Theorem 1idl 34304

Description: Two ways of expressing the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

Ref	Expression
1idl.1	⊢ 𝐺 = (1st ‘𝑅)
1idl.2	⊢ 𝐻 = (2nd ‘𝑅)
1idl.3	⊢ 𝑋 = ran 𝐺
1idl.4	⊢ 𝑈 = (GId‘𝐻)

Assertion

Ref	Expression
1idl	⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 ↔ 𝐼 = 𝑋))

Proof of Theorem 1idl

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1idl.1	. . . . . 6 ⊢ 𝐺 = (1st ‘𝑅)
2		1idl.3	. . . . . 6 ⊢ 𝑋 = ran 𝐺
3	1, 2	idlss 34294	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ 𝑋)
4	3	adantr 473	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝐼 ⊆ 𝑋)
5		1idl.2	. . . . . . . . 9 ⊢ 𝐻 = (2nd ‘𝑅)
6	1	rneqi 5553	. . . . . . . . . 10 ⊢ ran 𝐺 = ran (1st ‘𝑅)
7	2, 6	eqtri 2819	. . . . . . . . 9 ⊢ 𝑋 = ran (1st ‘𝑅)
8		1idl.4	. . . . . . . . 9 ⊢ 𝑈 = (GId‘𝐻)
9	5, 7, 8	rngolidm 34215	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑈𝐻𝑥) = 𝑥)
10	9	ad2ant2rl 756	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → (𝑈𝐻𝑥) = 𝑥)
11	1, 5, 2	idlrmulcl 34299	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → (𝑈𝐻𝑥) ∈ 𝐼)
12	10, 11	eqeltrrd 2877	. . . . . 6 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → 𝑥 ∈ 𝐼)
13	12	expr 449	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → (𝑥 ∈ 𝑋 → 𝑥 ∈ 𝐼))
14	13	ssrdv 3802	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝑋 ⊆ 𝐼)
15	4, 14	eqssd 3813	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝐼 = 𝑋)
16	15	ex 402	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 → 𝐼 = 𝑋))
17	7, 5, 8	rngo1cl 34217	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
18	17	adantr 473	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑈 ∈ 𝑋)
19		eleq2 2865	. . 3 ⊢ (𝐼 = 𝑋 → (𝑈 ∈ 𝐼 ↔ 𝑈 ∈ 𝑋))
20	18, 19	syl5ibrcom 239	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼 = 𝑋 → 𝑈 ∈ 𝐼))
21	16, 20	impbid 204	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 ↔ 𝐼 = 𝑋))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   = wceq 1653   ∈ wcel 2157   ⊆ wss 3767  ran crn 5311  ‘cfv 6099  (class class class)co 6876  1^st c1st 7397  2^nd c2nd 7398  GIdcgi 27862  RingOpscrngo 34172  Idlcidl 34285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-fo 6105  df-fv 6107  df-riota 6837  df-ov 6879  df-1st 7399  df-2nd 7400  df-grpo 27865  df-gid 27866  df-ablo 27917  df-ass 34121  df-exid 34123  df-mgmOLD 34127  df-sgrOLD 34139  df-mndo 34145  df-rngo 34173  df-idl 34288

This theorem is referenced by:  0rngo  34305  divrngidl  34306  maxidln1  34322