Theorem rnplrnml 10404

Description: In a unital ring the range of the addition equals the range of the multiplication. (Contributed by FL, 24-Jan-2010.)

Hypotheses

Ref	Expression
rnplrnml0.1	|- H = (2nd` R)
rnplrnml0.2	|- G = (1st` R)

Assertion

Ref	Expression
rnplrnml	|- (R e. Ring -> ran G = ran H)

Proof of Theorem rnplrnml

Step	Hyp	Ref	Expression
1		rnplrnml0.2	. . . 4 |- G = (1st` R)
2		rnplrnml0.1	. . . 4 |- H = (2nd` R)
3		eqid 1884	. . . 4 |- ran G = ran G
4	1, 2, 3	ringsm 9467	. . 3 |- (R e. Ring -> H:(ran G X. ran G)-->ran G)
5	1, 2, 3	ringi 9466	. . . 4 |- (R e. Ring -> ((G e. Abel /\ H:(ran G X. ran G)-->ran G) /\ (A.y e. ran GA.x e. ran GA.z e. ran G(((yHx)Hz) = (yH(xHz)) /\ (yH(xGz)) = ((yHx)G(yHz)) /\ ((yGx)Hz) = ((yHz)G(xHz))) /\ E.y e. ran GA.x e. ran G((xHy) = x /\ (yHx) = x))))
6		simprr 451	. . . 4 |- (((G e. Abel /\ H:(ran G X. ran G)-->ran G) /\ (A.y e. ran GA.x e. ran GA.z e. ran G(((yHx)Hz) = (yH(xHz)) /\ (yH(xGz)) = ((yHx)G(yHz)) /\ ((yGx)Hz) = ((yHz)G(xHz))) /\ E.y e. ran GA.x e. ran G((xHy) = x /\ (yHx) = x))) -> E.y e. ran GA.x e. ran G((xHy) = x /\ (yHx) = x))
7	5, 6	syl 12	. . 3 |- (R e. Ring -> E.y e. ran GA.x e. ran G((xHy) = x /\ (yHx) = x))
8		rngmgmbs4 10401	. . 3 |- ((H:(ran G X. ran G)-->ran G /\ E.y e. ran GA.x e. ran G((xHy) = x /\ (yHx) = x)) -> ran H = ran G)
9	4, 7, 8	syl11anc 524	. 2 |- (R e. Ring -> ran H = ran G)
10	9	eqcomd 1889	1 |- (R e. Ring -> ran G = ran H)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   X. cxp 3984  ran crn 3987  -->wf 3994  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Abelcabl 9407  Ringcring 9463

This theorem is referenced by:  ringidmlem 10409  ring1cl 10415  rngdmdmrn 14767  rngunval2 14774  isdivrng2 16111

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-ring 9464

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