Theorem grpidinvNEW 17113

Description: A group has a left and right identity element, and every member has a left and right inverse.

Hypotheses

Ref	Expression
grplem1.1NEW	|- B = (base` G)
grplem1.2NEW	|- P = (+g` G)

Assertion

Ref	Expression
grpidinvNEW	|- (G e. GrpNEW -> E.u e. B A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)))

Distinct variable groups:   x,u,y,B   u,P,x,y   y,G,u,x

Proof of Theorem grpidinvNEW

Step	Hyp	Ref	Expression
1		grplem1.1NEW	. . 3 |- B = (base` G)
2		grplem1.2NEW	. . 3 |- P = (+g` G)
3	1, 2	grplidinvNEW 17108	. 2 |- (G e. GrpNEW -> E.u e. B A.z e. B ((uPz) = z /\ E.w e. B (wPz) = u))
4		opreq2 4890	. . . . . . . . . . 11 |- (z = x -> (uPz) = (uPx))
5		id 73	. . . . . . . . . . 11 |- (z = x -> z = x)
6	4, 5	eqeq12d 1899	. . . . . . . . . 10 |- (z = x -> ((uPz) = z <-> (uPx) = x))
7	6	rcla4cva 2379	. . . . . . . . 9 |- ((A.z e. B (uPz) = z /\ x e. B) -> (uPx) = x)
8		simpl 346	. . . . . . . . . 10 |- (((uPz) = z /\ E.w e. B (wPz) = u) -> (uPz) = z)
9	8	ralimi 2168	. . . . . . . . 9 |- (A.z e. B ((uPz) = z /\ E.w e. B (wPz) = u) -> A.z e. B (uPz) = z)
10	7, 9	sylan 497	. . . . . . . 8 |- ((A.z e. B ((uPz) = z /\ E.w e. B (wPz) = u) /\ x e. B) -> (uPx) = x)
11	10	adantll 428	. . . . . . 7 |- (((u e. B /\ A.z e. B ((uPz) = z /\ E.w e. B (wPz) = u)) /\ x e. B) -> (uPx) = x)
12	11	adantll 428	. . . . . 6 |- (((G e. GrpNEW /\ (u e. B /\ A.z e. B ((uPz) = z /\ E.w e. B (wPz) = u))) /\ x e. B) -> (uPx) = x)
13		simpl 346	. . . . . . . . 9 |- ((G e. GrpNEW /\ (u e. B /\ A.z e. B ((uPz) = z /\ E.w e. B (wPz) = u))) -> G e. GrpNEW)
14	13	anim1i 361	. . . . . . . 8 |- (((G e. GrpNEW /\ (u e. B /\ A.z e. B ((uPz) = z /\ E.w e. B (wPz) = u))) /\ x e. B) -> (G e. GrpNEW /\ x e. B))
15		id 73	. . . . . . . . . . . 12 |- ((G e. GrpNEW /\ u e. B) -> (G e. GrpNEW /\ u e. B))
16	15	adantrr 431	. . . . . . . . . . 11 |- ((G e. GrpNEW /\ (u e. B /\ A.z e. B ((uPz) = z /\ E.w e. B (wPz) = u))) -> (G e. GrpNEW /\ u e. B))
17	16	adantr 425	. . . . . . . . . 10 |- (((G e. GrpNEW /\ (u e. B /\ A.z e. B ((uPz) = z /\ E.w e. B (wPz) = u))) /\ x e. B) -> (G e. GrpNEW /\ u e. B))
18	9	adantl 424	. . . . . . . . . . 11 |- ((u e. B /\ A.z e. B ((uPz) = z /\ E.w e. B (wPz) = u)) -> A.z e. B (uPz) = z)
19	18	ad2antlr 441	. . . . . . . . . 10 |- (((G e. GrpNEW /\ (u e. B /\ A.z e. B ((uPz) = z /\ E.w e. B (wPz) = u))) /\ x e. B) -> A.z e. B (uPz) = z)
20		simpr 350	. . . . . . . . . . . . 13 |- (((uPz) = z /\ E.w e. B (wPz) = u) -> E.w e. B (wPz) = u)
21	20	ralimi 2168	. . . . . . . . . . . 12 |- (A.z e. B ((uPz) = z /\ E.w e. B (wPz) = u) -> A.z e. B E.w e. B (wPz) = u)
22	21	adantl 424	. . . . . . . . . . 11 |- ((u e. B /\ A.z e. B ((uPz) = z /\ E.w e. B (wPz) = u)) -> A.z e. B E.w e. B (wPz) = u)
23	22	ad2antlr 441	. . . . . . . . . 10 |- (((G e. GrpNEW /\ (u e. B /\ A.z e. B ((uPz) = z /\ E.w e. B (wPz) = u))) /\ x e. B) -> A.z e. B E.w e. B (wPz) = u)
24	17, 19, 23	jca32 312	. . . . . . . . 9 |- (((G e. GrpNEW /\ (u e. B /\ A.z e. B ((uPz) = z /\ E.w e. B (wPz) = u))) /\ x e. B) -> ((G e. GrpNEW /\ u e. B) /\ (A.z e. B (uPz) = z /\ A.z e. B E.w e. B (wPz) = u)))
25		biid 187	. . . . . . . . . 10 |- (A.z e. B (uPz) = z <-> A.z e. B (uPz) = z)
26		biid 187	. . . . . . . . . 10 |- (A.z e. B E.w e. B (wPz) = u <-> A.z e. B E.w e. B (wPz) = u)
27	1, 2, 25, 26	grpidinvlem3NEW 17111	. . . . . . . . 9 |- ((((G e. GrpNEW /\ u e. B) /\ (A.z e. B (uPz) = z /\ A.z e. B E.w e. B (wPz) = u)) /\ x e. B) -> E.y e. B ((yPx) = u /\ (xPy) = u))
28	24, 27	sylancom 531	. . . . . . . 8 |- (((G e. GrpNEW /\ (u e. B /\ A.z e. B ((uPz) = z /\ E.w e. B (wPz) = u))) /\ x e. B) -> E.y e. B ((yPx) = u /\ (xPy) = u))
29	1, 2	grpidinvlem4NEW 17112	. . . . . . . 8 |- (((G e. GrpNEW /\ x e. B) /\ E.y e. B ((yPx) = u /\ (xPy) = u)) -> (xPu) = (uPx))
30	14, 28, 29	syl11anc 524	. . . . . . 7 |- (((G e. GrpNEW /\ (u e. B /\ A.z e. B ((uPz) = z /\ E.w e. B (wPz) = u))) /\ x e. B) -> (xPu) = (uPx))
31	30, 12	eqtrd 1925	. . . . . 6 |- (((G e. GrpNEW /\ (u e. B /\ A.z e. B ((uPz) = z /\ E.w e. B (wPz) = u))) /\ x e. B) -> (xPu) = x)
32	12, 31, 28	jca31 311	. . . . 5 |- (((G e. GrpNEW /\ (u e. B /\ A.z e. B ((uPz) = z /\ E.w e. B (wPz) = u))) /\ x e. B) -> (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)))
33	32	r19.21aiva 2176	. . . 4 |- ((G e. GrpNEW /\ (u e. B /\ A.z e. B ((uPz) = z /\ E.w e. B (wPz) = u))) -> A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)))
34	33	exp32 408	. . 3 |- (G e. GrpNEW -> (u e. B -> (A.z e. B ((uPz) = z /\ E.w e. B (wPz) = u) -> A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)))))
35	34	reximdvai 2201	. 2 |- (G e. GrpNEW -> (E.u e. B A.z e. B ((uPz) = z /\ E.w e. B (wPz) = u) -> E.u e. B A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u))))
36	3, 35	mpd 29	1 |- (G e. GrpNEW -> E.u e. B A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  ` cfv 3998  (class class class)co 4884  basecbs 16758  +_gcplusg 17080  GrpNEWcgrp 17081

This theorem is referenced by:  grpideuNEW 17114  grpidinv2NEW 17119

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-rep 3428  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-tru 1262  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-fv 4014  df-opr 4886  df-struct 16708  df-grpNEW 17089

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