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Theorem 1idl 35185
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 35175 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼𝑋)
43adantr 481 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈𝐼) → 𝐼𝑋)
5 1idl.2 . . . . . . . . 9 𝐻 = (2nd𝑅)
61rneqi 5800 . . . . . . . . . 10 ran 𝐺 = ran (1st𝑅)
72, 6eqtri 2841 . . . . . . . . 9 𝑋 = ran (1st𝑅)
8 1idl.4 . . . . . . . . 9 𝑈 = (GId‘𝐻)
95, 7, 8rngolidm 35096 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝑥𝑋) → (𝑈𝐻𝑥) = 𝑥)
109ad2ant2rl 745 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈𝐼𝑥𝑋)) → (𝑈𝐻𝑥) = 𝑥)
111, 5, 2idlrmulcl 35180 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈𝐼𝑥𝑋)) → (𝑈𝐻𝑥) ∈ 𝐼)
1210, 11eqeltrrd 2911 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝑈𝐼𝑥𝑋)) → 𝑥𝐼)
1312expr 457 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈𝐼) → (𝑥𝑋𝑥𝐼))
1413ssrdv 3970 . . . 4 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈𝐼) → 𝑋𝐼)
154, 14eqssd 3981 . . 3 (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝑈𝐼) → 𝐼 = 𝑋)
1615ex 413 . 2 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈𝐼𝐼 = 𝑋))
177, 5, 8rngo1cl 35098 . . . 4 (𝑅 ∈ RingOps → 𝑈𝑋)
1817adantr 481 . . 3 ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑈𝑋)
19 eleq2 2898 . . 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 1528  wcel 2105  wss 3933  ran crn 5549  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  GIdcgi 28194  RingOpscrngo 35053  Idlcidl 35166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-riota 7103  df-ov 7148  df-1st 7678  df-2nd 7679  df-grpo 28197  df-gid 28198  df-ablo 28249  df-ass 35002  df-exid 35004  df-mgmOLD 35008  df-sgrOLD 35020  df-mndo 35026  df-rngo 35054  df-idl 35169
This theorem is referenced by:  0rngo  35186  divrngidl  35187  maxidln1  35203
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