Theorem igenval2 38387

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 38384	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → (𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅))
4	1, 2	igenss 38383	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) → 𝑆 ⊆ (𝑅 IdlGen 𝑆))
5		igenmin 38385	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝑗) → (𝑅 IdlGen 𝑆) ⊆ 𝑗)
6	5	3expia 1122	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝑗 ∈ (Idl‘𝑅)) → (𝑆 ⊆ 𝑗 → (𝑅 IdlGen 𝑆) ⊆ 𝑗))
7	6	ralrimiva 3129	. . . . 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 1129	. . 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 2824	. . . 4 ⊢ ((𝑅 IdlGen 𝑆) = 𝐼 → ((𝑅 IdlGen 𝑆) ∈ (Idl‘𝑅) ↔ 𝐼 ∈ (Idl‘𝑅)))
11		sseq2 3948	. . . 4 ⊢ ((𝑅 IdlGen 𝑆) = 𝐼 → (𝑆 ⊆ (𝑅 IdlGen 𝑆) ↔ 𝑆 ⊆ 𝐼))
12		sseq1 3947	. . . . . 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 3160	. . . 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 1439	. . 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 38385	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
18	17	3adant3r3 1186	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
19	18	adantlr 716	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ⊆ 𝑋) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑆 ⊆ 𝐼 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑆 ⊆ 𝑗 → 𝐼 ⊆ 𝑗))) → (𝑅 IdlGen 𝑆) ⊆ 𝐼)
20		ssint 4906	. . . . . . . 8 ⊢ (𝐼 ⊆ ∩ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖} ↔ ∀𝑗 ∈ {𝑖 ∈ (Idl‘𝑅) ∣ 𝑆 ⊆ 𝑖}𝐼 ⊆ 𝑗)
21		sseq2 3948	. . . . . . . . 9 ⊢ (𝑖 = 𝑗 → (𝑆 ⊆ 𝑖 ↔ 𝑆 ⊆ 𝑗))
22	21	ralrab 3640	. . . . . . . 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 1136	. . . . . 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 38382	. . . . . 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 3959	. . . 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 3939	. . 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 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  {crab 3389   ⊆ wss 3889  ∩ cint 4889  ran crn 5632  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940  RingOpscrngo 38215  Idlcidl 38328   IdlGen cigen 38380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fo 6504  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-grpo 30564  df-gid 30565  df-ablo 30616  df-rngo 38216  df-idl 38331  df-igen 38381

This theorem is referenced by:  prnc  38388