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Theorem rngorn1eq 22000
 Description: In a unital ring the range of the addition equals the range of the multiplication. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)
Hypotheses
Ref Expression
rnplrnml0.1
rnplrnml0.2
Assertion
Ref Expression
rngorn1eq

Proof of Theorem rngorn1eq
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnplrnml0.2 . . . 4
2 rnplrnml0.1 . . . 4
3 eqid 2435 . . . 4
41, 2, 3rngosm 21961 . . 3
51, 2, 3rngoi 21960 . . . 4
6 simprr 734 . . . 4
75, 6syl 16 . . 3
8 rngmgmbs4 21997 . . 3
94, 7, 8syl2anc 643 . 2
109eqcomd 2440 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2697  wrex 2698   cxp 4868   crn 4871  wf 5442  cfv 5446  (class class class)co 6073  c1st 6339  c2nd 6340  cablo 21861  crngo 21955 This theorem is referenced by:  rngoidmlem  22003  rngo1cl  22009  isdrngo2  26565 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-ov 6076  df-1st 6341  df-2nd 6342  df-rngo 21956
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