Theorem rnplrnml3 14768

Description: In a unital ring the domain of the first operand of the addition equals the domain of the second operand of the addition.

Hypothesis

Ref	Expression
rnplrnml3.1	|- G = (1st` R)

Assertion

Ref	Expression
rnplrnml3	|- (R e. Ring -> dom dom G = ran dom G)

Proof of Theorem rnplrnml3

Step	Hyp	Ref	Expression
1		rnplrnml3.1	. . 3 |- G = (1st` R)
2	1	ringgrp 9476	. 2 |- (R e. Ring -> G e. Grp)
3		eqid 1884	. . . 4 |- ran G = ran G
4	3	grpfo 9323	. . 3 |- (G e. Grp -> G:(ran G X. ran G)-onto->ran G)
5		fof 4617	. . 3 |- (G:(ran G X. ran G)-onto->ran G -> G:(ran G X. ran G)-->ran G)
6		fdm 4567	. . . 4 |- (G:(ran G X. ran G)-->ran G -> dom G = (ran G X. ran G))
7		dmeq 4157	. . . . . . 7 |- (dom G = (ran G X. ran G) -> dom dom G = dom (ran G X. ran G))
8		dmxpid 4179	. . . . . . 7 |- dom (ran G X. ran G) = ran G
9	7, 8	syl6eq 1944	. . . . . 6 |- (dom G = (ran G X. ran G) -> dom dom G = ran G)
10		eqtr 1904	. . . . . . . . . . 11 |- ((ran dom G = ran (ran G X. ran G) /\ ran (ran G X. ran G) = ran G) -> ran dom G = ran G)
11	10	ex 402	. . . . . . . . . 10 |- (ran dom G = ran (ran G X. ran G) -> (ran (ran G X. ran G) = ran G -> ran dom G = ran G))
12		rnxp 4342	. . . . . . . . . 10 |- (ran G =/= (/) -> ran (ran G X. ran G) = ran G)
13	11, 12	syl5com 63	. . . . . . . . 9 |- (ran G =/= (/) -> (ran dom G = ran (ran G X. ran G) -> ran dom G = ran G))
14		rneq 4186	. . . . . . . . 9 |- (dom G = (ran G X. ran G) -> ran dom G = ran (ran G X. ran G))
15	13, 14	syl5 20	. . . . . . . 8 |- (ran G =/= (/) -> (dom G = (ran G X. ran G) -> ran dom G = ran G))
16		eqtr 1904	. . . . . . . . . 10 |- ((dom dom G = ran G /\ ran G = ran dom G) -> dom dom G = ran dom G)
17	16	expcom 403	. . . . . . . . 9 |- (ran G = ran dom G -> (dom dom G = ran G -> dom dom G = ran dom G))
18	17	eqcoms 1887	. . . . . . . 8 |- (ran dom G = ran G -> (dom dom G = ran G -> dom dom G = ran dom G))
19	15, 18	syl6 25	. . . . . . 7 |- (ran G =/= (/) -> (dom G = (ran G X. ran G) -> (dom dom G = ran G -> dom dom G = ran dom G)))
20	19	com13 37	. . . . . 6 |- (dom dom G = ran G -> (dom G = (ran G X. ran G) -> (ran G =/= (/) -> dom dom G = ran dom G)))
21	9, 20	mpcom 60	. . . . 5 |- (dom G = (ran G X. ran G) -> (ran G =/= (/) -> dom dom G = ran dom G))
22	3	grpn0 9326	. . . . 5 |- (G e. Grp -> ran G =/= (/))
23	21, 22	syl5 20	. . . 4 |- (dom G = (ran G X. ran G) -> (G e. Grp -> dom dom G = ran dom G))
24	6, 23	syl 12	. . 3 |- (G:(ran G X. ran G)-->ran G -> (G e. Grp -> dom dom G = ran dom G))
25	4, 5, 24	3syl 24	. 2 |- (G e. Grp -> (G e. Grp -> dom dom G = ran dom G))
26	2, 2, 25	sylc 83	1 |- (R e. Ring -> dom dom G = ran dom G)

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300   =/= wne 2017  (/)c0 2875   X. cxp 3984  dom cdm 3986  ran crn 3987  -->wf 3994  -onto->wfo 3996  ` cfv 3998  1stc1st 5018  Grpcgr 9311  Ringcring 9463

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-rab 2112  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-grp 9316  df-abl 9408  df-ring 9464

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