Theorem grprcanNEW 17122

Description: Right cancellation law for groups.

Hypotheses

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

Assertion

Ref	Expression
grprcanNEW	|- ((G e. GrpNEW /\ (X e. B /\ Y e. B /\ Z e. B)) -> ((XPZ) = (YPZ) <-> X = Y))

Proof of Theorem grprcanNEW

Step	Hyp	Ref	Expression
1		grprcan.1NEW	. . . . . . . 8 |- B = (base` G)
2		grprcan.2NEW	. . . . . . . 8 |- P = (+g` G)
3		eqid 1884	. . . . . . . 8 |- (0g` G) = (0g` G)
4	1, 2, 3	grpidinv2NEW 17119	. . . . . . 7 |- ((G e. GrpNEW /\ Z e. B) -> ((((0g` G)PZ) = Z /\ (ZP(0g` G)) = Z) /\ E.y e. B ((yPZ) = (0g` G) /\ (ZPy) = (0g` G))))
5		simpr 350	. . . . . . . . 9 |- (((yPZ) = (0g` G) /\ (ZPy) = (0g` G)) -> (ZPy) = (0g` G))
6	5	reximi 2198	. . . . . . . 8 |- (E.y e. B ((yPZ) = (0g` G) /\ (ZPy) = (0g` G)) -> E.y e. B (ZPy) = (0g` G))
7	6	adantl 424	. . . . . . 7 |- (((((0g` G)PZ) = Z /\ (ZP(0g` G)) = Z) /\ E.y e. B ((yPZ) = (0g` G) /\ (ZPy) = (0g` G))) -> E.y e. B (ZPy) = (0g` G))
8	4, 7	syl 12	. . . . . 6 |- ((G e. GrpNEW /\ Z e. B) -> E.y e. B (ZPy) = (0g` G))
9	8	ad2ant2rl 447	. . . . 5 |- (((G e. GrpNEW /\ X e. B) /\ (Y e. B /\ Z e. B)) -> E.y e. B (ZPy) = (0g` G))
10		opreq1 4889	. . . . . . . . . . . 12 |- ((XPZ) = (YPZ) -> ((XPZ)Py) = ((YPZ)Py))
11	10	ad2antll 443	. . . . . . . . . . 11 |- ((((G e. GrpNEW /\ X e. B) /\ (Y e. B /\ Z e. B)) /\ (y e. B /\ (XPZ) = (YPZ))) -> ((XPZ)Py) = ((YPZ)Py))
12	1, 2	grpassNEW 17107	. . . . . . . . . . . . . . 15 |- ((G e. GrpNEW /\ (X e. B /\ Z e. B /\ y e. B)) -> ((XPZ)Py) = (XP(ZPy)))
13	12	3exp2 1086	. . . . . . . . . . . . . 14 |- (G e. GrpNEW -> (X e. B -> (Z e. B -> (y e. B -> ((XPZ)Py) = (XP(ZPy))))))
14	13	imp41 395	. . . . . . . . . . . . 13 |- ((((G e. GrpNEW /\ X e. B) /\ Z e. B) /\ y e. B) -> ((XPZ)Py) = (XP(ZPy)))
15	14	adantlrl 434	. . . . . . . . . . . 12 |- ((((G e. GrpNEW /\ X e. B) /\ (Y e. B /\ Z e. B)) /\ y e. B) -> ((XPZ)Py) = (XP(ZPy)))
16	15	adantrr 431	. . . . . . . . . . 11 |- ((((G e. GrpNEW /\ X e. B) /\ (Y e. B /\ Z e. B)) /\ (y e. B /\ (XPZ) = (YPZ))) -> ((XPZ)Py) = (XP(ZPy)))
17	1, 2	grpassNEW 17107	. . . . . . . . . . . . . . 15 |- ((G e. GrpNEW /\ (Y e. B /\ Z e. B /\ y e. B)) -> ((YPZ)Py) = (YP(ZPy)))
18	17	3exp2 1086	. . . . . . . . . . . . . 14 |- (G e. GrpNEW -> (Y e. B -> (Z e. B -> (y e. B -> ((YPZ)Py) = (YP(ZPy))))))
19	18	imp42 396	. . . . . . . . . . . . 13 |- (((G e. GrpNEW /\ (Y e. B /\ Z e. B)) /\ y e. B) -> ((YPZ)Py) = (YP(ZPy)))
20	19	adantllr 433	. . . . . . . . . . . 12 |- ((((G e. GrpNEW /\ X e. B) /\ (Y e. B /\ Z e. B)) /\ y e. B) -> ((YPZ)Py) = (YP(ZPy)))
21	20	adantrr 431	. . . . . . . . . . 11 |- ((((G e. GrpNEW /\ X e. B) /\ (Y e. B /\ Z e. B)) /\ (y e. B /\ (XPZ) = (YPZ))) -> ((YPZ)Py) = (YP(ZPy)))
22	11, 16, 21	3eqtr3d 1934	. . . . . . . . . 10 |- ((((G e. GrpNEW /\ X e. B) /\ (Y e. B /\ Z e. B)) /\ (y e. B /\ (XPZ) = (YPZ))) -> (XP(ZPy)) = (YP(ZPy)))
23	22	adantrrl 438	. . . . . . . . 9 |- ((((G e. GrpNEW /\ X e. B) /\ (Y e. B /\ Z e. B)) /\ (y e. B /\ ((ZPy) = (0g` G) /\ (XPZ) = (YPZ)))) -> (XP(ZPy)) = (YP(ZPy)))
24		opreq2 4890	. . . . . . . . . . 11 |- ((ZPy) = (0g` G) -> (XP(ZPy)) = (XP(0g` G)))
25	24	ad2antrl 442	. . . . . . . . . 10 |- ((y e. B /\ ((ZPy) = (0g` G) /\ (XPZ) = (YPZ))) -> (XP(ZPy)) = (XP(0g` G)))
26	25	adantl 424	. . . . . . . . 9 |- ((((G e. GrpNEW /\ X e. B) /\ (Y e. B /\ Z e. B)) /\ (y e. B /\ ((ZPy) = (0g` G) /\ (XPZ) = (YPZ)))) -> (XP(ZPy)) = (XP(0g` G)))
27		opreq2 4890	. . . . . . . . . . 11 |- ((ZPy) = (0g` G) -> (YP(ZPy)) = (YP(0g` G)))
28	27	ad2antrl 442	. . . . . . . . . 10 |- ((y e. B /\ ((ZPy) = (0g` G) /\ (XPZ) = (YPZ))) -> (YP(ZPy)) = (YP(0g` G)))
29	28	adantl 424	. . . . . . . . 9 |- ((((G e. GrpNEW /\ X e. B) /\ (Y e. B /\ Z e. B)) /\ (y e. B /\ ((ZPy) = (0g` G) /\ (XPZ) = (YPZ)))) -> (YP(ZPy)) = (YP(0g` G)))
30	23, 26, 29	3eqtr3d 1934	. . . . . . . 8 |- ((((G e. GrpNEW /\ X e. B) /\ (Y e. B /\ Z e. B)) /\ (y e. B /\ ((ZPy) = (0g` G) /\ (XPZ) = (YPZ)))) -> (XP(0g` G)) = (YP(0g` G)))
31	1, 2, 3	grpridNEW 17121	. . . . . . . . 9 |- ((G e. GrpNEW /\ X e. B) -> (XP(0g` G)) = X)
32	31	ad2antrr 440	. . . . . . . 8 |- ((((G e. GrpNEW /\ X e. B) /\ (Y e. B /\ Z e. B)) /\ (y e. B /\ ((ZPy) = (0g` G) /\ (XPZ) = (YPZ)))) -> (XP(0g` G)) = X)
33	1, 2, 3	grpridNEW 17121	. . . . . . . . . 10 |- ((G e. GrpNEW /\ Y e. B) -> (YP(0g` G)) = Y)
34	33	ad2ant2r 445	. . . . . . . . 9 |- (((G e. GrpNEW /\ X e. B) /\ (Y e. B /\ Z e. B)) -> (YP(0g` G)) = Y)
35	34	adantr 425	. . . . . . . 8 |- ((((G e. GrpNEW /\ X e. B) /\ (Y e. B /\ Z e. B)) /\ (y e. B /\ ((ZPy) = (0g` G) /\ (XPZ) = (YPZ)))) -> (YP(0g` G)) = Y)
36	30, 32, 35	3eqtr3d 1934	. . . . . . 7 |- ((((G e. GrpNEW /\ X e. B) /\ (Y e. B /\ Z e. B)) /\ (y e. B /\ ((ZPy) = (0g` G) /\ (XPZ) = (YPZ)))) -> X = Y)
37	36	exp45 417	. . . . . 6 |- (((G e. GrpNEW /\ X e. B) /\ (Y e. B /\ Z e. B)) -> (y e. B -> ((ZPy) = (0g` G) -> ((XPZ) = (YPZ) -> X = Y))))
38	37	r19.23adv 2215	. . . . 5 |- (((G e. GrpNEW /\ X e. B) /\ (Y e. B /\ Z e. B)) -> (E.y e. B (ZPy) = (0g` G) -> ((XPZ) = (YPZ) -> X = Y)))
39	9, 38	mpd 29	. . . 4 |- (((G e. GrpNEW /\ X e. B) /\ (Y e. B /\ Z e. B)) -> ((XPZ) = (YPZ) -> X = Y))
40		opreq1 4889	. . . 4 |- (X = Y -> (XPZ) = (YPZ))
41	39, 40	impbid1 575	. . 3 |- (((G e. GrpNEW /\ X e. B) /\ (Y e. B /\ Z e. B)) -> ((XPZ) = (YPZ) <-> X = Y))
42	41	exp43 415	. 2 |- (G e. GrpNEW -> (X e. B -> (Y e. B -> (Z e. B -> ((XPZ) = (YPZ) <-> X = Y)))))
43	42	3imp2 1083	1 |- ((G e. GrpNEW /\ (X e. B /\ Y e. B /\ Z e. B)) -> ((XPZ) = (YPZ) <-> X = Y))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wrex 2106  ` cfv 3998  (class class class)co 4884  basecbs 16758  +_gcplusg 17080  GrpNEWcgrp 17081  0_gc0g 17082

This theorem is referenced by:  grpinveuNEW 17123  grpidNEW 17124  ringrzNEW 17157  divrngidNEW 17166

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

Copyright terms: Public domain