Theorem caovmo 5646

Description: Uniqueness of inverse element in commutative, associative operation with identity. Remark in proof of Proposition 9-2.4 of [Gleason] p. 119. (Contributed by set.mm contributors, 4-Mar-1996.)

Hypotheses

Ref	Expression
caovmo.1	⊢ A ∈ V
caovmo.2	⊢ B ∈ S
caovmo.dom	⊢ dom F = (S × S)
caovmo.3	⊢ ¬ ∅ ∈ S
caovmo.com	⊢ (xFy) = (yFx)
caovmo.ass	⊢ ((xFy)Fz) = (xF(yFz))
caovmo.id	⊢ (x ∈ S → (xFB) = x)

Assertion

Ref	Expression
caovmo	⊢ ∃*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 caovmo

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2413	. . . . 5 ⊢ (w = v → (w ∈ S ↔ v ∈ S))
2		oveq2 5532	. . . . . 6 ⊢ (w = v → (AFw) = (AFv))
3	2	eqeq1d 2361	. . . . 5 ⊢ (w = v → ((AFw) = B ↔ (AFv) = B))
4	1, 3	anbi12d 691	. . . 4 ⊢ (w = v → ((w ∈ S ∧ (AFw) = B) ↔ (v ∈ S ∧ (AFv) = B)))
5	4	mo4 2237	. . 3 ⊢ (∃*w(w ∈ S ∧ (AFw) = B) ↔ ∀w∀v(((w ∈ S ∧ (AFw) = B) ∧ (v ∈ S ∧ (AFv) = B)) → w = v))
6		caovmo.1	. . . . . . . . 9 ⊢ A ∈ V
7		vex 2863	. . . . . . . . 9 ⊢ w ∈ V
8		vex 2863	. . . . . . . . 9 ⊢ v ∈ V
9		caovmo.ass	. . . . . . . . 9 ⊢ ((xFy)Fz) = (xF(yFz))
10	6, 7, 8, 9	caovass 5628	. . . . . . . 8 ⊢ ((AFw)Fv) = (AF(wFv))
11		caovmo.com	. . . . . . . . 9 ⊢ (xFy) = (yFx)
12	6, 7, 8, 11, 9	caov12 5637	. . . . . . . 8 ⊢ (AF(wFv)) = (wF(AFv))
13	10, 12	eqtri 2373	. . . . . . 7 ⊢ ((AFw)Fv) = (wF(AFv))
14		oveq2 5532	. . . . . . . 8 ⊢ ((AFv) = B → (wF(AFv)) = (wFB))
15		oveq1 5531	. . . . . . . . . 10 ⊢ (x = w → (xFB) = (wFB))
16		id 19	. . . . . . . . . 10 ⊢ (x = w → x = w)
17	15, 16	eqeq12d 2367	. . . . . . . . 9 ⊢ (x = w → ((xFB) = x ↔ (wFB) = w))
18		caovmo.id	. . . . . . . . 9 ⊢ (x ∈ S → (xFB) = x)
19	17, 18	vtoclga 2921	. . . . . . . 8 ⊢ (w ∈ S → (wFB) = w)
20	14, 19	sylan9eqr 2407	. . . . . . 7 ⊢ ((w ∈ S ∧ (AFv) = B) → (wF(AFv)) = w)
21	13, 20	syl5eq 2397	. . . . . 6 ⊢ ((w ∈ S ∧ (AFv) = B) → ((AFw)Fv) = w)
22	21	ad2ant2rl 729	. . . . 5 ⊢ (((w ∈ S ∧ (AFw) = B) ∧ (v ∈ S ∧ (AFv) = B)) → ((AFw)Fv) = w)
23		oveq1 5531	. . . . . . 7 ⊢ ((AFw) = B → ((AFw)Fv) = (BFv))
24		caovmo.2	. . . . . . . . . 10 ⊢ B ∈ S
25	24	elexi 2869	. . . . . . . . 9 ⊢ B ∈ V
26	25, 8, 11	caovcom 5626	. . . . . . . 8 ⊢ (BFv) = (vFB)
27		oveq1 5531	. . . . . . . . . 10 ⊢ (x = v → (xFB) = (vFB))
28		id 19	. . . . . . . . . 10 ⊢ (x = v → x = v)
29	27, 28	eqeq12d 2367	. . . . . . . . 9 ⊢ (x = v → ((xFB) = x ↔ (vFB) = v))
30	29, 18	vtoclga 2921	. . . . . . . 8 ⊢ (v ∈ S → (vFB) = v)
31	26, 30	syl5eq 2397	. . . . . . 7 ⊢ (v ∈ S → (BFv) = v)
32	23, 31	sylan9eq 2405	. . . . . 6 ⊢ (((AFw) = B ∧ v ∈ S) → ((AFw)Fv) = v)
33	32	ad2ant2lr 728	. . . . 5 ⊢ (((w ∈ S ∧ (AFw) = B) ∧ (v ∈ S ∧ (AFv) = B)) → ((AFw)Fv) = v)
34	22, 33	eqtr3d 2387	. . . 4 ⊢ (((w ∈ S ∧ (AFw) = B) ∧ (v ∈ S ∧ (AFv) = B)) → w = v)
35	34	ax-gen 1546	. . 3 ⊢ ∀v(((w ∈ S ∧ (AFw) = B) ∧ (v ∈ S ∧ (AFv) = B)) → w = v)
36	5, 35	mpgbir 1550	. 2 ⊢ ∃*w(w ∈ S ∧ (AFw) = B)
37		eleq1 2413	. . . . . 6 ⊢ ((AFw) = B → ((AFw) ∈ S ↔ B ∈ S))
38	24, 37	mpbiri 224	. . . . 5 ⊢ ((AFw) = B → (AFw) ∈ S)
39		caovmo.dom	. . . . . . 7 ⊢ dom F = (S × S)
40		caovmo.3	. . . . . . 7 ⊢ ¬ ∅ ∈ S
41	7, 39, 40	ndmovrcl 5617	. . . . . 6 ⊢ ((AFw) ∈ S → (A ∈ S ∧ w ∈ S))
42	41	simprd 449	. . . . 5 ⊢ ((AFw) ∈ S → w ∈ S)
43	38, 42	syl 15	. . . 4 ⊢ ((AFw) = B → w ∈ S)
44	43	ancri 535	. . 3 ⊢ ((AFw) = B → (w ∈ S ∧ (AFw) = B))
45	44	moimi 2251	. 2 ⊢ (∃*w(w ∈ S ∧ (AFw) = B) → ∃*w(AFw) = B)
46	36, 45	ax-mp 5	1 ⊢ ∃*w(AFw) = B

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  Vcvv 2860  ∅c0 3551   × cxp 4771  dom cdm 4773  (class class class)co 5526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-ima 4728  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-fv 4796  df-ov 5527

This theorem is referenced by: (None)