Theorem igenval2 36224

Description: The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

Ref	Expression
igenval2.1	⊢ 𝐺 = (1st ‘𝑅)
igenval2.2	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
igenval2	⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ((𝑅 IdlGen 𝑆) = 𝐼 ↔ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))))

Distinct variable groups:   𝑅,𝑗   𝑆,𝑗   𝑗,𝐼

Allowed substitution hints:   𝐺(𝑗)   𝑋(𝑗)

Proof of Theorem igenval2

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		igenval2.1	. . . . 5 ⊢ 𝐺 = (1st ‘𝑅)
2		igenval2.2	. . . . 5 ⊢ 𝑋 = ran 𝐺
3	1, 2	igenidl 36221	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅))
4	1, 2	igenss 36220	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆))
5		igenmin 36222	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝑗) → (𝑅 IdlGen 𝑆) ⊆ 𝑗)
6	5	3expia 1120	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) → (𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))
7	6	ralrimiva 3103	. . . . 5 ⊢ (𝑅 ∈ RingOps → ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))
8	7	adantr 481	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))
9	3, 4, 8	3jca 1127	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ((𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ (𝑅 IdlGen 𝑆) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗)))
10		eleq1 2826	. . . 4 ⊢ ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅) ↔ 𝐼 ∈ (Idl‘𝑅)))
11		sseq2 3947	. . . 4 ⊢ ((𝑅 IdlGen 𝑆) = 𝐼 → (𝑆 ⊆ (𝑅 IdlGen 𝑆) ↔ 𝑆 ⊆ 𝐼))
12		sseq1 3946	. . . . . 6 ⊢ ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑅 IdlGen 𝑆) ⊆ 𝑗 ↔ 𝐼 ⊆ 𝑗))
13	12	imbi2d 341	. . . . 5 ⊢ ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗) ↔ (𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗)))
14	13	ralbidv 3112	. . . 4 ⊢ ((𝑅 IdlGen 𝑆) = 𝐼 → (∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗) ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗)))
15	10, 11, 14	3anbi123d 1435	. . 3 ⊢ ((𝑅 IdlGen 𝑆) = 𝐼 → (((𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ (𝑅 IdlGen 𝑆) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗)) ↔ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))))
16	9, 15	syl5ibcom 244	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ((𝑅 IdlGen 𝑆) = 𝐼 → (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))))
17		igenmin 36222	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
18	17	3adant3r3 1183	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
19	18	adantlr 712	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
20		ssint 4895	. . . . . . . 8 ⊢ (𝐼 ⊆ ∩ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖} ↔ ∀𝑗 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖}𝐼 ⊆ 𝑗)
21		sseq2 3947	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (𝑆 ⊆ 𝑖 ↔ 𝑆 ⊆ 𝑗))
22	21	ralrab 3630	. . . . . . . 8 ⊢ (∀𝑗 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖}𝐼 ⊆ 𝑗 ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))
23	20, 22	sylbbr 235	. . . . . . 7 ⊢ (∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗) → 𝐼 ⊆ ∩ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖})
24	23	3ad2ant3 1134	. . . . . 6 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗)) → 𝐼 ⊆ ∩ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖})
25	24	adantl 482	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))) → 𝐼 ⊆ ∩ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖})
26	1, 2	igenval 36219	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) = ∩ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖})
27	26	adantr 481	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))) → (𝑅 IdlGen 𝑆) = ∩ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖})
28	25, 27	sseqtrrd 3962	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))) → 𝐼 ⊆ (𝑅 IdlGen 𝑆))
29	19, 28	eqssd 3938	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))) → (𝑅 IdlGen 𝑆) = 𝐼)
30	29	ex 413	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ((𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗)) → (𝑅 IdlGen 𝑆) = 𝐼))
31	16, 30	impbid 211	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ((𝑅 IdlGen 𝑆) = 𝐼 ↔ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068   ⊆ wss 3887  ∩ cint 4879  ran crn 5590  ‘cfv 6433  (class class class)co 7275  1^st c1st 7829  RingOpscrngo 36052  Idlcidl 36165   IdlGen cigen 36217

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-grpo 28855  df-gid 28856  df-ablo 28907  df-rngo 36053  df-idl 36168  df-igen 36218

This theorem is referenced by:  prnc  36225