Theorem ccatass 11138

Description: Associative law for concatenation of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)

Assertion

Ref	Expression
ccatass	|- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( ( S ++ T ) ++ U ) = ( S ++ ( T ++ U ) ) )

Proof of Theorem ccatass

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ccatcl 11123	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) e. Word B )
2		ccatcl 11123	. . . . 5 |- ( ( ( S ++ T ) e. Word B /\ U e. Word B ) -> ( ( S ++ T ) ++ U ) e. Word B )
3	1, 2	stoic3 1473	. . . 4 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( ( S ++ T ) ++ U ) e. Word B )
4		wrdfn 11081	. . . 4 |- ( ( ( S ++ T ) ++ U ) e. Word B -> ( ( S ++ T ) ++ U ) Fn ( 0..^ (♯ ` ( ( S ++ T ) ++ U ) ) ) )
5	3, 4	syl 14	. . 3 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( ( S ++ T ) ++ U ) Fn ( 0..^ (♯ ` ( ( S ++ T ) ++ U ) ) ) )
6		ccatlen 11125	. . . . . . 7 |- ( ( ( S ++ T ) e. Word B /\ U e. Word B ) -> (♯ ` ( ( S ++ T ) ++ U ) ) = ( (♯ ` ( S ++ T ) ) + (♯ ` U ) ) )
7	1, 6	stoic3 1473	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` ( ( S ++ T ) ++ U ) ) = ( (♯ ` ( S ++ T ) ) + (♯ ` U ) ) )
8		ccatlen 11125	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` ( S ++ T ) ) = ( (♯ ` S ) + (♯ ` T ) ) )
9	8	3adant3 1041	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` ( S ++ T ) ) = ( (♯ ` S ) + (♯ ` T ) ) )
10	9	oveq1d 6015	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( (♯ ` ( S ++ T ) ) + (♯ ` U ) ) = ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) )
11	7, 10	eqtrd 2262	. . . . 5 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` ( ( S ++ T ) ++ U ) ) = ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) )
12	11	oveq2d 6016	. . . 4 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( 0..^ (♯ ` ( ( S ++ T ) ++ U ) ) ) = ( 0..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) )
13	12	fneq2d 5411	. . 3 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( ( ( S ++ T ) ++ U ) Fn ( 0..^ (♯ ` ( ( S ++ T ) ++ U ) ) ) <-> ( ( S ++ T ) ++ U ) Fn ( 0..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) )
14	5, 13	mpbid 147	. 2 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( ( S ++ T ) ++ U ) Fn ( 0..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) )
15		simp1 1021	. . . . 5 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> S e. Word B )
16		ccatcl 11123	. . . . . 6 |- ( ( T e. Word B /\ U e. Word B ) -> ( T ++ U ) e. Word B )
17	16	3adant1 1039	. . . . 5 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( T ++ U ) e. Word B )
18		ccatcl 11123	. . . . 5 |- ( ( S e. Word B /\ ( T ++ U ) e. Word B ) -> ( S ++ ( T ++ U ) ) e. Word B )
19	15, 17, 18	syl2anc 411	. . . 4 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( S ++ ( T ++ U ) ) e. Word B )
20		wrdfn 11081	. . . 4 |- ( ( S ++ ( T ++ U ) ) e. Word B -> ( S ++ ( T ++ U ) ) Fn ( 0..^ (♯ ` ( S ++ ( T ++ U ) ) ) ) )
21	19, 20	syl 14	. . 3 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( S ++ ( T ++ U ) ) Fn ( 0..^ (♯ ` ( S ++ ( T ++ U ) ) ) ) )
22		ccatlen 11125	. . . . . . . 8 |- ( ( T e. Word B /\ U e. Word B ) -> (♯ ` ( T ++ U ) ) = ( (♯ ` T ) + (♯ ` U ) ) )
23	22	3adant1 1039	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` ( T ++ U ) ) = ( (♯ ` T ) + (♯ ` U ) ) )
24	23	oveq2d 6016	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( (♯ ` S ) + (♯ ` ( T ++ U ) ) ) = ( (♯ ` S ) + ( (♯ ` T ) + (♯ ` U ) ) ) )
25		ccatlen 11125	. . . . . . 7 |- ( ( S e. Word B /\ ( T ++ U ) e. Word B ) -> (♯ ` ( S ++ ( T ++ U ) ) ) = ( (♯ ` S ) + (♯ ` ( T ++ U ) ) ) )
26	15, 17, 25	syl2anc 411	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` ( S ++ ( T ++ U ) ) ) = ( (♯ ` S ) + (♯ ` ( T ++ U ) ) ) )
27		lencl 11070	. . . . . . . . 9 |- ( S e. Word B -> (♯ ` S ) e. NN0 )
28	27	3ad2ant1 1042	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` S ) e. NN0 )
29	28	nn0cnd 9420	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` S ) e. CC )
30		lencl 11070	. . . . . . . . 9 |- ( T e. Word B -> (♯ ` T ) e. NN0 )
31	30	3ad2ant2 1043	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` T ) e. NN0 )
32	31	nn0cnd 9420	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` T ) e. CC )
33		lencl 11070	. . . . . . . . 9 |- ( U e. Word B -> (♯ ` U ) e. NN0 )
34	33	3ad2ant3 1044	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` U ) e. NN0 )
35	34	nn0cnd 9420	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` U ) e. CC )
36	29, 32, 35	addassd 8165	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) = ( (♯ ` S ) + ( (♯ ` T ) + (♯ ` U ) ) ) )
37	24, 26, 36	3eqtr4d 2272	. . . . 5 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` ( S ++ ( T ++ U ) ) ) = ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) )
38	37	oveq2d 6016	. . . 4 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( 0..^ (♯ ` ( S ++ ( T ++ U ) ) ) ) = ( 0..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) )
39	38	fneq2d 5411	. . 3 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( ( S ++ ( T ++ U ) ) Fn ( 0..^ (♯ ` ( S ++ ( T ++ U ) ) ) ) <-> ( S ++ ( T ++ U ) ) Fn ( 0..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) )
40	21, 39	mpbid 147	. 2 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( S ++ ( T ++ U ) ) Fn ( 0..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) )
41	28	nn0zd 9563	. . . 4 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` S ) e. ZZ )
42		fzospliti 10370	. . . . 5 |- ( ( x e. ( 0..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) /\ (♯ ` S ) e. ZZ ) -> ( x e. ( 0..^ (♯ ` S ) ) \/ x e. ( (♯ ` S )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) )
43	42	ex 115	. . . 4 |- ( x e. ( 0..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) -> ( (♯ ` S ) e. ZZ -> ( x e. ( 0..^ (♯ ` S ) ) \/ x e. ( (♯ ` S )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) ) )
44	41, 43	mpan9 281	. . 3 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( 0..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> ( x e. ( 0..^ (♯ ` S ) ) \/ x e. ( (♯ ` S )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) )
45		simp2 1022	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> T e. Word B )
46		id 19	. . . . . 6 |- ( x e. ( 0..^ (♯ ` S ) ) -> x e. ( 0..^ (♯ ` S ) ) )
47		ccatval1 11127	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` x ) = ( S ` x ) )
48	15, 45, 46, 47	syl2an3an 1332	. . . . 5 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` x ) = ( S ` x ) )
49	1	3adant3 1041	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( S ++ T ) e. Word B )
50	49	adantr 276	. . . . . 6 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( S ++ T ) e. Word B )
51		simpl3 1026	. . . . . 6 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> U e. Word B )
52	41	uzidd 9733	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` S ) e. ( ZZ>= ` (♯ ` S ) ) )
53		uzaddcl 9777	. . . . . . . . . 10 |- ( ( (♯ ` S ) e. ( ZZ>= ` (♯ ` S ) ) /\ (♯ ` T ) e. NN0 ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` (♯ ` S ) ) )
54	52, 31, 53	syl2anc 411	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` (♯ ` S ) ) )
55		fzoss2 10366	. . . . . . . . 9 |- ( ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` (♯ ` S ) ) -> ( 0..^ (♯ ` S ) ) C_ ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
56	54, 55	syl 14	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( 0..^ (♯ ` S ) ) C_ ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
57	9	oveq2d 6016	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( 0..^ (♯ ` ( S ++ T ) ) ) = ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
58	56, 57	sseqtrrd 3263	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( 0..^ (♯ ` S ) ) C_ ( 0..^ (♯ ` ( S ++ T ) ) ) )
59	58	sselda 3224	. . . . . 6 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> x e. ( 0..^ (♯ ` ( S ++ T ) ) ) )
60		ccatval1 11127	. . . . . 6 |- ( ( ( S ++ T ) e. Word B /\ U e. Word B /\ x e. ( 0..^ (♯ ` ( S ++ T ) ) ) ) -> ( ( ( S ++ T ) ++ U ) ` x ) = ( ( S ++ T ) ` x ) )
61	50, 51, 59, 60	syl3anc 1271	. . . . 5 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( ( S ++ T ) ++ U ) ` x ) = ( ( S ++ T ) ` x ) )
62		ccatval1 11127	. . . . . 6 |- ( ( S e. Word B /\ ( T ++ U ) e. Word B /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ ( T ++ U ) ) ` x ) = ( S ` x ) )
63	15, 17, 46, 62	syl2an3an 1332	. . . . 5 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ ( T ++ U ) ) ` x ) = ( S ` x ) )
64	48, 61, 63	3eqtr4d 2272	. . . 4 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( ( S ++ T ) ++ U ) ` x ) = ( ( S ++ ( T ++ U ) ) ` x ) )
65	31	nn0zd 9563	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` T ) e. ZZ )
66	41, 65	zaddcld 9569	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ZZ )
67		fzospliti 10370	. . . . . . 7 |- ( ( x e. ( (♯ ` S )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) /\ ( (♯ ` S ) + (♯ ` T ) ) e. ZZ ) -> ( x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) \/ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) )
68	67	ex 115	. . . . . 6 |- ( x e. ( (♯ ` S )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) -> ( ( (♯ ` S ) + (♯ ` T ) ) e. ZZ -> ( x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) \/ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) ) )
69	66, 68	mpan9 281	. . . . 5 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( (♯ ` S )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> ( x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) \/ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) )
70		id 19	. . . . . . . . 9 |- ( x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) -> x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
71		ccatval2 11128	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( S ++ T ) ` x ) = ( T ` ( x - (♯ ` S ) ) ) )
72	15, 45, 70, 71	syl2an3an 1332	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( S ++ T ) ` x ) = ( T ` ( x - (♯ ` S ) ) ) )
73		simpl2 1025	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> T e. Word B )
74		simpl3 1026	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> U e. Word B )
75		fzosubel3 10397	. . . . . . . . . . 11 |- ( ( x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) /\ (♯ ` T ) e. ZZ ) -> ( x - (♯ ` S ) ) e. ( 0..^ (♯ ` T ) ) )
76	75	ex 115	. . . . . . . . . 10 |- ( x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) -> ( (♯ ` T ) e. ZZ -> ( x - (♯ ` S ) ) e. ( 0..^ (♯ ` T ) ) ) )
77	65, 76	mpan9 281	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( x - (♯ ` S ) ) e. ( 0..^ (♯ ` T ) ) )
78		ccatval1 11127	. . . . . . . . 9 |- ( ( T e. Word B /\ U e. Word B /\ ( x - (♯ ` S ) ) e. ( 0..^ (♯ ` T ) ) ) -> ( ( T ++ U ) ` ( x - (♯ ` S ) ) ) = ( T ` ( x - (♯ ` S ) ) ) )
79	73, 74, 77, 78	syl3anc 1271	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( T ++ U ) ` ( x - (♯ ` S ) ) ) = ( T ` ( x - (♯ ` S ) ) ) )
80	72, 79	eqtr4d 2265	. . . . . . 7 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( S ++ T ) ` x ) = ( ( T ++ U ) ` ( x - (♯ ` S ) ) ) )
81	49	adantr 276	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( S ++ T ) e. Word B )
82		fzoss1 10365	. . . . . . . . . . . 12 |- ( (♯ ` S ) e. ( ZZ>= ` 0 ) -> ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) C_ ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
83		nn0uz 9753	. . . . . . . . . . . 12 |- NN0 = ( ZZ>= ` 0 )
84	82, 83	eleq2s 2324	. . . . . . . . . . 11 |- ( (♯ ` S ) e. NN0 -> ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) C_ ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
85	28, 84	syl 14	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) C_ ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
86	85, 57	sseqtrrd 3263	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) C_ ( 0..^ (♯ ` ( S ++ T ) ) ) )
87	86	sselda 3224	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> x e. ( 0..^ (♯ ` ( S ++ T ) ) ) )
88	81, 74, 87, 60	syl3anc 1271	. . . . . . 7 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( ( S ++ T ) ++ U ) ` x ) = ( ( S ++ T ) ` x ) )
89		simpl1 1024	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> S e. Word B )
90	17	adantr 276	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( T ++ U ) e. Word B )
91	66	uzidd 9733	. . . . . . . . . . . 12 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` ( (♯ ` S ) + (♯ ` T ) ) ) )
92		uzaddcl 9777	. . . . . . . . . . . 12 |- ( ( ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` ( (♯ ` S ) + (♯ ` T ) ) ) /\ (♯ ` U ) e. NN0 ) -> ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) e. ( ZZ>= ` ( (♯ ` S ) + (♯ ` T ) ) ) )
93	91, 34, 92	syl2anc 411	. . . . . . . . . . 11 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) e. ( ZZ>= ` ( (♯ ` S ) + (♯ ` T ) ) ) )
94		fzoss2 10366	. . . . . . . . . . 11 |- ( ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) e. ( ZZ>= ` ( (♯ ` S ) + (♯ ` T ) ) ) -> ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) C_ ( (♯ ` S )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) )
95	93, 94	syl 14	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) C_ ( (♯ ` S )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) )
96	24, 36	eqtr4d 2265	. . . . . . . . . . 11 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( (♯ ` S ) + (♯ ` ( T ++ U ) ) ) = ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) )
97	96	oveq2d 6016	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` ( T ++ U ) ) ) ) = ( (♯ ` S )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) )
98	95, 97	sseqtrrd 3263	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) C_ ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` ( T ++ U ) ) ) ) )
99	98	sselda 3224	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` ( T ++ U ) ) ) ) )
100		ccatval2 11128	. . . . . . . 8 |- ( ( S e. Word B /\ ( T ++ U ) e. Word B /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` ( T ++ U ) ) ) ) ) -> ( ( S ++ ( T ++ U ) ) ` x ) = ( ( T ++ U ) ` ( x - (♯ ` S ) ) ) )
101	89, 90, 99, 100	syl3anc 1271	. . . . . . 7 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( S ++ ( T ++ U ) ) ` x ) = ( ( T ++ U ) ` ( x - (♯ ` S ) ) ) )
102	80, 88, 101	3eqtr4d 2272	. . . . . 6 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( ( S ++ T ) ++ U ) ` x ) = ( ( S ++ ( T ++ U ) ) ` x ) )
103	9	oveq2d 6016	. . . . . . . . . . 11 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( x - (♯ ` ( S ++ T ) ) ) = ( x - ( (♯ ` S ) + (♯ ` T ) ) ) )
104	103	adantr 276	. . . . . . . . . 10 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> ( x - (♯ ` ( S ++ T ) ) ) = ( x - ( (♯ ` S ) + (♯ ` T ) ) ) )
105		elfzoelz 10339	. . . . . . . . . . . . 13 |- ( x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) -> x e. ZZ )
106	105	zcnd 9566	. . . . . . . . . . . 12 |- ( x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) -> x e. CC )
107	106	adantl 277	. . . . . . . . . . 11 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> x e. CC )
108	29	adantr 276	. . . . . . . . . . 11 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> (♯ ` S ) e. CC )
109	32	adantr 276	. . . . . . . . . . 11 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> (♯ ` T ) e. CC )
110	107, 108, 109	subsub4d 8484	. . . . . . . . . 10 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> ( ( x - (♯ ` S ) ) - (♯ ` T ) ) = ( x - ( (♯ ` S ) + (♯ ` T ) ) ) )
111	104, 110	eqtr4d 2265	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> ( x - (♯ ` ( S ++ T ) ) ) = ( ( x - (♯ ` S ) ) - (♯ ` T ) ) )
112	111	fveq2d 5630	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> ( U ` ( x - (♯ ` ( S ++ T ) ) ) ) = ( U ` ( ( x - (♯ ` S ) ) - (♯ ` T ) ) ) )
113		simpl2 1025	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> T e. Word B )
114		simpl3 1026	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> U e. Word B )
115	36	oveq2d 6016	. . . . . . . . . . . 12 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) = ( ( (♯ ` S ) + (♯ ` T ) )..^ ( (♯ ` S ) + ( (♯ ` T ) + (♯ ` U ) ) ) ) )
116	115	eleq2d 2299	. . . . . . . . . . 11 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) <-> x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( (♯ ` S ) + ( (♯ ` T ) + (♯ ` U ) ) ) ) ) )
117	116	biimpa 296	. . . . . . . . . 10 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( (♯ ` S ) + ( (♯ ` T ) + (♯ ` U ) ) ) ) )
118	34	nn0zd 9563	. . . . . . . . . . . . 13 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` U ) e. ZZ )
119	65, 118	zaddcld 9569	. . . . . . . . . . . 12 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( (♯ ` T ) + (♯ ` U ) ) e. ZZ )
120	41, 65, 119	3jca 1201	. . . . . . . . . . 11 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( (♯ ` S ) e. ZZ /\ (♯ ` T ) e. ZZ /\ ( (♯ ` T ) + (♯ ` U ) ) e. ZZ ) )
121	120	adantr 276	. . . . . . . . . 10 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> ( (♯ ` S ) e. ZZ /\ (♯ ` T ) e. ZZ /\ ( (♯ ` T ) + (♯ ` U ) ) e. ZZ ) )
122		fzosubel2 10396	. . . . . . . . . 10 |- ( ( x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( (♯ ` S ) + ( (♯ ` T ) + (♯ ` U ) ) ) ) /\ ( (♯ ` S ) e. ZZ /\ (♯ ` T ) e. ZZ /\ ( (♯ ` T ) + (♯ ` U ) ) e. ZZ ) ) -> ( x - (♯ ` S ) ) e. ( (♯ ` T )..^ ( (♯ ` T ) + (♯ ` U ) ) ) )
123	117, 121, 122	syl2anc 411	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> ( x - (♯ ` S ) ) e. ( (♯ ` T )..^ ( (♯ ` T ) + (♯ ` U ) ) ) )
124		ccatval2 11128	. . . . . . . . 9 |- ( ( T e. Word B /\ U e. Word B /\ ( x - (♯ ` S ) ) e. ( (♯ ` T )..^ ( (♯ ` T ) + (♯ ` U ) ) ) ) -> ( ( T ++ U ) ` ( x - (♯ ` S ) ) ) = ( U ` ( ( x - (♯ ` S ) ) - (♯ ` T ) ) ) )
125	113, 114, 123, 124	syl3anc 1271	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> ( ( T ++ U ) ` ( x - (♯ ` S ) ) ) = ( U ` ( ( x - (♯ ` S ) ) - (♯ ` T ) ) ) )
126	112, 125	eqtr4d 2265	. . . . . . 7 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> ( U ` ( x - (♯ ` ( S ++ T ) ) ) ) = ( ( T ++ U ) ` ( x - (♯ ` S ) ) ) )
127	49	adantr 276	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> ( S ++ T ) e. Word B )
128	9, 10	oveq12d 6018	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( (♯ ` ( S ++ T ) )..^ ( (♯ ` ( S ++ T ) ) + (♯ ` U ) ) ) = ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) )
129	128	eleq2d 2299	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( x e. ( (♯ ` ( S ++ T ) )..^ ( (♯ ` ( S ++ T ) ) + (♯ ` U ) ) ) <-> x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) )
130	129	biimpar 297	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> x e. ( (♯ ` ( S ++ T ) )..^ ( (♯ ` ( S ++ T ) ) + (♯ ` U ) ) ) )
131		ccatval2 11128	. . . . . . . 8 |- ( ( ( S ++ T ) e. Word B /\ U e. Word B /\ x e. ( (♯ ` ( S ++ T ) )..^ ( (♯ ` ( S ++ T ) ) + (♯ ` U ) ) ) ) -> ( ( ( S ++ T ) ++ U ) ` x ) = ( U ` ( x - (♯ ` ( S ++ T ) ) ) ) )
132	127, 114, 130, 131	syl3anc 1271	. . . . . . 7 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> ( ( ( S ++ T ) ++ U ) ` x ) = ( U ` ( x - (♯ ` ( S ++ T ) ) ) ) )
133		simpl1 1024	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> S e. Word B )
134	17	adantr 276	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> ( T ++ U ) e. Word B )
135		fzoss1 10365	. . . . . . . . . . 11 |- ( ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` (♯ ` S ) ) -> ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) C_ ( (♯ ` S )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) )
136	54, 135	syl 14	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) C_ ( (♯ ` S )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) )
137	136, 97	sseqtrrd 3263	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) C_ ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` ( T ++ U ) ) ) ) )
138	137	sselda 3224	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` ( T ++ U ) ) ) ) )
139	133, 134, 138, 100	syl3anc 1271	. . . . . . 7 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> ( ( S ++ ( T ++ U ) ) ` x ) = ( ( T ++ U ) ` ( x - (♯ ` S ) ) ) )
140	126, 132, 139	3eqtr4d 2272	. . . . . 6 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> ( ( ( S ++ T ) ++ U ) ` x ) = ( ( S ++ ( T ++ U ) ) ` x ) )
141	102, 140	jaodan 802	. . . . 5 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ ( x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) \/ x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) ) -> ( ( ( S ++ T ) ++ U ) ` x ) = ( ( S ++ ( T ++ U ) ) ` x ) )
142	69, 141	syldan 282	. . . 4 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( (♯ ` S )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> ( ( ( S ++ T ) ++ U ) ` x ) = ( ( S ++ ( T ++ U ) ) ` x ) )
143	64, 142	jaodan 802	. . 3 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ ( x e. ( 0..^ (♯ ` S ) ) \/ x e. ( (♯ ` S )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) ) -> ( ( ( S ++ T ) ++ U ) ` x ) = ( ( S ++ ( T ++ U ) ) ` x ) )
144	44, 143	syldan 282	. 2 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( 0..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) ) -> ( ( ( S ++ T ) ++ U ) ` x ) = ( ( S ++ ( T ++ U ) ) ` x ) )
145	14, 40, 144	eqfnfvd 5734	1 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( ( S ++ T ) ++ U ) = ( S ++ ( T ++ U ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   C_ wss 3197   Fn wfn 5312   ` cfv 5317  (class class class)co 6000   CCcc 7993   0cc0 7995   + caddc 7998   - cmin 8313   NN0cn0 9365   ZZcz 9442   ZZ>=cuz 9718  ..^cfzo 10334  ♯chash 10992  Word cword 11066   ++ cconcat 11120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-1o 6560  df-er 6678  df-en 6886  df-dom 6887  df-fin 6888  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-fz 10201  df-fzo 10335  df-ihash 10993  df-word 11067  df-concat 11121

This theorem is referenced by:  cats1catd  11295  cats2catd  11296