Theorem grpmnd 10393

Description: A group is a monoid. (Contributed by FL, 2-Nov-2009.)

Assertion

Ref	Expression
grpmnd	|- (G e. Grp -> G e. Mnd)

Proof of Theorem grpmnd

Step	Hyp	Ref	Expression
1		eqid 1884	. . . . 5 |- ran G = ran G
2	1	isgrp 9321	. . . 4 |- (G e. Grp -> (G e. Grp <-> (G:(ran G X. ran G)-->ran G /\ A.x e. ran GA.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) /\ E.w e. ran GA.x e. ran G((wGx) = x /\ E.y e. ran G(yGx) = w))))
3	2	biimpd 170	. . 3 |- (G e. Grp -> (G e. Grp -> (G:(ran G X. ran G)-->ran G /\ A.x e. ran GA.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) /\ E.w e. ran GA.x e. ran G((wGx) = x /\ E.y e. ran G(yGx) = w))))
4	1	grpidinv 9332	. . . . . . . . 9 |- (G e. Grp -> E.x e. ran GA.y e. ran G(((xGy) = y /\ (yGx) = y) /\ E.w e. ran G((wGy) = x /\ (yGw) = x)))
5		simpl 346	. . . . . . . . . . . 12 |- ((((xGy) = y /\ (yGx) = y) /\ E.w e. ran G((wGy) = x /\ (yGw) = x)) -> ((xGy) = y /\ (yGx) = y))
6	5	ralimi 2168	. . . . . . . . . . 11 |- (A.y e. ran G(((xGy) = y /\ (yGx) = y) /\ E.w e. ran G((wGy) = x /\ (yGw) = x)) -> A.y e. ran G((xGy) = y /\ (yGx) = y))
7	6	reximi 2198	. . . . . . . . . 10 |- (E.x e. ran GA.y e. ran G(((xGy) = y /\ (yGx) = y) /\ E.w e. ran G((wGy) = x /\ (yGw) = x)) -> E.x e. ran GA.y e. ran G((xGy) = y /\ (yGx) = y))
8	1	ismnd2 10392	. . . . . . . . . . . . . . 15 |- (G e. Grp -> (G e. Mnd <-> (G:(ran G X. ran G)-->ran G /\ A.x e. ran GA.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) /\ E.x e. ran GA.y e. ran G((yGx) = y /\ (xGy) = y))))
9		ancom 482	. . . . . . . . . . . . . . . . . 18 |- (((yGx) = y /\ (xGy) = y) <-> ((xGy) = y /\ (yGx) = y))
10	9	a1i 8	. . . . . . . . . . . . . . . . 17 |- (G e. Grp -> (((yGx) = y /\ (xGy) = y) <-> ((xGy) = y /\ (yGx) = y)))
11	10	rexralbidv 2142	. . . . . . . . . . . . . . . 16 |- (G e. Grp -> (E.x e. ran GA.y e. ran G((yGx) = y /\ (xGy) = y) <-> E.x e. ran GA.y e. ran G((xGy) = y /\ (yGx) = y)))
12	11	3anbi3d 1174	. . . . . . . . . . . . . . 15 |- (G e. Grp -> ((G:(ran G X. ran G)-->ran G /\ A.x e. ran GA.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) /\ E.x e. ran GA.y e. ran G((yGx) = y /\ (xGy) = y)) <-> (G:(ran G X. ran G)-->ran G /\ A.x e. ran GA.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) /\ E.x e. ran GA.y e. ran G((xGy) = y /\ (yGx) = y))))
13	8, 12	bitrd 587	. . . . . . . . . . . . . 14 |- (G e. Grp -> (G e. Mnd <-> (G:(ran G X. ran G)-->ran G /\ A.x e. ran GA.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) /\ E.x e. ran GA.y e. ran G((xGy) = y /\ (yGx) = y))))
14	13	biimprcd 173	. . . . . . . . . . . . 13 |- ((G:(ran G X. ran G)-->ran G /\ A.x e. ran GA.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) /\ E.x e. ran GA.y e. ran G((xGy) = y /\ (yGx) = y)) -> (G e. Grp -> G e. Mnd))
15	14	3exp 1066	. . . . . . . . . . . 12 |- (G:(ran G X. ran G)-->ran G -> (A.x e. ran GA.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) -> (E.x e. ran GA.y e. ran G((xGy) = y /\ (yGx) = y) -> (G e. Grp -> G e. Mnd))))
16	15	impcom 378	. . . . . . . . . . 11 |- ((A.x e. ran GA.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) /\ G:(ran G X. ran G)-->ran G) -> (E.x e. ran GA.y e. ran G((xGy) = y /\ (yGx) = y) -> (G e. Grp -> G e. Mnd)))
17	16	com3l 38	. . . . . . . . . 10 |- (E.x e. ran GA.y e. ran G((xGy) = y /\ (yGx) = y) -> (G e. Grp -> ((A.x e. ran GA.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) /\ G:(ran G X. ran G)-->ran G) -> G e. Mnd)))
18	7, 17	syl 12	. . . . . . . . 9 |- (E.x e. ran GA.y e. ran G(((xGy) = y /\ (yGx) = y) /\ E.w e. ran G((wGy) = x /\ (yGw) = x)) -> (G e. Grp -> ((A.x e. ran GA.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) /\ G:(ran G X. ran G)-->ran G) -> G e. Mnd)))
19	4, 18	mpcom 60	. . . . . . . 8 |- (G e. Grp -> ((A.x e. ran GA.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) /\ G:(ran G X. ran G)-->ran G) -> G e. Mnd))
20	19	com12 14	. . . . . . 7 |- ((A.x e. ran GA.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) /\ G:(ran G X. ran G)-->ran G) -> (G e. Grp -> G e. Mnd))
21	20	ex 402	. . . . . 6 |- (A.x e. ran GA.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) -> (G:(ran G X. ran G)-->ran G -> (G e. Grp -> G e. Mnd)))
22	21	a1i 8	. . . . 5 |- (E.w e. ran GA.x e. ran G((wGx) = x /\ E.y e. ran G(yGx) = w) -> (A.x e. ran GA.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) -> (G:(ran G X. ran G)-->ran G -> (G e. Grp -> G e. Mnd))))
23	22	com13 37	. . . 4 |- (G:(ran G X. ran G)-->ran G -> (A.x e. ran GA.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) -> (E.w e. ran GA.x e. ran G((wGx) = x /\ E.y e. ran G(yGx) = w) -> (G e. Grp -> G e. Mnd))))
24	23	3imp 1061	. . 3 |- ((G:(ran G X. ran G)-->ran G /\ A.x e. ran GA.y e. ran GA.z e. ran G((xGy)Gz) = (xG(yGz)) /\ E.w e. ran GA.x e. ran G((wGx) = x /\ E.y e. ran G(yGx) = w)) -> (G e. Grp -> G e. Mnd))
25	3, 24	syli 65	. 2 |- (G e. Grp -> (G e. Grp -> G e. Mnd))
26	25	pm2.43i 78	1 |- (G e. Grp -> G e. Mnd)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   X. cxp 3984  ran crn 3987  -->wf 3994  (class class class)co 4884  Grpcgr 9311  Mndcmnd 10384

This theorem is referenced by:  ltlga 14729  fprodneg 14741  fprodsub 14742  clfsebs4 14744  zintdom 14787  svli2 14826  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-grp 9316  df-ass 10360  df-exid 10362  df-mgm 10366  df-sgr 10378  df-mnd 10385

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