Theorem caoprmo 3084

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 ∈ V
caoprmo.2	⊢ B ∈ S
caoprmo.dom	⊢ dom F = (S × S)
caoprmo.3	⊢ ¬ ∅ ∈ S
caoprmo.com	⊢ (xFy) = (yFx)
caoprmo.ass	⊢ ((xFy)Fz) = (xF(yFz))
caoprmo.id	⊢ (x ∈ S → (xFB) = x)

Assertion

Ref	Expression
caoprmo	⊢ ∃*w(AFw) = B

Distinct variable group(s):   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 1149	. . . . . 6 ⊢ (w = v → (w ∈ S ↔ v ∈ S))
2		opreq2 3007	. . . . . . 7 ⊢ (w = v → (AFw) = (AFv))
3	2	cleq1d 1109	. . . . . 6 ⊢ (w = v → ((AFw) = B ↔ (AFv) = B))
4	1, 3	anbi12d 476	. . . . 5 ⊢ (w = v → ((w ∈ S ∧ (AFw) = B) ↔ (v ∈ S ∧ (AFv) = B)))
5	4	mo4 1029	. . . 4 ⊢ (∃*w(w ∈ S ∧ (AFw) = B) ↔ ∀w∀v(((w ∈ S ∧ (AFw) = B) ∧ (v ∈ S ∧ (AFv) = B)) → w = v))
6		opreq2 3007	. . . . . . . . . 10 ⊢ ((AFv) = B → (wF(AFv)) = (wFB))
7		opreq1 3006	. . . . . . . . . . . 12 ⊢ (x = w → (xFB) = (wFB))
8		id 9	. . . . . . . . . . . 12 ⊢ (x = w → x = w)
9	7, 8	cleq12d 1115	. . . . . . . . . . 11 ⊢ (x = w → ((xFB) = x ↔ (wFB) = w))
10		caoprmo.id	. . . . . . . . . . 11 ⊢ (x ∈ S → (xFB) = x)
11	9, 10	vtoclga 1387	. . . . . . . . . 10 ⊢ (w ∈ S → (wFB) = w)
12	6, 11	sylan9eqr 1145	. . . . . . . . 9 ⊢ ((w ∈ S ∧ (AFv) = B) → (wF(AFv)) = w)
13		caoprmo.1	. . . . . . . . . . 11 ⊢ A ∈ V
14		visset 1350	. . . . . . . . . . 11 ⊢ w ∈ V
15		visset 1350	. . . . . . . . . . 11 ⊢ v ∈ V
16		caoprmo.ass	. . . . . . . . . . 11 ⊢ ((xFy)Fz) = (xF(yFz))
17	13, 14, 15, 16	caoprass 3068	. . . . . . . . . 10 ⊢ ((AFw)Fv) = (AF(wFv))
18		caoprmo.com	. . . . . . . . . . 11 ⊢ (xFy) = (yFx)
19	13, 14, 15, 18, 16	caopr12 3075	. . . . . . . . . 10 ⊢ (AF(wFv)) = (wF(AFv))
20	17, 19	eqtr 1119	. . . . . . . . 9 ⊢ ((AFw)Fv) = (wF(AFv))
21	12, 20	syl5eq 1136	. . . . . . . 8 ⊢ ((w ∈ S ∧ (AFv) = B) → ((AFw)Fv) = w)
22	21	adantrl 311	. . . . . . 7 ⊢ ((w ∈ S ∧ (v ∈ S ∧ (AFv) = B)) → ((AFw)Fv) = w)
23	22	adantlr 310	. . . . . 6 ⊢ (((w ∈ S ∧ (AFw) = B) ∧ (v ∈ S ∧ (AFv) = B)) → ((AFw)Fv) = w)
24		opreq1 3006	. . . . . . . . 9 ⊢ ((AFw) = B → ((AFw)Fv) = (BFv))
25		opreq1 3006	. . . . . . . . . . . 12 ⊢ (x = v → (xFB) = (vFB))
26		id 9	. . . . . . . . . . . 12 ⊢ (x = v → x = v)
27	25, 26	cleq12d 1115	. . . . . . . . . . 11 ⊢ (x = v → ((xFB) = x ↔ (vFB) = v))
28	27, 10	vtoclga 1387	. . . . . . . . . 10 ⊢ (v ∈ S → (vFB) = v)
29		caoprmo.2	. . . . . . . . . . . 12 ⊢ B ∈ S
30	29	elisseti 1355	. . . . . . . . . . 11 ⊢ B ∈ V
31	30, 15, 18	caoprcom 3067	. . . . . . . . . 10 ⊢ (BFv) = (vFB)
32	28, 31	syl5eq 1136	. . . . . . . . 9 ⊢ (v ∈ S → (BFv) = v)
33	24, 32	sylan9eq 1144	. . . . . . . 8 ⊢ (((AFw) = B ∧ v ∈ S) → ((AFw)Fv) = v)
34	33	adantrr 312	. . . . . . 7 ⊢ (((AFw) = B ∧ (v ∈ S ∧ (AFv) = B)) → ((AFw)Fv) = v)
35	34	adantll 309	. . . . . 6 ⊢ (((w ∈ S ∧ (AFw) = B) ∧ (v ∈ S ∧ (AFv) = B)) → ((AFw)Fv) = v)
36	23, 35	eqtr3d 1130	. . . . 5 ⊢ (((w ∈ S ∧ (AFw) = B) ∧ (v ∈ S ∧ (AFv) = B)) → w = v)
37	36	ax-gen 677	. . . 4 ⊢ ∀v(((w ∈ S ∧ (AFw) = B) ∧ (v ∈ S ∧ (AFv) = B)) → w = v)
38	5, 37	mpgbir 686	. . 3 ⊢ ∃*w(w ∈ S ∧ (AFw) = B)
39		immo 1043	. . 3 ⊢ (∀w((AFw) = B → (w ∈ S ∧ (AFw) = B)) → (∃*w(w ∈ S ∧ (AFw) = B) → ∃*w(AFw) = B))
40	38, 39	mpi 44	. 2 ⊢ (∀w((AFw) = B → (w ∈ S ∧ (AFw) = B)) → ∃*w(AFw) = B)
41		eleq1 1149	. . . . 5 ⊢ ((AFw) = B → ((AFw) ∈ S ↔ B ∈ S))
42	29, 41	mpbiri 169	. . . 4 ⊢ ((AFw) = B → (AFw) ∈ S)
43		caoprmo.dom	. . . . . 6 ⊢ dom F = (S × S)
44		caoprmo.3	. . . . . 6 ⊢ ¬ ∅ ∈ S
45	14, 43, 44	ndmoprrcl 3060	. . . . 5 ⊢ ((AFw) ∈ S → (A ∈ S ∧ w ∈ S))
46	45	pm3.27d 262	. . . 4 ⊢ ((AFw) ∈ S → w ∈ S)
47	42, 46	syl 12	. . 3 ⊢ ((AFw) = B → w ∈ S)
48	47	ancri 245	. 2 ⊢ ((AFw) = B → (w ∈ S ∧ (AFw) = B))
49	40, 48	mpg 684	1 ⊢ ∃*w(AFw) = B

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∀wal 672   = weq 797  ∃*wmo 1008   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707   × cxp 2408  dom cdm 2410  (class class class)co 3001

This theorem is referenced by:  recmulpq 3864

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003

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