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Theorem 1idl 36894
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 36884 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼𝑋)
43adantr 482 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈𝐼) → 𝐼𝑋)
5 1idl.2 . . . . . . . . 9 𝐻 = (2nd𝑅)
61rneqi 5937 . . . . . . . . . 10 ran 𝐺 = ran (1st𝑅)
72, 6eqtri 2761 . . . . . . . . 9 𝑋 = ran (1st𝑅)
8 1idl.4 . . . . . . . . 9 𝑈 = (GId‘𝐻)
95, 7, 8rngolidm 36805 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (𝑈𝐻𝑥) = 𝑥)
109ad2ant2rl 748 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈𝐼𝑥𝑋)) → (𝑈𝐻𝑥) = 𝑥)
111, 5, 2idlrmulcl 36889 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈𝐼𝑥𝑋)) → (𝑈𝐻𝑥) ∈ 𝐼)
1210, 11eqeltrrd 2835 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈𝐼𝑥𝑋)) → 𝑥𝐼)
1312expr 458 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈𝐼) → (𝑥𝑋𝑥𝐼))
1413ssrdv 3989 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈𝐼) → 𝑋𝐼)
154, 14eqssd 4000 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈𝐼) → 𝐼 = 𝑋)
1615ex 414 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈𝐼𝐼 = 𝑋))
177, 5, 8rngo1cl 36807 . . . 4 (𝑅 ∈ RingOps → 𝑈𝑋)
1817adantr 482 . . 3 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑈𝑋)
19 eleq2 2823 . . 3 (𝐼 = 𝑋 → (𝑈𝐼𝑈𝑋))
2018, 19syl5ibrcom 246 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼 = 𝑋𝑈𝐼))
2116, 20impbid 211 1 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈𝐼𝐼 = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wss 3949  ran crn 5678  cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  GIdcgi 29743  RingOpscrngo 36762  Idlcidl 36875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-riota 7365  df-ov 7412  df-1st 7975  df-2nd 7976  df-grpo 29746  df-gid 29747  df-ablo 29798  df-ass 36711  df-exid 36713  df-mgmOLD 36717  df-sgrOLD 36729  df-mndo 36735  df-rngo 36763  df-idl 36878
This theorem is referenced by:  0rngo  36895  divrngidl  36896  maxidln1  36912
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