Theorem igenval2 38129

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 38126	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅))
4	1, 2	igenss 38125	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆))
5		igenmin 38127	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝑗) → (𝑅 IdlGen 𝑆) ⊆ 𝑗)
6	5	3expia 1121	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) → (𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))
7	6	ralrimiva 3125	. . . . 5 ⊢ (𝑅 ∈ RingOps → ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))
8	7	adantr 480	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))
9	3, 4, 8	3jca 1128	. . 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 2821	. . . 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 340	. . . . 5 ⊢ ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗) ↔ (𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗)))
14	13	ralbidv 3156	. . . 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 1438	. . 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 245	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → ((𝑅 IdlGen 𝑆) = 𝐼 → (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))))
17		igenmin 38127	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
18	17	3adant3r3 1185	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
19	18	adantlr 715	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
20		ssint 4916	. . . . . . . 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 3649	. . . . . . . 8 ⊢ (∀𝑗 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖}𝐼 ⊆ 𝑗 ↔ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))
23	20, 22	sylbbr 236	. . . . . . 7 ⊢ (∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗) → 𝐼 ⊆ ∩ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖})
24	23	3ad2ant3 1135	. . . . . 6 ⊢ ((𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗)) → 𝐼 ⊆ ∩ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖})
25	24	adantl 481	. . . . 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 38124	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) = ∩ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖})
27	26	adantr 480	. . . . 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 412	. 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 212	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3048  {crab 3396   ⊆ wss 3898  ∩ cint 4899  ran crn 5622  ‘cfv 6488  (class class class)co 7354  1^st c1st 7927  RingOpscrngo 37957  Idlcidl 38070   IdlGen cigen 38122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-fo 6494  df-fv 6496  df-riota 7311  df-ov 7357  df-oprab 7358  df-mpo 7359  df-1st 7929  df-2nd 7930  df-grpo 30477  df-gid 30478  df-ablo 30529  df-rngo 37958  df-idl 38073  df-igen 38123

This theorem is referenced by:  prnc  38130