Theorem caoprmo 4010

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 1510	. . . . 5 |- (w = v -> (w e. S <-> v e. S))
2		opreq2 3908	. . . . . 6 |- (w = v -> (AFw) = (AFv))
3	2	eqeq1d 1459	. . . . 5 |- (w = v -> ((AFw) = B <-> (AFv) = B))
4	1, 3	anbi12d 626	. . . 4 |- (w = v -> ((w e. S /\ (AFw) = B) <-> (v e. S /\ (AFv) = B)))
5	4	mo4 1380	. . 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 3908	. . . . . . . 8 |- ((AFv) = B -> (wF(AFv)) = (wFB))
7		opreq1 3907	. . . . . . . . . 10 |- (x = w -> (xFB) = (wFB))
8		id 59	. . . . . . . . . 10 |- (x = w -> x = w)
9	7, 8	eqeq12d 1465	. . . . . . . . 9 |- (x = w -> ((xFB) = x <-> (wFB) = w))
10		caoprmo.id	. . . . . . . . 9 |- (x e. S -> (xFB) = x)
11	9, 10	vtoclga 1827	. . . . . . . 8 |- (w e. S -> (wFB) = w)
12	6, 11	sylan9eqr 1505	. . . . . . 7 |- ((w e. S /\ (AFv) = B) -> (wF(AFv)) = w)
13		caoprmo.1	. . . . . . . . 9 |- A e. V
14		visset 1788	. . . . . . . . 9 |- w e. V
15		visset 1788	. . . . . . . . 9 |- v e. V
16		caoprmo.ass	. . . . . . . . 9 |- ((xFy)Fz) = (xF(yFz))
17	13, 14, 15, 16	caoprass 3994	. . . . . . . 8 |- ((AFw)Fv) = (AF(wFv))
18		caoprmo.com	. . . . . . . . 9 |- (xFy) = (yFx)
19	13, 14, 15, 18, 16	caopr12 4001	. . . . . . . 8 |- (AF(wFv)) = (wF(AFv))
20	17, 19	eqtr 1471	. . . . . . 7 |- ((AFw)Fv) = (wF(AFv))
21	12, 20	syl5eq 1495	. . . . . 6 |- ((w e. S /\ (AFv) = B) -> ((AFw)Fv) = w)
22	21	ad2ant2rl 411	. . . . 5 |- (((w e. S /\ (AFw) = B) /\ (v e. S /\ (AFv) = B)) -> ((AFw)Fv) = w)
23		opreq1 3907	. . . . . . 7 |- ((AFw) = B -> ((AFw)Fv) = (BFv))
24		opreq1 3907	. . . . . . . . . 10 |- (x = v -> (xFB) = (vFB))
25		id 59	. . . . . . . . . 10 |- (x = v -> x = v)
26	24, 25	eqeq12d 1465	. . . . . . . . 9 |- (x = v -> ((xFB) = x <-> (vFB) = v))
27	26, 10	vtoclga 1827	. . . . . . . 8 |- (v e. S -> (vFB) = v)
28		caoprmo.2	. . . . . . . . . 10 |- B e. S
29	28	elisseti 1793	. . . . . . . . 9 |- B e. V
30	29, 15, 18	caoprcom 3993	. . . . . . . 8 |- (BFv) = (vFB)
31	27, 30	syl5eq 1495	. . . . . . 7 |- (v e. S -> (BFv) = v)
32	23, 31	sylan9eq 1503	. . . . . 6 |- (((AFw) = B /\ v e. S) -> ((AFw)Fv) = v)
33	32	ad2ant2lr 410	. . . . 5 |- (((w e. S /\ (AFw) = B) /\ (v e. S /\ (AFv) = B)) -> ((AFw)Fv) = v)
34	22, 33	eqtr3d 1485	. . . 4 |- (((w e. S /\ (AFw) = B) /\ (v e. S /\ (AFv) = B)) -> w = v)
35	34	ax-gen 955	. . 3 |- A.v(((w e. S /\ (AFw) = B) /\ (v e. S /\ (AFv) = B)) -> w = v)
36	5, 35	mpgbir 964	. 2 |- E*w(w e. S /\ (AFw) = B)
37		eleq1 1510	. . . . . 6 |- ((AFw) = B -> ((AFw) e. S <-> B e. S))
38	28, 37	mpbiri 194	. . . . 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 3986	. . . . . 6 |- ((AFw) e. S -> (A e. S /\ w e. S))
42	41	pm3.27d 325	. . . . 5 |- ((AFw) e. S -> w e. S)
43	38, 42	syl 10	. . . 4 |- ((AFw) = B -> w e. S)
44	43	ancri 297	. . 3 |- ((AFw) = B -> (w e. S /\ (AFw) = B))
45	44	immoi 1395	. 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 223  A.wal 950   = wceq 1099   e. wcel 1105  E*wmo 1358  Vcvv 1786  (/)c0 2251   X. cxp 3131  dom cdm 3133  (class class class)co 3902

This theorem is referenced by:  recmulpq 4993

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 774  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-rex 1626  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-op 2387  df-uni 2472  df-br 2588  df-opab 2635  df-xp 3147  df-cnv 3149  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fv 3161  df-opr 3904

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