Theorem 1idl 37434

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 37424	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼 ⊆ 𝑋)
4	3	adantr 480	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝐼 ⊆ 𝑋)
5		1idl.2	. . . . . . . . 9 ⊢ 𝐻 = (2nd ‘𝑅)
6	1	rneqi 5933	. . . . . . . . . 10 ⊢ ran 𝐺 = ran (1st ‘𝑅)
7	2, 6	eqtri 2755	. . . . . . . . 9 ⊢ 𝑋 = ran (1st ‘𝑅)
8		1idl.4	. . . . . . . . 9 ⊢ 𝑈 = (GId‘𝐻)
9	5, 7, 8	rngolidm 37345	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑈𝐻𝑥) = 𝑥)
10	9	ad2ant2rl 748	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → (𝑈𝐻𝑥) = 𝑥)
11	1, 5, 2	idlrmulcl 37429	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → (𝑈𝐻𝑥) ∈ 𝐼)
12	10, 11	eqeltrrd 2829	. . . . . 6 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → 𝑥 ∈ 𝐼)
13	12	expr 456	. . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → (𝑥 ∈ 𝑋 → 𝑥 ∈ 𝐼))
14	13	ssrdv 3984	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝑋 ⊆ 𝐼)
15	4, 14	eqssd 3995	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝐼 = 𝑋)
16	15	ex 412	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 → 𝐼 = 𝑋))
17	7, 5, 8	rngo1cl 37347	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
18	17	adantr 480	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑈 ∈ 𝑋)
19		eleq2 2817	. . 3 ⊢ (𝐼 = 𝑋 → (𝑈 ∈ 𝐼 ↔ 𝑈 ∈ 𝑋))
20	18, 19	syl5ibrcom 246	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼 = 𝑋 → 𝑈 ∈ 𝐼))
21	16, 20	impbid 211	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 ↔ 𝐼 = 𝑋))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099   ⊆ wss 3944  ran crn 5673  ‘cfv 6542  (class class class)co 7414  1^st c1st 7985  2^nd c2nd 7986  GIdcgi 30287  RingOpscrngo 37302  Idlcidl 37415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  df-riota 7370  df-ov 7417  df-1st 7987  df-2nd 7988  df-grpo 30290  df-gid 30291  df-ablo 30342  df-ass 37251  df-exid 37253  df-mgmOLD 37257  df-sgrOLD 37269  df-mndo 37275  df-rngo 37303  df-idl 37418

This theorem is referenced by:  0rngo  37435  divrngidl  37436  maxidln1  37452