Theorem caoprmo 5003

Description: Uniqueness of inverse element in commutative, associative operation with identity. Remark in proof of Proposition 9-2.4 of [Gleason] p. 119.

Hypotheses

Ref	Expression
caoprmo.1	|- A e. _V
caoprmo.2	|- B e. S
caoprmo.dom	|- dom F = (S X. S)
caoprmo.3	|- -. (/) e. S
caoprmo.com	|- (xFy) = (yFx)
caoprmo.ass	|- ((xFy)Fz) = (xF(yFz))
caoprmo.id	|- (x e. S -> (xFB) = x)

Assertion

Ref	Expression
caoprmo	|- E*w(AFw) = B

Distinct variable groups:   x,y,z,F   x,S,y,z   x,A,y,z   x,B,y,z,w   w,S   w,A   w,B   w,F

Proof of Theorem caoprmo

Step	Hyp	Ref	Expression
1		eleq1 1957	. . . . 5 |- (w = v -> (w e. S <-> v e. S))
2		opreq2 4890	. . . . . 6 |- (w = v -> (AFw) = (AFv))
3	2	eqeq1d 1892	. . . . 5 |- (w = v -> ((AFw) = B <-> (AFv) = B))
4	1, 3	anbi12d 690	. . . 4 |- (w = v -> ((w e. S /\ (AFw) = B) <-> (v e. S /\ (AFv) = B)))
5	4	mo4 1799	. . 3 |- (E*w(w e. S /\ (AFw) = B) <-> A.wA.v(((w e. S /\ (AFw) = B) /\ (v e. S /\ (AFv) = B)) -> w = v))
6		opreq2 4890	. . . . . . . 8 |- ((AFv) = B -> (wF(AFv)) = (wFB))
7		opreq1 4889	. . . . . . . . . 10 |- (x = w -> (xFB) = (wFB))
8		id 73	. . . . . . . . . 10 |- (x = w -> x = w)
9	7, 8	eqeq12d 1899	. . . . . . . . 9 |- (x = w -> ((xFB) = x <-> (wFB) = w))
10		caoprmo.id	. . . . . . . . 9 |- (x e. S -> (xFB) = x)
11	9, 10	vtoclga 2352	. . . . . . . 8 |- (w e. S -> (wFB) = w)
12	6, 11	sylan9eqr 1951	. . . . . . 7 |- ((w e. S /\ (AFv) = B) -> (wF(AFv)) = w)
13		caoprmo.1	. . . . . . . . 9 |- A e. _V
14		visset 2295	. . . . . . . . 9 |- w e. _V
15		visset 2295	. . . . . . . . 9 |- v e. _V
16		caoprmo.ass	. . . . . . . . 9 |- ((xFy)Fz) = (xF(yFz))
17	13, 14, 15, 16	caoprass 4987	. . . . . . . 8 |- ((AFw)Fv) = (AF(wFv))
18		caoprmo.com	. . . . . . . . 9 |- (xFy) = (yFx)
19	13, 14, 15, 18, 16	caopr12 4994	. . . . . . . 8 |- (AF(wFv)) = (wF(AFv))
20	17, 19	eqtri 1908	. . . . . . 7 |- ((AFw)Fv) = (wF(AFv))
21	12, 20	syl5eq 1940	. . . . . 6 |- ((w e. S /\ (AFv) = B) -> ((AFw)Fv) = w)
22	21	ad2ant2rl 447	. . . . 5 |- (((w e. S /\ (AFw) = B) /\ (v e. S /\ (AFv) = B)) -> ((AFw)Fv) = w)
23		opreq1 4889	. . . . . . 7 |- ((AFw) = B -> ((AFw)Fv) = (BFv))
24		opreq1 4889	. . . . . . . . . 10 |- (x = v -> (xFB) = (vFB))
25		id 73	. . . . . . . . . 10 |- (x = v -> x = v)
26	24, 25	eqeq12d 1899	. . . . . . . . 9 |- (x = v -> ((xFB) = x <-> (vFB) = v))
27	26, 10	vtoclga 2352	. . . . . . . 8 |- (v e. S -> (vFB) = v)
28		caoprmo.2	. . . . . . . . . 10 |- B e. S
29	28	elisseti 2301	. . . . . . . . 9 |- B e. _V
30	29, 15, 18	caoprcom 4986	. . . . . . . 8 |- (BFv) = (vFB)
31	27, 30	syl5eq 1940	. . . . . . 7 |- (v e. S -> (BFv) = v)
32	23, 31	sylan9eq 1948	. . . . . 6 |- (((AFw) = B /\ v e. S) -> ((AFw)Fv) = v)
33	32	ad2ant2lr 446	. . . . 5 |- (((w e. S /\ (AFw) = B) /\ (v e. S /\ (AFv) = B)) -> ((AFw)Fv) = v)
34	22, 33	eqtr3d 1927	. . . 4 |- (((w e. S /\ (AFw) = B) /\ (v e. S /\ (AFv) = B)) -> w = v)
35	34	ax-gen 1305	. . 3 |- A.v(((w e. S /\ (AFw) = B) /\ (v e. S /\ (AFv) = B)) -> w = v)
36	5, 35	mpgbir 1334	. 2 |- E*w(w e. S /\ (AFw) = B)
37		eleq1 1957	. . . . . 6 |- ((AFw) = B -> ((AFw) e. S <-> B e. S))
38	28, 37	mpbiri 211	. . . . 5 |- ((AFw) = B -> (AFw) e. S)
39		caoprmo.dom	. . . . . . 7 |- dom F = (S X. S)
40		caoprmo.3	. . . . . . 7 |- -. (/) e. S
41	14, 39, 40	ndmoprrcl 4979	. . . . . 6 |- ((AFw) e. S -> (A e. S /\ w e. S))
42	41	simprd 352	. . . . 5 |- ((AFw) e. S -> w e. S)
43	38, 42	syl 12	. . . 4 |- ((AFw) = B -> w e. S)
44	43	ancri 321	. . 3 |- ((AFw) = B -> (w e. S /\ (AFw) = B))
45	44	immoi 1814	. 2 |- (E*w(w e. S /\ (AFw) = B) -> E*w(AFw) = B)
46	36, 45	ax-mp 7	1 |- E*w(AFw) = B

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E*wmo 1772  _Vcvv 2292  (/)c0 2875   X. cxp 3984  dom cdm 3986  (class class class)co 4884

This theorem is referenced by:  recmulpq 6222

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-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

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-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886

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