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Theorem 1idl 35174
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 35164 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼𝑋)
43adantr 481 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈𝐼) → 𝐼𝑋)
5 1idl.2 . . . . . . . . 9 𝐻 = (2nd𝑅)
61rneqi 5805 . . . . . . . . . 10 ran 𝐺 = ran (1st𝑅)
72, 6eqtri 2848 . . . . . . . . 9 𝑋 = ran (1st𝑅)
8 1idl.4 . . . . . . . . 9 𝑈 = (GId‘𝐻)
95, 7, 8rngolidm 35085 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (𝑈𝐻𝑥) = 𝑥)
109ad2ant2rl 745 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈𝐼𝑥𝑋)) → (𝑈𝐻𝑥) = 𝑥)
111, 5, 2idlrmulcl 35169 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈𝐼𝑥𝑋)) → (𝑈𝐻𝑥) ∈ 𝐼)
1210, 11eqeltrrd 2918 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈𝐼𝑥𝑋)) → 𝑥𝐼)
1312expr 457 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈𝐼) → (𝑥𝑋𝑥𝐼))
1413ssrdv 3976 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈𝐼) → 𝑋𝐼)
154, 14eqssd 3987 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈𝐼) → 𝐼 = 𝑋)
1615ex 413 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈𝐼𝐼 = 𝑋))
177, 5, 8rngo1cl 35087 . . . 4 (𝑅 ∈ RingOps → 𝑈𝑋)
1817adantr 481 . . 3 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑈𝑋)
19 eleq2 2905 . . 3 (𝐼 = 𝑋 → (𝑈𝐼𝑈𝑋))
2018, 19syl5ibrcom 248 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝐼 = 𝑋𝑈𝐼))
2116, 20impbid 213 1 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈𝐼𝐼 = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wss 3939  ran crn 5554  cfv 6351  (class class class)co 7151  1st c1st 7681  2nd c2nd 7682  GIdcgi 28183  RingOpscrngo 35042  Idlcidl 35155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fo 6357  df-fv 6359  df-riota 7109  df-ov 7154  df-1st 7683  df-2nd 7684  df-grpo 28186  df-gid 28187  df-ablo 28238  df-ass 34991  df-exid 34993  df-mgmOLD 34997  df-sgrOLD 35009  df-mndo 35015  df-rngo 35043  df-idl 35158
This theorem is referenced by:  0rngo  35175  divrngidl  35176  maxidln1  35192
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