Theorem grpidinv2NEW 17119

Description: A group's properties using the explicit identity element.

Hypotheses

Ref	Expression
grpidinv2.1NEW	|- B = (base` G)
grpidinv2.2NEW	|- P = (+g` G)
grpidinv2.3NEW	|- U = (0g` G)

Assertion

Ref	Expression
grpidinv2NEW	|- ((G e. GrpNEW /\ A e. B) -> (((UPA) = A /\ (APU) = A) /\ E.y e. B ((yPA) = U /\ (APy) = U)))

Distinct variable groups:   y,A   y,B   y,G   y,P   y,U

Proof of Theorem grpidinv2NEW

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . . 6 |- (x = A -> (UPx) = (UPA))
2		id 73	. . . . . 6 |- (x = A -> x = A)
3	1, 2	eqeq12d 1899	. . . . 5 |- (x = A -> ((UPx) = x <-> (UPA) = A))
4		opreq1 4889	. . . . . 6 |- (x = A -> (xPU) = (APU))
5	4, 2	eqeq12d 1899	. . . . 5 |- (x = A -> ((xPU) = x <-> (APU) = A))
6	3, 5	anbi12d 690	. . . 4 |- (x = A -> (((UPx) = x /\ (xPU) = x) <-> ((UPA) = A /\ (APU) = A)))
7		opreq2 4890	. . . . . . 7 |- (x = A -> (yPx) = (yPA))
8	7	eqeq1d 1892	. . . . . 6 |- (x = A -> ((yPx) = U <-> (yPA) = U))
9		opreq1 4889	. . . . . . 7 |- (x = A -> (xPy) = (APy))
10	9	eqeq1d 1892	. . . . . 6 |- (x = A -> ((xPy) = U <-> (APy) = U))
11	8, 10	anbi12d 690	. . . . 5 |- (x = A -> (((yPx) = U /\ (xPy) = U) <-> ((yPA) = U /\ (APy) = U)))
12	11	rexbidv 2124	. . . 4 |- (x = A -> (E.y e. B ((yPx) = U /\ (xPy) = U) <-> E.y e. B ((yPA) = U /\ (APy) = U)))
13	6, 12	anbi12d 690	. . 3 |- (x = A -> ((((UPx) = x /\ (xPU) = x) /\ E.y e. B ((yPx) = U /\ (xPy) = U)) <-> (((UPA) = A /\ (APU) = A) /\ E.y e. B ((yPA) = U /\ (APy) = U))))
14	13	rcla4cva 2379	. 2 |- ((A.x e. B (((UPx) = x /\ (xPU) = x) /\ E.y e. B ((yPx) = U /\ (xPy) = U)) /\ A e. B) -> (((UPA) = A /\ (APU) = A) /\ E.y e. B ((yPA) = U /\ (APy) = U)))
15		grpidinv2.1NEW	. . . . . 6 |- B = (base` G)
16		grpidinv2.2NEW	. . . . . 6 |- P = (+g` G)
17	15, 16	grpidinvNEW 17113	. . . . 5 |- (G e. GrpNEW -> E.u e. B A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)))
18	15, 16	grpideuNEW 17114	. . . . 5 |- (G e. GrpNEW -> E!u e. B A.x e. B (uPx) = x)
19		ssid 2634	. . . . . 6 |- B C_ B
20		simpll 448	. . . . . . . . 9 |- ((((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)) -> (uPx) = x)
21	20	ralimi 2168	. . . . . . . 8 |- (A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)) -> A.x e. B (uPx) = x)
22	21	a1i 8	. . . . . . 7 |- (u e. B -> (A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)) -> A.x e. B (uPx) = x))
23	22	rgen 2159	. . . . . 6 |- A.u e. B (A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)) -> A.x e. B (uPx) = x)
24		reiotass2 5111	. . . . . 6 |- (((B C_ B /\ A.u e. B (A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)) -> A.x e. B (uPx) = x)) /\ (E.u e. B A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)) /\ E!u e. B A.x e. B (uPx) = x)) -> (iotau(u e. B /\ A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)))) = (iotau(u e. B /\ A.x e. B (uPx) = x)))
25	19, 23, 24	mpanl12 773	. . . . 5 |- ((E.u e. B A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)) /\ E!u e. B A.x e. B (uPx) = x) -> (iotau(u e. B /\ A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)))) = (iotau(u e. B /\ A.x e. B (uPx) = x)))
26	17, 18, 25	syl11anc 524	. . . 4 |- (G e. GrpNEW -> (iotau(u e. B /\ A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)))) = (iotau(u e. B /\ A.x e. B (uPx) = x)))
27		grpidinv2.3NEW	. . . . 5 |- U = (0g` G)
28	15, 16, 27	grpidvalNEW 17117	. . . 4 |- (G e. GrpNEW -> U = (iotau(u e. B /\ A.x e. B (uPx) = x)))
29	26, 28	eqtr4d 1928	. . 3 |- (G e. GrpNEW -> (iotau(u e. B /\ A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)))) = U)
30	15, 27	grpidclNEW 17118	. . . 4 |- (G e. GrpNEW -> U e. B)
31		reuss2 2870	. . . . . 6 |- (((B C_ B /\ A.u e. B (A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)) -> A.x e. B (uPx) = x)) /\ (E.u e. B A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)) /\ E!u e. B A.x e. B (uPx) = x)) -> E!u e. B A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)))
32	19, 23, 31	mpanl12 773	. . . . 5 |- ((E.u e. B A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)) /\ E!u e. B A.x e. B (uPx) = x) -> E!u e. B A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)))
33	17, 18, 32	syl11anc 524	. . . 4 |- (G e. GrpNEW -> E!u e. B A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)))
34		opreq1 4889	. . . . . . . . 9 |- (u = U -> (uPx) = (UPx))
35	34	eqeq1d 1892	. . . . . . . 8 |- (u = U -> ((uPx) = x <-> (UPx) = x))
36		opreq2 4890	. . . . . . . . 9 |- (u = U -> (xPu) = (xPU))
37	36	eqeq1d 1892	. . . . . . . 8 |- (u = U -> ((xPu) = x <-> (xPU) = x))
38	35, 37	anbi12d 690	. . . . . . 7 |- (u = U -> (((uPx) = x /\ (xPu) = x) <-> ((UPx) = x /\ (xPU) = x)))
39		eqeq2 1893	. . . . . . . . 9 |- (u = U -> ((yPx) = u <-> (yPx) = U))
40		eqeq2 1893	. . . . . . . . 9 |- (u = U -> ((xPy) = u <-> (xPy) = U))
41	39, 40	anbi12d 690	. . . . . . . 8 |- (u = U -> (((yPx) = u /\ (xPy) = u) <-> ((yPx) = U /\ (xPy) = U)))
42	41	rexbidv 2124	. . . . . . 7 |- (u = U -> (E.y e. B ((yPx) = u /\ (xPy) = u) <-> E.y e. B ((yPx) = U /\ (xPy) = U)))
43	38, 42	anbi12d 690	. . . . . 6 |- (u = U -> ((((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)) <-> (((UPx) = x /\ (xPU) = x) /\ E.y e. B ((yPx) = U /\ (xPy) = U))))
44	43	ralbidv 2123	. . . . 5 |- (u = U -> (A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)) <-> A.x e. B (((UPx) = x /\ (xPU) = x) /\ E.y e. B ((yPx) = U /\ (xPy) = U))))
45	44	reiota2 5110	. . . 4 |- ((U e. B /\ E!u e. B A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u))) -> (A.x e. B (((UPx) = x /\ (xPU) = x) /\ E.y e. B ((yPx) = U /\ (xPy) = U)) <-> (iotau(u e. B /\ A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)))) = U))
46	30, 33, 45	syl11anc 524	. . 3 |- (G e. GrpNEW -> (A.x e. B (((UPx) = x /\ (xPU) = x) /\ E.y e. B ((yPx) = U /\ (xPy) = U)) <-> (iotau(u e. B /\ A.x e. B (((uPx) = x /\ (xPu) = x) /\ E.y e. B ((yPx) = u /\ (xPy) = u)))) = U))
47	29, 46	mpbird 213	. 2 |- (G e. GrpNEW -> A.x e. B (((UPx) = x /\ (xPU) = x) /\ E.y e. B ((yPx) = U /\ (xPy) = U)))
48	14, 47	sylan 497	1 |- ((G e. GrpNEW /\ A e. B) -> (((UPA) = A /\ (APU) = A) /\ E.y e. B ((yPA) = U /\ (APy) = U)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  E!wreu 2107   C_ wss 2593  ` cfv 3998  (class class class)co 4884  iotacio 5087  basecbs 16758  +_gcplusg 17080  GrpNEWcgrp 17081  0_gc0g 17082

This theorem is referenced by:  grplidNEW 17120  grpridNEW 17121  grprcanNEW 17122  grpinveuNEW 17123  grpinvNEW 17128

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-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  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-mpt 5006  df-iota 5089  df-struct 16708  df-grpNEW 17089  df-0g 17090

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