Theorem 1idl 38015

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 38005	. . . . 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 5903	. . . . . . . . . 10 ⊢ ran 𝐺 = ran (1st ‘𝑅)
7	2, 6	eqtri 2753	. . . . . . . . 9 ⊢ 𝑋 = ran (1st ‘𝑅)
8		1idl.4	. . . . . . . . 9 ⊢ 𝑈 = (GId‘𝐻)
9	5, 7, 8	rngolidm 37926	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝑥 ∈ 𝑋) → (𝑈𝐻𝑥) = 𝑥)
10	9	ad2ant2rl 749	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → (𝑈𝐻𝑥) = 𝑥)
11	1, 5, 2	idlrmulcl 38010	. . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋)) → (𝑈𝐻𝑥) ∈ 𝐼)
12	10, 11	eqeltrrd 2830	. . . . . 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 3954	. . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈 ∈ 𝐼) → 𝑋 ⊆ 𝐼)
15	4, 14	eqssd 3966	. . 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 37928	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋)
18	17	adantr 480	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑈 ∈ 𝑋)
19		eleq2 2818	. . 3 ⊢ (𝐼 = 𝑋 → (𝑈 ∈ 𝐼 ↔ 𝑈 ∈ 𝑋))
20	18, 19	syl5ibrcom 247	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼 = 𝑋 → 𝑈 ∈ 𝐼))
21	16, 20	impbid 212	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈 ∈ 𝐼 ↔ 𝐼 = 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ⊆ wss 3916  ran crn 5641  ‘cfv 6513  (class class class)co 7389  1^st c1st 7968  2^nd c2nd 7969  GIdcgi 30425  RingOpscrngo 37883  Idlcidl 37996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-fo 6519  df-fv 6521  df-riota 7346  df-ov 7392  df-1st 7970  df-2nd 7971  df-grpo 30428  df-gid 30429  df-ablo 30480  df-ass 37832  df-exid 37834  df-mgmOLD 37838  df-sgrOLD 37850  df-mndo 37856  df-rngo 37884  df-idl 37999

This theorem is referenced by:  0rngo  38016  divrngidl  38017  maxidln1  38033