Theorem gsumccat 17378

Description: Homomorphic property of composites. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.)

Hypotheses

Ref	Expression
gsumwcl.b	|- B = ( Base ` G )
gsumccat.p	|- .+ = ( +g ` G )

Assertion

Ref	Expression
gsumccat	|- ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) -> ( G gsumg ( W ++ X ) ) = ( ( G gsumg W ) .+ ( G gsumg X ) ) )

Proof of Theorem gsumccat

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

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . 4 |- ( W = (/) -> ( W ++ X ) = ( (/) ++ X ) )
2	1	oveq2d 6666	. . 3 |- ( W = (/) -> ( G gsumg ( W ++ X ) ) = ( G gsumg ( (/) ++ X ) ) )
3		oveq2 6658	. . . . 5 |- ( W = (/) -> ( G gsumg W ) = ( G gsumg (/) ) )
4		eqid 2622	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
5	4	gsum0 17278	. . . . 5 |- ( G gsumg (/) ) = ( 0g ` G )
6	3, 5	syl6eq 2672	. . . 4 |- ( W = (/) -> ( G gsumg W ) = ( 0g ` G ) )
7	6	oveq1d 6665	. . 3 |- ( W = (/) -> ( ( G gsumg W ) .+ ( G gsumg X ) ) = ( ( 0g ` G ) .+ ( G gsumg X ) ) )
8	2, 7	eqeq12d 2637	. 2 |- ( W = (/) -> ( ( G gsumg ( W ++ X ) ) = ( ( G gsumg W ) .+ ( G gsumg X ) ) <-> ( G gsumg ( (/) ++ X ) ) = ( ( 0g ` G ) .+ ( G gsumg X ) ) ) )
9		oveq2 6658	. . . . 5 |- ( X = (/) -> ( W ++ X ) = ( W ++ (/) ) )
10	9	oveq2d 6666	. . . 4 |- ( X = (/) -> ( G gsumg ( W ++ X ) ) = ( G gsumg ( W ++ (/) ) ) )
11		oveq2 6658	. . . . . 6 |- ( X = (/) -> ( G gsumg X ) = ( G gsumg (/) ) )
12	11, 5	syl6eq 2672	. . . . 5 |- ( X = (/) -> ( G gsumg X ) = ( 0g ` G ) )
13	12	oveq2d 6666	. . . 4 |- ( X = (/) -> ( ( G gsumg W ) .+ ( G gsumg X ) ) = ( ( G gsumg W ) .+ ( 0g ` G ) ) )
14	10, 13	eqeq12d 2637	. . 3 |- ( X = (/) -> ( ( G gsumg ( W ++ X ) ) = ( ( G gsumg W ) .+ ( G gsumg X ) ) <-> ( G gsumg ( W ++ (/) ) ) = ( ( G gsumg W ) .+ ( 0g ` G ) ) ) )
15		gsumwcl.b	. . . . . 6 |- B = ( Base ` G )
16		gsumccat.p	. . . . . 6 |- .+ = ( +g ` G )
17		simpl1 1064	. . . . . 6 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> G e. Mnd )
18		lennncl 13325	. . . . . . . . . . 11 |- ( ( W e. Word B /\ W =/= (/) ) -> ( # ` W ) e. NN )
19	18	3ad2antl2 1224	. . . . . . . . . 10 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ W =/= (/) ) -> ( # ` W ) e. NN )
20	19	adantrr 753	. . . . . . . . 9 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( # ` W ) e. NN )
21		lennncl 13325	. . . . . . . . . . 11 |- ( ( X e. Word B /\ X =/= (/) ) -> ( # ` X ) e. NN )
22	21	3ad2antl3 1225	. . . . . . . . . 10 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ X =/= (/) ) -> ( # ` X ) e. NN )
23	22	adantrl 752	. . . . . . . . 9 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( # ` X ) e. NN )
24	20, 23	nnaddcld 11067	. . . . . . . 8 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( ( # ` W ) + ( # ` X ) ) e. NN )
25		nnm1nn0 11334	. . . . . . . 8 |- ( ( ( # ` W ) + ( # ` X ) ) e. NN -> ( ( ( # ` W ) + ( # ` X ) ) - 1 ) e. NN0 )
26	24, 25	syl 17	. . . . . . 7 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( ( ( # ` W ) + ( # ` X ) ) - 1 ) e. NN0 )
27		nn0uz 11722	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
28	26, 27	syl6eleq 2711	. . . . . 6 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( ( ( # ` W ) + ( # ` X ) ) - 1 ) e. ( ZZ>= ` 0 ) )
29		simpl2 1065	. . . . . . . . 9 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> W e. Word B )
30		simpl3 1066	. . . . . . . . 9 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> X e. Word B )
31		ccatcl 13359	. . . . . . . . 9 |- ( ( W e. Word B /\ X e. Word B ) -> ( W ++ X ) e. Word B )
32	29, 30, 31	syl2anc 693	. . . . . . . 8 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( W ++ X ) e. Word B )
33		wrdf 13310	. . . . . . . 8 |- ( ( W ++ X ) e. Word B -> ( W ++ X ) : ( 0..^ ( # ` ( W ++ X ) ) ) --> B )
34	32, 33	syl 17	. . . . . . 7 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( W ++ X ) : ( 0..^ ( # ` ( W ++ X ) ) ) --> B )
35		ccatlen 13360	. . . . . . . . . . 11 |- ( ( W e. Word B /\ X e. Word B ) -> ( # ` ( W ++ X ) ) = ( ( # ` W ) + ( # ` X ) ) )
36	29, 30, 35	syl2anc 693	. . . . . . . . . 10 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( # ` ( W ++ X ) ) = ( ( # ` W ) + ( # ` X ) ) )
37	36	oveq2d 6666	. . . . . . . . 9 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( 0..^ ( # ` ( W ++ X ) ) ) = ( 0..^ ( ( # ` W ) + ( # ` X ) ) ) )
38	20	nnzd 11481	. . . . . . . . . . 11 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( # ` W ) e. ZZ )
39	23	nnzd 11481	. . . . . . . . . . 11 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( # ` X ) e. ZZ )
40	38, 39	zaddcld 11486	. . . . . . . . . 10 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( ( # ` W ) + ( # ` X ) ) e. ZZ )
41		fzoval 12471	. . . . . . . . . 10 |- ( ( ( # ` W ) + ( # ` X ) ) e. ZZ -> ( 0..^ ( ( # ` W ) + ( # ` X ) ) ) = ( 0 ... ( ( ( # ` W ) + ( # ` X ) ) - 1 ) ) )
42	40, 41	syl 17	. . . . . . . . 9 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( 0..^ ( ( # ` W ) + ( # ` X ) ) ) = ( 0 ... ( ( ( # ` W ) + ( # ` X ) ) - 1 ) ) )
43	37, 42	eqtrd 2656	. . . . . . . 8 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( 0..^ ( # ` ( W ++ X ) ) ) = ( 0 ... ( ( ( # ` W ) + ( # ` X ) ) - 1 ) ) )
44	43	feq2d 6031	. . . . . . 7 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( ( W ++ X ) : ( 0..^ ( # ` ( W ++ X ) ) ) --> B <-> ( W ++ X ) : ( 0 ... ( ( ( # ` W ) + ( # ` X ) ) - 1 ) ) --> B ) )
45	34, 44	mpbid 222	. . . . . 6 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( W ++ X ) : ( 0 ... ( ( ( # ` W ) + ( # ` X ) ) - 1 ) ) --> B )
46	15, 16, 17, 28, 45	gsumval2 17280	. . . . 5 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( G gsumg ( W ++ X ) ) = ( seq 0 ( .+ , ( W ++ X ) ) ` ( ( ( # ` W ) + ( # ` X ) ) - 1 ) ) )
47		nnm1nn0 11334	. . . . . . . . . 10 |- ( ( # ` W ) e. NN -> ( ( # ` W ) - 1 ) e. NN0 )
48	20, 47	syl 17	. . . . . . . . 9 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( ( # ` W ) - 1 ) e. NN0 )
49	48, 27	syl6eleq 2711	. . . . . . . 8 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( ( # ` W ) - 1 ) e. ( ZZ>= ` 0 ) )
50		wrdf 13310	. . . . . . . . . 10 |- ( W e. Word B -> W : ( 0..^ ( # ` W ) ) --> B )
51	29, 50	syl 17	. . . . . . . . 9 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> W : ( 0..^ ( # ` W ) ) --> B )
52		fzoval 12471	. . . . . . . . . . 11 |- ( ( # ` W ) e. ZZ -> ( 0..^ ( # ` W ) ) = ( 0 ... ( ( # ` W ) - 1 ) ) )
53	38, 52	syl 17	. . . . . . . . . 10 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( 0..^ ( # ` W ) ) = ( 0 ... ( ( # ` W ) - 1 ) ) )
54	53	feq2d 6031	. . . . . . . . 9 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( W : ( 0..^ ( # ` W ) ) --> B <-> W : ( 0 ... ( ( # ` W ) - 1 ) ) --> B ) )
55	51, 54	mpbid 222	. . . . . . . 8 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> W : ( 0 ... ( ( # ` W ) - 1 ) ) --> B )
56	15, 16, 17, 49, 55	gsumval2 17280	. . . . . . 7 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( G gsumg W ) = ( seq 0 ( .+ , W ) ` ( ( # ` W ) - 1 ) ) )
57		nnm1nn0 11334	. . . . . . . . . 10 |- ( ( # ` X ) e. NN -> ( ( # ` X ) - 1 ) e. NN0 )
58	23, 57	syl 17	. . . . . . . . 9 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( ( # ` X ) - 1 ) e. NN0 )
59	58, 27	syl6eleq 2711	. . . . . . . 8 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( ( # ` X ) - 1 ) e. ( ZZ>= ` 0 ) )
60		wrdf 13310	. . . . . . . . . 10 |- ( X e. Word B -> X : ( 0..^ ( # ` X ) ) --> B )
61	30, 60	syl 17	. . . . . . . . 9 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> X : ( 0..^ ( # ` X ) ) --> B )
62		fzoval 12471	. . . . . . . . . . 11 |- ( ( # ` X ) e. ZZ -> ( 0..^ ( # ` X ) ) = ( 0 ... ( ( # ` X ) - 1 ) ) )
63	39, 62	syl 17	. . . . . . . . . 10 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( 0..^ ( # ` X ) ) = ( 0 ... ( ( # ` X ) - 1 ) ) )
64	63	feq2d 6031	. . . . . . . . 9 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( X : ( 0..^ ( # ` X ) ) --> B <-> X : ( 0 ... ( ( # ` X ) - 1 ) ) --> B ) )
65	61, 64	mpbid 222	. . . . . . . 8 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> X : ( 0 ... ( ( # ` X ) - 1 ) ) --> B )
66	15, 16, 17, 59, 65	gsumval2 17280	. . . . . . 7 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( G gsumg X ) = ( seq 0 ( .+ , X ) ` ( ( # ` X ) - 1 ) ) )
67	56, 66	oveq12d 6668	. . . . . 6 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( ( G gsumg W ) .+ ( G gsumg X ) ) = ( ( seq 0 ( .+ , W ) ` ( ( # ` W ) - 1 ) ) .+ ( seq 0 ( .+ , X ) ` ( ( # ` X ) - 1 ) ) ) )
68	15, 16	mndcl 17301	. . . . . . . . . 10 |- ( ( G e. Mnd /\ x e. B /\ y e. B ) -> ( x .+ y ) e. B )
69	68	3expb 1266	. . . . . . . . 9 |- ( ( G e. Mnd /\ ( x e. B /\ y e. B ) ) -> ( x .+ y ) e. B )
70	17, 69	sylan 488	. . . . . . . 8 |- ( ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) /\ ( x e. B /\ y e. B ) ) -> ( x .+ y ) e. B )
71	15, 16	mndass 17302	. . . . . . . . 9 |- ( ( G e. Mnd /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
72	17, 71	sylan 488	. . . . . . . 8 |- ( ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) )
73		uzid 11702	. . . . . . . . . . 11 |- ( ( # ` W ) e. ZZ -> ( # ` W ) e. ( ZZ>= ` ( # ` W ) ) )
74	38, 73	syl 17	. . . . . . . . . 10 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( # ` W ) e. ( ZZ>= ` ( # ` W ) ) )
75		uzaddcl 11744	. . . . . . . . . 10 |- ( ( ( # ` W ) e. ( ZZ>= ` ( # ` W ) ) /\ ( ( # ` X ) - 1 ) e. NN0 ) -> ( ( # ` W ) + ( ( # ` X ) - 1 ) ) e. ( ZZ>= ` ( # ` W ) ) )
76	74, 58, 75	syl2anc 693	. . . . . . . . 9 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( ( # ` W ) + ( ( # ` X ) - 1 ) ) e. ( ZZ>= ` ( # ` W ) ) )
77	20	nncnd 11036	. . . . . . . . . 10 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( # ` W ) e. CC )
78	23	nncnd 11036	. . . . . . . . . 10 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( # ` X ) e. CC )
79		1cnd 10056	. . . . . . . . . 10 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> 1 e. CC )
80	77, 78, 79	addsubassd 10412	. . . . . . . . 9 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( ( ( # ` W ) + ( # ` X ) ) - 1 ) = ( ( # ` W ) + ( ( # ` X ) - 1 ) ) )
81		ax-1cn 9994	. . . . . . . . . . 11 |- 1 e. CC
82		npcan 10290	. . . . . . . . . . 11 |- ( ( ( # ` W ) e. CC /\ 1 e. CC ) -> ( ( ( # ` W ) - 1 ) + 1 ) = ( # ` W ) )
83	77, 81, 82	sylancl 694	. . . . . . . . . 10 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( ( ( # ` W ) - 1 ) + 1 ) = ( # ` W ) )
84	83	fveq2d 6195	. . . . . . . . 9 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( ZZ>= ` ( ( ( # ` W ) - 1 ) + 1 ) ) = ( ZZ>= ` ( # ` W ) ) )
85	76, 80, 84	3eltr4d 2716	. . . . . . . 8 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( ( ( # ` W ) + ( # ` X ) ) - 1 ) e. ( ZZ>= ` ( ( ( # ` W ) - 1 ) + 1 ) ) )
86	45	ffvelrnda 6359	. . . . . . . 8 |- ( ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) /\ x e. ( 0 ... ( ( ( # ` W ) + ( # ` X ) ) - 1 ) ) ) -> ( ( W ++ X ) ` x ) e. B )
87	70, 72, 85, 49, 86	seqsplit 12834	. . . . . . 7 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( seq 0 ( .+ , ( W ++ X ) ) ` ( ( ( # ` W ) + ( # ` X ) ) - 1 ) ) = ( ( seq 0 ( .+ , ( W ++ X ) ) ` ( ( # ` W ) - 1 ) ) .+ ( seq ( ( ( # ` W ) - 1 ) + 1 ) ( .+ , ( W ++ X ) ) ` ( ( ( # ` W ) + ( # ` X ) ) - 1 ) ) ) )
88		simpll2 1101	. . . . . . . . . 10 |- ( ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) /\ x e. ( 0 ... ( ( # ` W ) - 1 ) ) ) -> W e. Word B )
89		simpll3 1102	. . . . . . . . . 10 |- ( ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) /\ x e. ( 0 ... ( ( # ` W ) - 1 ) ) ) -> X e. Word B )
90	53	eleq2d 2687	. . . . . . . . . . 11 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( x e. ( 0..^ ( # ` W ) ) <-> x e. ( 0 ... ( ( # ` W ) - 1 ) ) ) )
91	90	biimpar 502	. . . . . . . . . 10 |- ( ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) /\ x e. ( 0 ... ( ( # ` W ) - 1 ) ) ) -> x e. ( 0..^ ( # ` W ) ) )
92		ccatval1 13361	. . . . . . . . . 10 |- ( ( W e. Word B /\ X e. Word B /\ x e. ( 0..^ ( # ` W ) ) ) -> ( ( W ++ X ) ` x ) = ( W ` x ) )
93	88, 89, 91, 92	syl3anc 1326	. . . . . . . . 9 |- ( ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) /\ x e. ( 0 ... ( ( # ` W ) - 1 ) ) ) -> ( ( W ++ X ) ` x ) = ( W ` x ) )
94	49, 93	seqfveq 12825	. . . . . . . 8 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( seq 0 ( .+ , ( W ++ X ) ) ` ( ( # ` W ) - 1 ) ) = ( seq 0 ( .+ , W ) ` ( ( # ` W ) - 1 ) ) )
95	77	addid2d 10237	. . . . . . . . . . . 12 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( 0 + ( # ` W ) ) = ( # ` W ) )
96	83, 95	eqtr4d 2659	. . . . . . . . . . 11 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( ( ( # ` W ) - 1 ) + 1 ) = ( 0 + ( # ` W ) ) )
97	96	seqeq1d 12807	. . . . . . . . . 10 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> seq ( ( ( # ` W ) - 1 ) + 1 ) ( .+ , ( W ++ X ) ) = seq ( 0 + ( # ` W ) ) ( .+ , ( W ++ X ) ) )
98	77, 78	addcomd 10238	. . . . . . . . . . . 12 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( ( # ` W ) + ( # ` X ) ) = ( ( # ` X ) + ( # ` W ) ) )
99	98	oveq1d 6665	. . . . . . . . . . 11 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( ( ( # ` W ) + ( # ` X ) ) - 1 ) = ( ( ( # ` X ) + ( # ` W ) ) - 1 ) )
100	78, 77, 79	addsubd 10413	. . . . . . . . . . 11 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( ( ( # ` X ) + ( # ` W ) ) - 1 ) = ( ( ( # ` X ) - 1 ) + ( # ` W ) ) )
101	99, 100	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( ( ( # ` W ) + ( # ` X ) ) - 1 ) = ( ( ( # ` X ) - 1 ) + ( # ` W ) ) )
102	97, 101	fveq12d 6197	. . . . . . . . 9 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( seq ( ( ( # ` W ) - 1 ) + 1 ) ( .+ , ( W ++ X ) ) ` ( ( ( # ` W ) + ( # ` X ) ) - 1 ) ) = ( seq ( 0 + ( # ` W ) ) ( .+ , ( W ++ X ) ) ` ( ( ( # ` X ) - 1 ) + ( # ` W ) ) ) )
103		simpll2 1101	. . . . . . . . . . . 12 |- ( ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) /\ x e. ( 0 ... ( ( # ` X ) - 1 ) ) ) -> W e. Word B )
104		simpll3 1102	. . . . . . . . . . . 12 |- ( ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) /\ x e. ( 0 ... ( ( # ` X ) - 1 ) ) ) -> X e. Word B )
105	63	eleq2d 2687	. . . . . . . . . . . . 13 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( x e. ( 0..^ ( # ` X ) ) <-> x e. ( 0 ... ( ( # ` X ) - 1 ) ) ) )
106	105	biimpar 502	. . . . . . . . . . . 12 |- ( ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) /\ x e. ( 0 ... ( ( # ` X ) - 1 ) ) ) -> x e. ( 0..^ ( # ` X ) ) )
107		ccatval3 13363	. . . . . . . . . . . 12 |- ( ( W e. Word B /\ X e. Word B /\ x e. ( 0..^ ( # ` X ) ) ) -> ( ( W ++ X ) ` ( x + ( # ` W ) ) ) = ( X ` x ) )
108	103, 104, 106, 107	syl3anc 1326	. . . . . . . . . . 11 |- ( ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) /\ x e. ( 0 ... ( ( # ` X ) - 1 ) ) ) -> ( ( W ++ X ) ` ( x + ( # ` W ) ) ) = ( X ` x ) )
109	108	eqcomd 2628	. . . . . . . . . 10 |- ( ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) /\ x e. ( 0 ... ( ( # ` X ) - 1 ) ) ) -> ( X ` x ) = ( ( W ++ X ) ` ( x + ( # ` W ) ) ) )
110	59, 38, 109	seqshft2 12827	. . . . . . . . 9 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( seq 0 ( .+ , X ) ` ( ( # ` X ) - 1 ) ) = ( seq ( 0 + ( # ` W ) ) ( .+ , ( W ++ X ) ) ` ( ( ( # ` X ) - 1 ) + ( # ` W ) ) ) )
111	102, 110	eqtr4d 2659	. . . . . . . 8 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( seq ( ( ( # ` W ) - 1 ) + 1 ) ( .+ , ( W ++ X ) ) ` ( ( ( # ` W ) + ( # ` X ) ) - 1 ) ) = ( seq 0 ( .+ , X ) ` ( ( # ` X ) - 1 ) ) )
112	94, 111	oveq12d 6668	. . . . . . 7 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( ( seq 0 ( .+ , ( W ++ X ) ) ` ( ( # ` W ) - 1 ) ) .+ ( seq ( ( ( # ` W ) - 1 ) + 1 ) ( .+ , ( W ++ X ) ) ` ( ( ( # ` W ) + ( # ` X ) ) - 1 ) ) ) = ( ( seq 0 ( .+ , W ) ` ( ( # ` W ) - 1 ) ) .+ ( seq 0 ( .+ , X ) ` ( ( # ` X ) - 1 ) ) ) )
113	87, 112	eqtrd 2656	. . . . . 6 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( seq 0 ( .+ , ( W ++ X ) ) ` ( ( ( # ` W ) + ( # ` X ) ) - 1 ) ) = ( ( seq 0 ( .+ , W ) ` ( ( # ` W ) - 1 ) ) .+ ( seq 0 ( .+ , X ) ` ( ( # ` X ) - 1 ) ) ) )
114	67, 113	eqtr4d 2659	. . . . 5 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( ( G gsumg W ) .+ ( G gsumg X ) ) = ( seq 0 ( .+ , ( W ++ X ) ) ` ( ( ( # ` W ) + ( # ` X ) ) - 1 ) ) )
115	46, 114	eqtr4d 2659	. . . 4 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ ( W =/= (/) /\ X =/= (/) ) ) -> ( G gsumg ( W ++ X ) ) = ( ( G gsumg W ) .+ ( G gsumg X ) ) )
116	115	anassrs 680	. . 3 |- ( ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ W =/= (/) ) /\ X =/= (/) ) -> ( G gsumg ( W ++ X ) ) = ( ( G gsumg W ) .+ ( G gsumg X ) ) )
117		simpl2 1065	. . . . . 6 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ W =/= (/) ) -> W e. Word B )
118		ccatrid 13370	. . . . . 6 |- ( W e. Word B -> ( W ++ (/) ) = W )
119	117, 118	syl 17	. . . . 5 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ W =/= (/) ) -> ( W ++ (/) ) = W )
120	119	oveq2d 6666	. . . 4 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ W =/= (/) ) -> ( G gsumg ( W ++ (/) ) ) = ( G gsumg W ) )
121		simpl1 1064	. . . . 5 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ W =/= (/) ) -> G e. Mnd )
122	15	gsumwcl 17377	. . . . . . 7 |- ( ( G e. Mnd /\ W e. Word B ) -> ( G gsumg W ) e. B )
123	122	3adant3 1081	. . . . . 6 |- ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) -> ( G gsumg W ) e. B )
124	123	adantr 481	. . . . 5 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ W =/= (/) ) -> ( G gsumg W ) e. B )
125	15, 16, 4	mndrid 17312	. . . . 5 |- ( ( G e. Mnd /\ ( G gsumg W ) e. B ) -> ( ( G gsumg W ) .+ ( 0g ` G ) ) = ( G gsumg W ) )
126	121, 124, 125	syl2anc 693	. . . 4 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ W =/= (/) ) -> ( ( G gsumg W ) .+ ( 0g ` G ) ) = ( G gsumg W ) )
127	120, 126	eqtr4d 2659	. . 3 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ W =/= (/) ) -> ( G gsumg ( W ++ (/) ) ) = ( ( G gsumg W ) .+ ( 0g ` G ) ) )
128	14, 116, 127	pm2.61ne 2879	. 2 |- ( ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) /\ W =/= (/) ) -> ( G gsumg ( W ++ X ) ) = ( ( G gsumg W ) .+ ( G gsumg X ) ) )
129		ccatlid 13369	. . . . 5 |- ( X e. Word B -> ( (/) ++ X ) = X )
130	129	3ad2ant3 1084	. . . 4 |- ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) -> ( (/) ++ X ) = X )
131	130	oveq2d 6666	. . 3 |- ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) -> ( G gsumg ( (/) ++ X ) ) = ( G gsumg X ) )
132		simp1 1061	. . . 4 |- ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) -> G e. Mnd )
133	15	gsumwcl 17377	. . . . 5 |- ( ( G e. Mnd /\ X e. Word B ) -> ( G gsumg X ) e. B )
134	133	3adant2 1080	. . . 4 |- ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) -> ( G gsumg X ) e. B )
135	15, 16, 4	mndlid 17311	. . . 4 |- ( ( G e. Mnd /\ ( G gsumg X ) e. B ) -> ( ( 0g ` G ) .+ ( G gsumg X ) ) = ( G gsumg X ) )
136	132, 134, 135	syl2anc 693	. . 3 |- ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) -> ( ( 0g ` G ) .+ ( G gsumg X ) ) = ( G gsumg X ) )
137	131, 136	eqtr4d 2659	. 2 |- ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) -> ( G gsumg ( (/) ++ X ) ) = ( ( 0g ` G ) .+ ( G gsumg X ) ) )
138	8, 128, 137	pm2.61ne 2879	1 |- ( ( G e. Mnd /\ W e. Word B /\ X e. Word B ) -> ( G gsumg ( W ++ X ) ) = ( ( G gsumg W ) .+ ( G gsumg X ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   seqcseq 12801   #chash 13117  Word cword 13291   ++ cconcat 13293   Basecbs 15857   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294

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-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-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336

This theorem is referenced by:  gsumws2  17379  gsumccatsn  17380  gsumspl  17381  gsumwspan  17383  frmdgsum  17399  frmdup1  17401  gsumwrev  17796  psgnunilem5  17914  psgnuni  17919  frgpuplem  18185  frgpup1  18188  psgnghm  19926  mrsubccat  31415  gsumws3  38499  gsumws4  38500