Theorem igenval2 38816

Description: Obsolete theorem, use rspprop 21433 instead. The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (New usage is discouraged.) (Proof modification is discouraged.)

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 38813	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅))
4	1, 2	igenss 38812	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆))
5		igenmin 38814	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝑗) → (𝑅 IdlGen 𝑆) ⊆ 𝑗)
6	5	3expia 1139	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) → (𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))
7	6	ralrimiva 3154	. . . . 5 ⊢ (𝑅 ∈ RingOps → ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))
8	7	adantr 486	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))
9	3, 4, 8	3jca 1146	. . 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 2848	. . . 4 ⊢ ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅) ↔ 𝐼 ∈ (Idl‘𝑅)))
11		sseq2 3957	. . . 4 ⊢ ((𝑅 IdlGen 𝑆) = 𝐼 → (𝑆 ⊆ (𝑅 IdlGen 𝑆) ↔ 𝑆 ⊆ 𝐼))
12		sseq1 3956	. . . . . 6 ⊢ ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑅 IdlGen 𝑆) ⊆ 𝑗 ↔ 𝐼 ⊆ 𝑗))
13	12	imbi2d 343	. . . . 5 ⊢ ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗) ↔ (𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗)))
14	13	ralbidv 3185	. . . 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 1464	. . 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 248	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ((𝑅 IdlGen 𝑆) = 𝐼 → (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))))
17		igenmin 38814	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
18	17	3adant3r3 1203	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
19	18	adantlr 728	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
20		ssint 4924	. . . . . . . 8 ⊢ (𝐼 ⊆ ∩ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖} ↔ ∀𝑗 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖}𝐼 ⊆ 𝑗)
21		sseq2 3957	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (𝑆 ⊆ 𝑖 ↔ 𝑆 ⊆ 𝑗))
22	21	ralrab 3652	. . . . . . . 8 ⊢ (∀𝑗 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖}𝐼 ⊆ 𝑗 ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))
23	20, 22	sylbbr 239	. . . . . . 7 ⊢ (∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗) → 𝐼 ⊆ ∩ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖})
24	23	3ad2ant3 1153	. . . . . 6 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗)) → 𝐼 ⊆ ∩ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖})
25	24	adantl 487	. . . . 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 38811	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) = ∩ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖})
27	26	adantr 486	. . . . 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 3968	. . . 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 3948	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))) → (𝑅 IdlGen 𝑆) = 𝐼)
30	29	ex 418	. 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 215	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ((𝑅 IdlGen 𝑆) = 𝐼 ↔ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412   ⊆ wss 3899  ∩ cint 4907  ran crn 5656  ‘cfv 6533  (class class class)co 7413  1^st c1st 7984  RingOpscrngo 38644  Idlcidl 38757   IdlGen cigen 38809

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fo 6539  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7986  df-2nd 7987  df-grpo 30974  df-gid 30975  df-ablo 31026  df-rngo 38645  df-idl 38760  df-igen 38810

This theorem is used by:  prnc  38817