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Theorem 1idl 36488
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 36478 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼𝑋)
43adantr 482 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈𝐼) → 𝐼𝑋)
5 1idl.2 . . . . . . . . 9 𝐻 = (2nd𝑅)
61rneqi 5893 . . . . . . . . . 10 ran 𝐺 = ran (1st𝑅)
72, 6eqtri 2765 . . . . . . . . 9 𝑋 = ran (1st𝑅)
8 1idl.4 . . . . . . . . 9 𝑈 = (GId‘𝐻)
95, 7, 8rngolidm 36399 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (𝑈𝐻𝑥) = 𝑥)
109ad2ant2rl 748 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈𝐼𝑥𝑋)) → (𝑈𝐻𝑥) = 𝑥)
111, 5, 2idlrmulcl 36483 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈𝐼𝑥𝑋)) → (𝑈𝐻𝑥) ∈ 𝐼)
1210, 11eqeltrrd 2839 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈𝐼𝑥𝑋)) → 𝑥𝐼)
1312expr 458 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈𝐼) → (𝑥𝑋𝑥𝐼))
1413ssrdv 3951 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈𝐼) → 𝑋𝐼)
154, 14eqssd 3962 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈𝐼) → 𝐼 = 𝑋)
1615ex 414 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈𝐼𝐼 = 𝑋))
177, 5, 8rngo1cl 36401 . . . 4 (𝑅 ∈ RingOps → 𝑈𝑋)
1817adantr 482 . . 3 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑈𝑋)
19 eleq2 2827 . . 3 (𝐼 = 𝑋 → (𝑈𝐼𝑈𝑋))
2018, 19syl5ibrcom 247 . 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 3911  ran crn 5635  cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  GIdcgi 29435  RingOpscrngo 36356  Idlcidl 36469
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 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505  df-riota 7314  df-ov 7361  df-1st 7922  df-2nd 7923  df-grpo 29438  df-gid 29439  df-ablo 29490  df-ass 36305  df-exid 36307  df-mgmOLD 36311  df-sgrOLD 36323  df-mndo 36329  df-rngo 36357  df-idl 36472
This theorem is referenced by:  0rngo  36489  divrngidl  36490  maxidln1  36506
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