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Theorem 1idl 35770
 Description: Two ways of expressing the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
1idl.1 𝐺 = (1st𝑅)
1idl.2 𝐻 = (2nd𝑅)
1idl.3 𝑋 = ran 𝐺
1idl.4 𝑈 = (GId‘𝐻)
Assertion
Ref Expression
1idl ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈𝐼𝐼 = 𝑋))

Proof of Theorem 1idl
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 1idl.1 . . . . . 6 𝐺 = (1st𝑅)
2 1idl.3 . . . . . 6 𝑋 = ran 𝐺
31, 2idlss 35760 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼𝑋)
43adantr 484 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈𝐼) → 𝐼𝑋)
5 1idl.2 . . . . . . . . 9 𝐻 = (2nd𝑅)
61rneqi 5782 . . . . . . . . . 10 ran 𝐺 = ran (1st𝑅)
72, 6eqtri 2781 . . . . . . . . 9 𝑋 = ran (1st𝑅)
8 1idl.4 . . . . . . . . 9 𝑈 = (GId‘𝐻)
95, 7, 8rngolidm 35681 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (𝑈𝐻𝑥) = 𝑥)
109ad2ant2rl 748 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈𝐼𝑥𝑋)) → (𝑈𝐻𝑥) = 𝑥)
111, 5, 2idlrmulcl 35765 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈𝐼𝑥𝑋)) → (𝑈𝐻𝑥) ∈ 𝐼)
1210, 11eqeltrrd 2853 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈𝐼𝑥𝑋)) → 𝑥𝐼)
1312expr 460 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈𝐼) → (𝑥𝑋𝑥𝐼))
1413ssrdv 3900 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈𝐼) → 𝑋𝐼)
154, 14eqssd 3911 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈𝐼) → 𝐼 = 𝑋)
1615ex 416 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈𝐼𝐼 = 𝑋))
177, 5, 8rngo1cl 35683 . . . 4 (𝑅 ∈ RingOps → 𝑈𝑋)
1817adantr 484 . . 3 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑈𝑋)
19 eleq2 2840 . . 3 (𝐼 = 𝑋 → (𝑈𝐼𝑈𝑋))
2018, 19syl5ibrcom 250 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼 = 𝑋𝑈𝐼))
2116, 20impbid 215 1 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈𝐼𝐼 = 𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ⊆ wss 3860  ran crn 5528  ‘cfv 6339  (class class class)co 7155  1st c1st 7696  2nd c2nd 7697  GIdcgi 28377  RingOpscrngo 35638  Idlcidl 35751 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-fo 6345  df-fv 6347  df-riota 7113  df-ov 7158  df-1st 7698  df-2nd 7699  df-grpo 28380  df-gid 28381  df-ablo 28432  df-ass 35587  df-exid 35589  df-mgmOLD 35593  df-sgrOLD 35605  df-mndo 35611  df-rngo 35639  df-idl 35754 This theorem is referenced by:  0rngo  35771  divrngidl  35772  maxidln1  35788
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