Theorem grposnOLD 33681

Description: The group operation for the singleton group. Obsolete, use grp1 17522. instead (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
grposnOLD.1	|- A e. _V

Assertion

Ref	Expression
grposnOLD	|- { <. <. A , A >. , A >. } e. GrpOp

Proof of Theorem grposnOLD

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 4908	. 2 |- { A } e. _V
2		opex 4932	. . . . 5 |- <. A , A >. e. _V
3		grposnOLD.1	. . . . 5 |- A e. _V
4	2, 3	f1osn 6176	. . . 4 |- { <. <. A , A >. , A >. } : { <. A , A >. } -1-1-onto-> { A }
5		f1of 6137	. . . 4 |- ( { <. <. A , A >. , A >. } : { <. A , A >. } -1-1-onto-> { A } -> { <. <. A , A >. , A >. } : { <. A , A >. } --> { A } )
6	4, 5	ax-mp 5	. . 3 |- { <. <. A , A >. , A >. } : { <. A , A >. } --> { A }
7	3, 3	xpsn 6407	. . . 4 |- ( { A } X. { A } ) = { <. A , A >. }
8	7	feq2i 6037	. . 3 |- ( { <. <. A , A >. , A >. } : ( { A } X. { A } ) --> { A } <-> { <. <. A , A >. , A >. } : { <. A , A >. } --> { A } )
9	6, 8	mpbir 221	. 2 |- { <. <. A , A >. , A >. } : ( { A } X. { A } ) --> { A }
10		velsn 4193	. . 3 |- ( x e. { A } <-> x = A )
11		velsn 4193	. . 3 |- ( y e. { A } <-> y = A )
12		velsn 4193	. . 3 |- ( z e. { A } <-> z = A )
13		oveq2 6658	. . . . . 6 |- ( z = A -> ( ( x { <. <. A , A >. , A >. } y ) { <. <. A , A >. , A >. } z ) = ( ( x { <. <. A , A >. , A >. } y ) { <. <. A , A >. , A >. } A ) )
14		oveq1 6657	. . . . . . . . 9 |- ( x = A -> ( x { <. <. A , A >. , A >. } y ) = ( A { <. <. A , A >. , A >. } y ) )
15		oveq2 6658	. . . . . . . . . 10 |- ( y = A -> ( A { <. <. A , A >. , A >. } y ) = ( A { <. <. A , A >. , A >. } A ) )
16		df-ov 6653	. . . . . . . . . . 11 |- ( A { <. <. A , A >. , A >. } A ) = ( { <. <. A , A >. , A >. } ` <. A , A >. )
17	2, 3	fvsn 6446	. . . . . . . . . . 11 |- ( { <. <. A , A >. , A >. } ` <. A , A >. ) = A
18	16, 17	eqtri 2644	. . . . . . . . . 10 |- ( A { <. <. A , A >. , A >. } A ) = A
19	15, 18	syl6eq 2672	. . . . . . . . 9 |- ( y = A -> ( A { <. <. A , A >. , A >. } y ) = A )
20	14, 19	sylan9eq 2676	. . . . . . . 8 |- ( ( x = A /\ y = A ) -> ( x { <. <. A , A >. , A >. } y ) = A )
21	20	oveq1d 6665	. . . . . . 7 |- ( ( x = A /\ y = A ) -> ( ( x { <. <. A , A >. , A >. } y ) { <. <. A , A >. , A >. } A ) = ( A { <. <. A , A >. , A >. } A ) )
22	21, 18	syl6eq 2672	. . . . . 6 |- ( ( x = A /\ y = A ) -> ( ( x { <. <. A , A >. , A >. } y ) { <. <. A , A >. , A >. } A ) = A )
23	13, 22	sylan9eqr 2678	. . . . 5 |- ( ( ( x = A /\ y = A ) /\ z = A ) -> ( ( x { <. <. A , A >. , A >. } y ) { <. <. A , A >. , A >. } z ) = A )
24	23	3impa 1259	. . . 4 |- ( ( x = A /\ y = A /\ z = A ) -> ( ( x { <. <. A , A >. , A >. } y ) { <. <. A , A >. , A >. } z ) = A )
25		oveq1 6657	. . . . . 6 |- ( x = A -> ( x { <. <. A , A >. , A >. } ( y { <. <. A , A >. , A >. } z ) ) = ( A { <. <. A , A >. , A >. } ( y { <. <. A , A >. , A >. } z ) ) )
26		oveq1 6657	. . . . . . . . 9 |- ( y = A -> ( y { <. <. A , A >. , A >. } z ) = ( A { <. <. A , A >. , A >. } z ) )
27		oveq2 6658	. . . . . . . . . 10 |- ( z = A -> ( A { <. <. A , A >. , A >. } z ) = ( A { <. <. A , A >. , A >. } A ) )
28	27, 18	syl6eq 2672	. . . . . . . . 9 |- ( z = A -> ( A { <. <. A , A >. , A >. } z ) = A )
29	26, 28	sylan9eq 2676	. . . . . . . 8 |- ( ( y = A /\ z = A ) -> ( y { <. <. A , A >. , A >. } z ) = A )
30	29	oveq2d 6666	. . . . . . 7 |- ( ( y = A /\ z = A ) -> ( A { <. <. A , A >. , A >. } ( y { <. <. A , A >. , A >. } z ) ) = ( A { <. <. A , A >. , A >. } A ) )
31	30, 18	syl6eq 2672	. . . . . 6 |- ( ( y = A /\ z = A ) -> ( A { <. <. A , A >. , A >. } ( y { <. <. A , A >. , A >. } z ) ) = A )
32	25, 31	sylan9eq 2676	. . . . 5 |- ( ( x = A /\ ( y = A /\ z = A ) ) -> ( x { <. <. A , A >. , A >. } ( y { <. <. A , A >. , A >. } z ) ) = A )
33	32	3impb 1260	. . . 4 |- ( ( x = A /\ y = A /\ z = A ) -> ( x { <. <. A , A >. , A >. } ( y { <. <. A , A >. , A >. } z ) ) = A )
34	24, 33	eqtr4d 2659	. . 3 |- ( ( x = A /\ y = A /\ z = A ) -> ( ( x { <. <. A , A >. , A >. } y ) { <. <. A , A >. , A >. } z ) = ( x { <. <. A , A >. , A >. } ( y { <. <. A , A >. , A >. } z ) ) )
35	10, 11, 12, 34	syl3anb 1369	. 2 |- ( ( x e. { A } /\ y e. { A } /\ z e. { A } ) -> ( ( x { <. <. A , A >. , A >. } y ) { <. <. A , A >. , A >. } z ) = ( x { <. <. A , A >. , A >. } ( y { <. <. A , A >. , A >. } z ) ) )
36	3	snid 4208	. 2 |- A e. { A }
37		oveq2 6658	. . . . 5 |- ( x = A -> ( A { <. <. A , A >. , A >. } x ) = ( A { <. <. A , A >. , A >. } A ) )
38	37, 18	syl6eq 2672	. . . 4 |- ( x = A -> ( A { <. <. A , A >. , A >. } x ) = A )
39		id 22	. . . 4 |- ( x = A -> x = A )
40	38, 39	eqtr4d 2659	. . 3 |- ( x = A -> ( A { <. <. A , A >. , A >. } x ) = x )
41	10, 40	sylbi 207	. 2 |- ( x e. { A } -> ( A { <. <. A , A >. , A >. } x ) = x )
42	36	a1i 11	. 2 |- ( x e. { A } -> A e. { A } )
43	10, 38	sylbi 207	. 2 |- ( x e. { A } -> ( A { <. <. A , A >. , A >. } x ) = A )
44	1, 9, 35, 36, 41, 42, 43	isgrpoi 27352	1 |- { <. <. A , A >. , A >. } e. GrpOp

Syntax hints:   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   <.cop 4183   X. cxp 5112   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   GrpOpcgr 27343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-grpo 27347

This theorem is referenced by:  gidsn  33751