Theorem gsumwspan 17383

Description: The submonoid generated by a set of elements is precisely the set of elements which can be expressed as finite products of the generator. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Hypotheses

Ref	Expression
gsumwspan.b	|- B = ( Base ` M )
gsumwspan.k	|- K = (mrCls ` (SubMnd ` M ) )

Assertion

Ref	Expression
gsumwspan	|- ( ( M e. Mnd /\ G C_ B ) -> ( K ` G ) = ran ( w e. Word G |-> ( M gsumg w ) ) )

Distinct variable groups:   w, G   w, B   w, M   w, K

Proof of Theorem gsumwspan

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

Step	Hyp	Ref	Expression
1		gsumwspan.b	. . . . . 6 |- B = ( Base ` M )
2	1	submacs 17365	. . . . 5 |- ( M e. Mnd -> (SubMnd ` M ) e. (ACS ` B ) )
3	2	acsmred 16317	. . . 4 |- ( M e. Mnd -> (SubMnd ` M ) e. (Moore ` B ) )
4	3	adantr 481	. . 3 |- ( ( M e. Mnd /\ G C_ B ) -> (SubMnd ` M ) e. (Moore ` B ) )
5		simpr 477	. . . . . . . 8 |- ( ( ( M e. Mnd /\ G C_ B ) /\ x e. G ) -> x e. G )
6	5	s1cld 13383	. . . . . . 7 |- ( ( ( M e. Mnd /\ G C_ B ) /\ x e. G ) -> <" x "> e. Word G )
7		ssel2 3598	. . . . . . . . . 10 |- ( ( G C_ B /\ x e. G ) -> x e. B )
8	7	adantll 750	. . . . . . . . 9 |- ( ( ( M e. Mnd /\ G C_ B ) /\ x e. G ) -> x e. B )
9	1	gsumws1 17376	. . . . . . . . 9 |- ( x e. B -> ( M gsumg <" x "> ) = x )
10	8, 9	syl 17	. . . . . . . 8 |- ( ( ( M e. Mnd /\ G C_ B ) /\ x e. G ) -> ( M gsumg <" x "> ) = x )
11	10	eqcomd 2628	. . . . . . 7 |- ( ( ( M e. Mnd /\ G C_ B ) /\ x e. G ) -> x = ( M gsumg <" x "> ) )
12		oveq2 6658	. . . . . . . . 9 |- ( w = <" x "> -> ( M gsumg w ) = ( M gsumg <" x "> ) )
13	12	eqeq2d 2632	. . . . . . . 8 |- ( w = <" x "> -> ( x = ( M gsumg w ) <-> x = ( M gsumg <" x "> ) ) )
14	13	rspcev 3309	. . . . . . 7 |- ( ( <" x "> e. Word G /\ x = ( M gsumg <" x "> ) ) -> E. w e. Word G x = ( M gsumg w ) )
15	6, 11, 14	syl2anc 693	. . . . . 6 |- ( ( ( M e. Mnd /\ G C_ B ) /\ x e. G ) -> E. w e. Word G x = ( M gsumg w ) )
16		vex 3203	. . . . . . 7 |- x e. _V
17		eqid 2622	. . . . . . . 8 |- ( w e. Word G |-> ( M gsumg w ) ) = ( w e. Word G |-> ( M gsumg w ) )
18	17	elrnmpt 5372	. . . . . . 7 |- ( x e. _V -> ( x e. ran ( w e. Word G |-> ( M gsumg w ) ) <-> E. w e. Word G x = ( M gsumg w ) ) )
19	16, 18	ax-mp 5	. . . . . 6 |- ( x e. ran ( w e. Word G |-> ( M gsumg w ) ) <-> E. w e. Word G x = ( M gsumg w ) )
20	15, 19	sylibr 224	. . . . 5 |- ( ( ( M e. Mnd /\ G C_ B ) /\ x e. G ) -> x e. ran ( w e. Word G |-> ( M gsumg w ) ) )
21	20	ex 450	. . . 4 |- ( ( M e. Mnd /\ G C_ B ) -> ( x e. G -> x e. ran ( w e. Word G |-> ( M gsumg w ) ) ) )
22	21	ssrdv 3609	. . 3 |- ( ( M e. Mnd /\ G C_ B ) -> G C_ ran ( w e. Word G |-> ( M gsumg w ) ) )
23		gsumwspan.k	. . . . . . . . . . 11 |- K = (mrCls ` (SubMnd ` M ) )
24	23	mrccl 16271	. . . . . . . . . 10 |- ( ( (SubMnd ` M ) e. (Moore ` B ) /\ G C_ B ) -> ( K ` G ) e. (SubMnd ` M ) )
25	3, 24	sylan 488	. . . . . . . . 9 |- ( ( M e. Mnd /\ G C_ B ) -> ( K ` G ) e. (SubMnd ` M ) )
26	25	adantr 481	. . . . . . . 8 |- ( ( ( M e. Mnd /\ G C_ B ) /\ w e. Word G ) -> ( K ` G ) e. (SubMnd ` M ) )
27	23	mrcssid 16277	. . . . . . . . . . 11 |- ( ( (SubMnd ` M ) e. (Moore ` B ) /\ G C_ B ) -> G C_ ( K ` G ) )
28	3, 27	sylan 488	. . . . . . . . . 10 |- ( ( M e. Mnd /\ G C_ B ) -> G C_ ( K ` G ) )
29		sswrd 13313	. . . . . . . . . 10 |- ( G C_ ( K ` G ) -> Word G C_ Word ( K ` G ) )
30	28, 29	syl 17	. . . . . . . . 9 |- ( ( M e. Mnd /\ G C_ B ) -> Word G C_ Word ( K ` G ) )
31	30	sselda 3603	. . . . . . . 8 |- ( ( ( M e. Mnd /\ G C_ B ) /\ w e. Word G ) -> w e. Word ( K ` G ) )
32		gsumwsubmcl 17375	. . . . . . . 8 |- ( ( ( K ` G ) e. (SubMnd ` M ) /\ w e. Word ( K ` G ) ) -> ( M gsumg w ) e. ( K ` G ) )
33	26, 31, 32	syl2anc 693	. . . . . . 7 |- ( ( ( M e. Mnd /\ G C_ B ) /\ w e. Word G ) -> ( M gsumg w ) e. ( K ` G ) )
34	33, 17	fmptd 6385	. . . . . 6 |- ( ( M e. Mnd /\ G C_ B ) -> ( w e. Word G |-> ( M gsumg w ) ) :Word G --> ( K ` G ) )
35		frn 6053	. . . . . 6 |- ( ( w e. Word G |-> ( M gsumg w ) ) :Word G --> ( K ` G ) -> ran ( w e. Word G |-> ( M gsumg w ) ) C_ ( K ` G ) )
36	34, 35	syl 17	. . . . 5 |- ( ( M e. Mnd /\ G C_ B ) -> ran ( w e. Word G |-> ( M gsumg w ) ) C_ ( K ` G ) )
37	3, 23	mrcssvd 16283	. . . . . 6 |- ( M e. Mnd -> ( K ` G ) C_ B )
38	37	adantr 481	. . . . 5 |- ( ( M e. Mnd /\ G C_ B ) -> ( K ` G ) C_ B )
39	36, 38	sstrd 3613	. . . 4 |- ( ( M e. Mnd /\ G C_ B ) -> ran ( w e. Word G |-> ( M gsumg w ) ) C_ B )
40		wrd0 13330	. . . . . 6 |- (/) e. Word G
41		eqid 2622	. . . . . . . . 9 |- ( 0g ` M ) = ( 0g ` M )
42	41	gsum0 17278	. . . . . . . 8 |- ( M gsumg (/) ) = ( 0g ` M )
43	42	eqcomi 2631	. . . . . . 7 |- ( 0g ` M ) = ( M gsumg (/) )
44	43	a1i 11	. . . . . 6 |- ( ( M e. Mnd /\ G C_ B ) -> ( 0g ` M ) = ( M gsumg (/) ) )
45		oveq2 6658	. . . . . . . 8 |- ( w = (/) -> ( M gsumg w ) = ( M gsumg (/) ) )
46	45	eqeq2d 2632	. . . . . . 7 |- ( w = (/) -> ( ( 0g ` M ) = ( M gsumg w ) <-> ( 0g ` M ) = ( M gsumg (/) ) ) )
47	46	rspcev 3309	. . . . . 6 |- ( ( (/) e. Word G /\ ( 0g ` M ) = ( M gsumg (/) ) ) -> E. w e. Word G ( 0g ` M ) = ( M gsumg w ) )
48	40, 44, 47	sylancr 695	. . . . 5 |- ( ( M e. Mnd /\ G C_ B ) -> E. w e. Word G ( 0g ` M ) = ( M gsumg w ) )
49		fvex 6201	. . . . . 6 |- ( 0g ` M ) e. _V
50	17	elrnmpt 5372	. . . . . 6 |- ( ( 0g ` M ) e. _V -> ( ( 0g ` M ) e. ran ( w e. Word G |-> ( M gsumg w ) ) <-> E. w e. Word G ( 0g ` M ) = ( M gsumg w ) ) )
51	49, 50	ax-mp 5	. . . . 5 |- ( ( 0g ` M ) e. ran ( w e. Word G |-> ( M gsumg w ) ) <-> E. w e. Word G ( 0g ` M ) = ( M gsumg w ) )
52	48, 51	sylibr 224	. . . 4 |- ( ( M e. Mnd /\ G C_ B ) -> ( 0g ` M ) e. ran ( w e. Word G |-> ( M gsumg w ) ) )
53		ccatcl 13359	. . . . . . . . 9 |- ( ( z e. Word G /\ v e. Word G ) -> ( z ++ v ) e. Word G )
54	53	adantl 482	. . . . . . . 8 |- ( ( ( M e. Mnd /\ G C_ B ) /\ ( z e. Word G /\ v e. Word G ) ) -> ( z ++ v ) e. Word G )
55		simpll 790	. . . . . . . . . 10 |- ( ( ( M e. Mnd /\ G C_ B ) /\ ( z e. Word G /\ v e. Word G ) ) -> M e. Mnd )
56		sswrd 13313	. . . . . . . . . . . 12 |- ( G C_ B -> Word G C_ Word B )
57	56	ad2antlr 763	. . . . . . . . . . 11 |- ( ( ( M e. Mnd /\ G C_ B ) /\ ( z e. Word G /\ v e. Word G ) ) -> Word G C_ Word B )
58		simprl 794	. . . . . . . . . . 11 |- ( ( ( M e. Mnd /\ G C_ B ) /\ ( z e. Word G /\ v e. Word G ) ) -> z e. Word G )
59	57, 58	sseldd 3604	. . . . . . . . . 10 |- ( ( ( M e. Mnd /\ G C_ B ) /\ ( z e. Word G /\ v e. Word G ) ) -> z e. Word B )
60		simprr 796	. . . . . . . . . . 11 |- ( ( ( M e. Mnd /\ G C_ B ) /\ ( z e. Word G /\ v e. Word G ) ) -> v e. Word G )
61	57, 60	sseldd 3604	. . . . . . . . . 10 |- ( ( ( M e. Mnd /\ G C_ B ) /\ ( z e. Word G /\ v e. Word G ) ) -> v e. Word B )
62		eqid 2622	. . . . . . . . . . 11 |- ( +g ` M ) = ( +g ` M )
63	1, 62	gsumccat 17378	. . . . . . . . . 10 |- ( ( M e. Mnd /\ z e. Word B /\ v e. Word B ) -> ( M gsumg ( z ++ v ) ) = ( ( M gsumg z ) ( +g ` M ) ( M gsumg v ) ) )
64	55, 59, 61, 63	syl3anc 1326	. . . . . . . . 9 |- ( ( ( M e. Mnd /\ G C_ B ) /\ ( z e. Word G /\ v e. Word G ) ) -> ( M gsumg ( z ++ v ) ) = ( ( M gsumg z ) ( +g ` M ) ( M gsumg v ) ) )
65	64	eqcomd 2628	. . . . . . . 8 |- ( ( ( M e. Mnd /\ G C_ B ) /\ ( z e. Word G /\ v e. Word G ) ) -> ( ( M gsumg z ) ( +g ` M ) ( M gsumg v ) ) = ( M gsumg ( z ++ v ) ) )
66		oveq2 6658	. . . . . . . . . 10 |- ( w = ( z ++ v ) -> ( M gsumg w ) = ( M gsumg ( z ++ v ) ) )
67	66	eqeq2d 2632	. . . . . . . . 9 |- ( w = ( z ++ v ) -> ( ( ( M gsumg z ) ( +g ` M ) ( M gsumg v ) ) = ( M gsumg w ) <-> ( ( M gsumg z ) ( +g ` M ) ( M gsumg v ) ) = ( M gsumg ( z ++ v ) ) ) )
68	67	rspcev 3309	. . . . . . . 8 |- ( ( ( z ++ v ) e. Word G /\ ( ( M gsumg z ) ( +g ` M ) ( M gsumg v ) ) = ( M gsumg ( z ++ v ) ) ) -> E. w e. Word G ( ( M gsumg z ) ( +g ` M ) ( M gsumg v ) ) = ( M gsumg w ) )
69	54, 65, 68	syl2anc 693	. . . . . . 7 |- ( ( ( M e. Mnd /\ G C_ B ) /\ ( z e. Word G /\ v e. Word G ) ) -> E. w e. Word G ( ( M gsumg z ) ( +g ` M ) ( M gsumg v ) ) = ( M gsumg w ) )
70		ovex 6678	. . . . . . . 8 |- ( ( M gsumg z ) ( +g ` M ) ( M gsumg v ) ) e. _V
71	17	elrnmpt 5372	. . . . . . . 8 |- ( ( ( M gsumg z ) ( +g ` M ) ( M gsumg v ) ) e. _V -> ( ( ( M gsumg z ) ( +g ` M ) ( M gsumg v ) ) e. ran ( w e. Word G |-> ( M gsumg w ) ) <-> E. w e. Word G ( ( M gsumg z ) ( +g ` M ) ( M gsumg v ) ) = ( M gsumg w ) ) )
72	70, 71	ax-mp 5	. . . . . . 7 |- ( ( ( M gsumg z ) ( +g ` M ) ( M gsumg v ) ) e. ran ( w e. Word G |-> ( M gsumg w ) ) <-> E. w e. Word G ( ( M gsumg z ) ( +g ` M ) ( M gsumg v ) ) = ( M gsumg w ) )
73	69, 72	sylibr 224	. . . . . 6 |- ( ( ( M e. Mnd /\ G C_ B ) /\ ( z e. Word G /\ v e. Word G ) ) -> ( ( M gsumg z ) ( +g ` M ) ( M gsumg v ) ) e. ran ( w e. Word G |-> ( M gsumg w ) ) )
74	73	ralrimivva 2971	. . . . 5 |- ( ( M e. Mnd /\ G C_ B ) -> A. z e. Word G A. v e. Word G ( ( M gsumg z ) ( +g ` M ) ( M gsumg v ) ) e. ran ( w e. Word G |-> ( M gsumg w ) ) )
75		oveq2 6658	. . . . . . . . 9 |- ( w = z -> ( M gsumg w ) = ( M gsumg z ) )
76	75	cbvmptv 4750	. . . . . . . 8 |- ( w e. Word G |-> ( M gsumg w ) ) = ( z e. Word G |-> ( M gsumg z ) )
77	76	rneqi 5352	. . . . . . 7 |- ran ( w e. Word G |-> ( M gsumg w ) ) = ran ( z e. Word G |-> ( M gsumg z ) )
78	77	raleqi 3142	. . . . . 6 |- ( A. x e. ran ( w e. Word G |-> ( M gsumg w ) ) A. y e. ran ( w e. Word G |-> ( M gsumg w ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsumg w ) ) <-> A. x e. ran ( z e. Word G |-> ( M gsumg z ) ) A. y e. ran ( w e. Word G |-> ( M gsumg w ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsumg w ) ) )
79		oveq2 6658	. . . . . . . . . . 11 |- ( w = v -> ( M gsumg w ) = ( M gsumg v ) )
80	79	cbvmptv 4750	. . . . . . . . . 10 |- ( w e. Word G |-> ( M gsumg w ) ) = ( v e. Word G |-> ( M gsumg v ) )
81	80	rneqi 5352	. . . . . . . . 9 |- ran ( w e. Word G |-> ( M gsumg w ) ) = ran ( v e. Word G |-> ( M gsumg v ) )
82	81	raleqi 3142	. . . . . . . 8 |- ( A. y e. ran ( w e. Word G |-> ( M gsumg w ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsumg w ) ) <-> A. y e. ran ( v e. Word G |-> ( M gsumg v ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsumg w ) ) )
83		eqid 2622	. . . . . . . . . 10 |- ( v e. Word G |-> ( M gsumg v ) ) = ( v e. Word G |-> ( M gsumg v ) )
84		oveq2 6658	. . . . . . . . . . 11 |- ( y = ( M gsumg v ) -> ( x ( +g ` M ) y ) = ( x ( +g ` M ) ( M gsumg v ) ) )
85	84	eleq1d 2686	. . . . . . . . . 10 |- ( y = ( M gsumg v ) -> ( ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsumg w ) ) <-> ( x ( +g ` M ) ( M gsumg v ) ) e. ran ( w e. Word G |-> ( M gsumg w ) ) ) )
86	83, 85	ralrnmpt 6368	. . . . . . . . 9 |- ( A. v e. Word G ( M gsumg v ) e. _V -> ( A. y e. ran ( v e. Word G |-> ( M gsumg v ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsumg w ) ) <-> A. v e. Word G ( x ( +g ` M ) ( M gsumg v ) ) e. ran ( w e. Word G |-> ( M gsumg w ) ) ) )
87		ovexd 6680	. . . . . . . . 9 |- ( v e. Word G -> ( M gsumg v ) e. _V )
88	86, 87	mprg 2926	. . . . . . . 8 |- ( A. y e. ran ( v e. Word G |-> ( M gsumg v ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsumg w ) ) <-> A. v e. Word G ( x ( +g ` M ) ( M gsumg v ) ) e. ran ( w e. Word G |-> ( M gsumg w ) ) )
89	82, 88	bitri 264	. . . . . . 7 |- ( A. y e. ran ( w e. Word G |-> ( M gsumg w ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsumg w ) ) <-> A. v e. Word G ( x ( +g ` M ) ( M gsumg v ) ) e. ran ( w e. Word G |-> ( M gsumg w ) ) )
90	89	ralbii 2980	. . . . . 6 |- ( A. x e. ran ( z e. Word G |-> ( M gsumg z ) ) A. y e. ran ( w e. Word G |-> ( M gsumg w ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsumg w ) ) <-> A. x e. ran ( z e. Word G |-> ( M gsumg z ) ) A. v e. Word G ( x ( +g ` M ) ( M gsumg v ) ) e. ran ( w e. Word G |-> ( M gsumg w ) ) )
91		eqid 2622	. . . . . . . 8 |- ( z e. Word G |-> ( M gsumg z ) ) = ( z e. Word G |-> ( M gsumg z ) )
92		oveq1 6657	. . . . . . . . . 10 |- ( x = ( M gsumg z ) -> ( x ( +g ` M ) ( M gsumg v ) ) = ( ( M gsumg z ) ( +g ` M ) ( M gsumg v ) ) )
93	92	eleq1d 2686	. . . . . . . . 9 |- ( x = ( M gsumg z ) -> ( ( x ( +g ` M ) ( M gsumg v ) ) e. ran ( w e. Word G |-> ( M gsumg w ) ) <-> ( ( M gsumg z ) ( +g ` M ) ( M gsumg v ) ) e. ran ( w e. Word G |-> ( M gsumg w ) ) ) )
94	93	ralbidv 2986	. . . . . . . 8 |- ( x = ( M gsumg z ) -> ( A. v e. Word G ( x ( +g ` M ) ( M gsumg v ) ) e. ran ( w e. Word G |-> ( M gsumg w ) ) <-> A. v e. Word G ( ( M gsumg z ) ( +g ` M ) ( M gsumg v ) ) e. ran ( w e. Word G |-> ( M gsumg w ) ) ) )
95	91, 94	ralrnmpt 6368	. . . . . . 7 |- ( A. z e. Word G ( M gsumg z ) e. _V -> ( A. x e. ran ( z e. Word G |-> ( M gsumg z ) ) A. v e. Word G ( x ( +g ` M ) ( M gsumg v ) ) e. ran ( w e. Word G |-> ( M gsumg w ) ) <-> A. z e. Word G A. v e. Word G ( ( M gsumg z ) ( +g ` M ) ( M gsumg v ) ) e. ran ( w e. Word G |-> ( M gsumg w ) ) ) )
96		ovexd 6680	. . . . . . 7 |- ( z e. Word G -> ( M gsumg z ) e. _V )
97	95, 96	mprg 2926	. . . . . 6 |- ( A. x e. ran ( z e. Word G |-> ( M gsumg z ) ) A. v e. Word G ( x ( +g ` M ) ( M gsumg v ) ) e. ran ( w e. Word G |-> ( M gsumg w ) ) <-> A. z e. Word G A. v e. Word G ( ( M gsumg z ) ( +g ` M ) ( M gsumg v ) ) e. ran ( w e. Word G |-> ( M gsumg w ) ) )
98	78, 90, 97	3bitri 286	. . . . 5 |- ( A. x e. ran ( w e. Word G |-> ( M gsumg w ) ) A. y e. ran ( w e. Word G |-> ( M gsumg w ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsumg w ) ) <-> A. z e. Word G A. v e. Word G ( ( M gsumg z ) ( +g ` M ) ( M gsumg v ) ) e. ran ( w e. Word G |-> ( M gsumg w ) ) )
99	74, 98	sylibr 224	. . . 4 |- ( ( M e. Mnd /\ G C_ B ) -> A. x e. ran ( w e. Word G |-> ( M gsumg w ) ) A. y e. ran ( w e. Word G |-> ( M gsumg w ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsumg w ) ) )
100	1, 41, 62	issubm 17347	. . . . 5 |- ( M e. Mnd -> ( ran ( w e. Word G |-> ( M gsumg w ) ) e. (SubMnd ` M ) <-> ( ran ( w e. Word G |-> ( M gsumg w ) ) C_ B /\ ( 0g ` M ) e. ran ( w e. Word G |-> ( M gsumg w ) ) /\ A. x e. ran ( w e. Word G |-> ( M gsumg w ) ) A. y e. ran ( w e. Word G |-> ( M gsumg w ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsumg w ) ) ) ) )
101	100	adantr 481	. . . 4 |- ( ( M e. Mnd /\ G C_ B ) -> ( ran ( w e. Word G |-> ( M gsumg w ) ) e. (SubMnd ` M ) <-> ( ran ( w e. Word G |-> ( M gsumg w ) ) C_ B /\ ( 0g ` M ) e. ran ( w e. Word G |-> ( M gsumg w ) ) /\ A. x e. ran ( w e. Word G |-> ( M gsumg w ) ) A. y e. ran ( w e. Word G |-> ( M gsumg w ) ) ( x ( +g ` M ) y ) e. ran ( w e. Word G |-> ( M gsumg w ) ) ) ) )
102	39, 52, 99, 101	mpbir3and 1245	. . 3 |- ( ( M e. Mnd /\ G C_ B ) -> ran ( w e. Word G |-> ( M gsumg w ) ) e. (SubMnd ` M ) )
103	23	mrcsscl 16280	. . 3 |- ( ( (SubMnd ` M ) e. (Moore ` B ) /\ G C_ ran ( w e. Word G |-> ( M gsumg w ) ) /\ ran ( w e. Word G |-> ( M gsumg w ) ) e. (SubMnd ` M ) ) -> ( K ` G ) C_ ran ( w e. Word G |-> ( M gsumg w ) ) )
104	4, 22, 102, 103	syl3anc 1326	. 2 |- ( ( M e. Mnd /\ G C_ B ) -> ( K ` G ) C_ ran ( w e. Word G |-> ( M gsumg w ) ) )
105	104, 36	eqssd 3620	1 |- ( ( M e. Mnd /\ G C_ B ) -> ( K ` G ) = ran ( w e. Word G |-> ( M gsumg w ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   |-> cmpt 4729   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650  Word cword 13291   ++ cconcat 13293   <"cs1 13294   Basecbs 15857   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101  Moorecmre 16242  mrClscmrc 16243   Mndcmnd 17294  SubMndcsubmnd 17334

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336

This theorem is referenced by:  psgneldm2  17924  psgnfitr  17937