, 51sylibr 223 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → (𝑧(2nd𝑅)𝑥) ∈ 𝐶)
534, 40, 5idlrmulcl 32990 . . . . . . . . . . . . . . . . . . 19 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ (𝑥𝑖𝑧 ∈ ran (1st𝑅))) → (𝑥(2nd𝑅)𝑧) ∈ 𝑖)
5453anass1rs 845 . . . . . . . . . . . . . . . . . 18 ((((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑥𝑖) → (𝑥(2nd𝑅)𝑧) ∈ 𝑖)
5554ex 449 . . . . . . . . . . . . . . . . 17 (((𝑅 ∈ RingOps ∧ 𝑖 ∈ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (𝑥𝑖 → (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
5655an32s 842 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑖 ∈ (Idl‘𝑅)) → (𝑥𝑖 → (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
573, 56sylan2 490 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑧 ∈ ran (1st𝑅)) ∧ (𝐶 ⊆ (Idl‘𝑅) ∧ 𝑖𝐶)) → (𝑥𝑖 → (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
5857an4s 865 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ (𝑧 ∈ ran (1st𝑅) ∧ 𝑖𝐶)) → (𝑥𝑖 → (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
5958anassrs 678 . . . . . . . . . . . . 13 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ 𝑖𝐶) → (𝑥𝑖 → (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
6059ralimdva 2945 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) → (∀𝑖𝐶 𝑥𝑖 → ∀𝑖𝐶 (𝑥(2nd𝑅)𝑧) ∈ 𝑖))
6160imp 444 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → ∀𝑖𝐶 (𝑥(2nd𝑅)𝑧) ∈ 𝑖)
62 ovex 6577 . . . . . . . . . . . 12 (𝑥(2nd𝑅)𝑧) ∈ V
6362elint2 4417 . . . . . . . . . . 11 ((𝑥(2nd𝑅)𝑧) ∈ 𝐶 ↔ ∀𝑖𝐶 (𝑥(2nd𝑅)𝑧) ∈ 𝑖)
6461, 63sylibr 223 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → (𝑥(2nd𝑅)𝑧) ∈ 𝐶)
6552, 64jca 553 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ 𝑧 ∈ ran (1st𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → ((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
6665an32s 842 . . . . . . . 8 ((((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) ∧ 𝑧 ∈ ran (1st𝑅)) → ((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
6766ralrimiva 2949 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐶 ⊆ (Idl‘𝑅)) ∧ ∀𝑖𝐶 𝑥𝑖) → ∀𝑧 ∈ ran (1st𝑅)((𝑧(2nd𝑅)𝑥) ∈ 𝐶 ∧ (𝑥(2nd𝑅)𝑧) ∈ 𝐶))
6839, 67jca 553 . . . . . 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 449 . . . . 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 231 . . . 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 2948 . . 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 1073 . 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 32983 . . 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 1075 . 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 1238 1 ((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 ∈ (Idl‘𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372  ∩ cint 4410  ran crn 5039  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  GIdcgi 26728  RingOpscrngo 32863  Idlcidl 32976 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-idl 32979 This theorem is referenced by:  inidl  32999  igenidl  33032
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