Theorem ccatass 11087

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 11072	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) e. Word B )
2		ccatcl 11072	. . . . 5 |- ( ( ( S ++ T ) e. Word B /\ U e. Word B ) -> ( ( S ++ T ) ++ U ) e. Word B )
3	1, 2	stoic3 1451	. . . 4 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( ( S ++ T ) ++ U ) e. Word B )
4		wrdfn 11031	. . . 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 11074	. . . . . . 7 |- ( ( ( S ++ T ) e. Word B /\ U e. Word B ) -> (♯ ` ( ( S ++ T ) ++ U ) ) = ( (♯ ` ( S ++ T ) ) + (♯ ` U ) ) )
7	1, 6	stoic3 1451	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` ( ( S ++ T ) ++ U ) ) = ( (♯ ` ( S ++ T ) ) + (♯ ` U ) ) )
8		ccatlen 11074	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` ( S ++ T ) ) = ( (♯ ` S ) + (♯ ` T ) ) )
9	8	3adant3 1020	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` ( S ++ T ) ) = ( (♯ ` S ) + (♯ ` T ) ) )
10	9	oveq1d 5972	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( (♯ ` ( S ++ T ) ) + (♯ ` U ) ) = ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) )
11	7, 10	eqtrd 2239	. . . . 5 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` ( ( S ++ T ) ++ U ) ) = ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) )
12	11	oveq2d 5973	. . . 4 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( 0..^ (♯ ` ( ( S ++ T ) ++ U ) ) ) = ( 0..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) )
13	12	fneq2d 5374	. . 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 1000	. . . . 5 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> S e. Word B )
16		ccatcl 11072	. . . . . 6 |- ( ( T e. Word B /\ U e. Word B ) -> ( T ++ U ) e. Word B )
17	16	3adant1 1018	. . . . 5 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( T ++ U ) e. Word B )
18		ccatcl 11072	. . . . 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 11031	. . . 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 11074	. . . . . . . 8 |- ( ( T e. Word B /\ U e. Word B ) -> (♯ ` ( T ++ U ) ) = ( (♯ ` T ) + (♯ ` U ) ) )
23	22	3adant1 1018	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` ( T ++ U ) ) = ( (♯ ` T ) + (♯ ` U ) ) )
24	23	oveq2d 5973	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( (♯ ` S ) + (♯ ` ( T ++ U ) ) ) = ( (♯ ` S ) + ( (♯ ` T ) + (♯ ` U ) ) ) )
25		ccatlen 11074	. . . . . . 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 11020	. . . . . . . . 9 |- ( S e. Word B -> (♯ ` S ) e. NN0 )
28	27	3ad2ant1 1021	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` S ) e. NN0 )
29	28	nn0cnd 9370	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` S ) e. CC )
30		lencl 11020	. . . . . . . . 9 |- ( T e. Word B -> (♯ ` T ) e. NN0 )
31	30	3ad2ant2 1022	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` T ) e. NN0 )
32	31	nn0cnd 9370	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` T ) e. CC )
33		lencl 11020	. . . . . . . . 9 |- ( U e. Word B -> (♯ ` U ) e. NN0 )
34	33	3ad2ant3 1023	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` U ) e. NN0 )
35	34	nn0cnd 9370	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` U ) e. CC )
36	29, 32, 35	addassd 8115	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) = ( (♯ ` S ) + ( (♯ ` T ) + (♯ ` U ) ) ) )
37	24, 26, 36	3eqtr4d 2249	. . . . 5 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` ( S ++ ( T ++ U ) ) ) = ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) )
38	37	oveq2d 5973	. . . 4 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( 0..^ (♯ ` ( S ++ ( T ++ U ) ) ) ) = ( 0..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) )
39	38	fneq2d 5374	. . 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 9513	. . . 4 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` S ) e. ZZ )
42		fzospliti 10320	. . . . 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 1001	. . . . . 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 11076	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` x ) = ( S ` x ) )
48	15, 45, 46, 47	syl2an3an 1311	. . . . 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 1020	. . . . . . 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 1005	. . . . . 6 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> U e. Word B )
52	41	uzidd 9683	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` S ) e. ( ZZ>= ` (♯ ` S ) ) )
53		uzaddcl 9727	. . . . . . . . . 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 10316	. . . . . . . . 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 5973	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( 0..^ (♯ ` ( S ++ T ) ) ) = ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
58	56, 57	sseqtrrd 3236	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( 0..^ (♯ ` S ) ) C_ ( 0..^ (♯ ` ( S ++ T ) ) ) )
59	58	sselda 3197	. . . . . 6 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> x e. ( 0..^ (♯ ` ( S ++ T ) ) ) )
60		ccatval1 11076	. . . . . 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 1250	. . . . 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 11076	. . . . . 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 1311	. . . . 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 2249	. . . 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 9513	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` T ) e. ZZ )
66	41, 65	zaddcld 9519	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ZZ )
67		fzospliti 10320	. . . . . . 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 11077	. . . . . . . . 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 1311	. . . . . . . 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 1004	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> T e. Word B )
74		simpl3 1005	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> U e. Word B )
75		fzosubel3 10347	. . . . . . . . . . 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 11076	. . . . . . . . 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 1250	. . . . . . . 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 2242	. . . . . . 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 10315	. . . . . . . . . . . 12 |- ( (♯ ` S ) e. ( ZZ>= ` 0 ) -> ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) C_ ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
83		nn0uz 9703	. . . . . . . . . . . 12 |- NN0 = ( ZZ>= ` 0 )
84	82, 83	eleq2s 2301	. . . . . . . . . . 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 3236	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) C_ ( 0..^ (♯ ` ( S ++ T ) ) ) )
87	86	sselda 3197	. . . . . . . 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 1250	. . . . . . 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 1003	. . . . . . . 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 9683	. . . . . . . . . . . 12 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` ( (♯ ` S ) + (♯ ` T ) ) ) )
92		uzaddcl 9727	. . . . . . . . . . . 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 10316	. . . . . . . . . . 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 2242	. . . . . . . . . . 11 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( (♯ ` S ) + (♯ ` ( T ++ U ) ) ) = ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) )
97	96	oveq2d 5973	. . . . . . . . . 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 3236	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) C_ ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` ( T ++ U ) ) ) ) )
99	98	sselda 3197	. . . . . . . 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 11077	. . . . . . . 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 1250	. . . . . . 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 2249	. . . . . 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 5973	. . . . . . . . . . 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 10289	. . . . . . . . . . . . 13 |- ( x e. ( ( (♯ ` S ) + (♯ ` T ) )..^ ( ( (♯ ` S ) + (♯ ` T ) ) + (♯ ` U ) ) ) -> x e. ZZ )
106	105	zcnd 9516	. . . . . . . . . . . 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 8434	. . . . . . . . . 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 2242	. . . . . . . . 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 5593	. . . . . . . 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 1004	. . . . . . . . 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 1005	. . . . . . . . 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 5973	. . . . . . . . . . . 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 2276	. . . . . . . . . . 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 9513	. . . . . . . . . . . . 13 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> (♯ ` U ) e. ZZ )
119	65, 118	zaddcld 9519	. . . . . . . . . . . 12 |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( (♯ ` T ) + (♯ ` U ) ) e. ZZ )
120	41, 65, 119	3jca 1180	. . . . . . . . . . 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 10346	. . . . . . . . . 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 11077	. . . . . . . . 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 1250	. . . . . . . 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 2242	. . . . . . 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 5975	. . . . . . . . . 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 2276	. . . . . . . . 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 11077	. . . . . . . 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 1250	. . . . . . 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 1003	. . . . . . . 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 10315	. . . . . . . . . . 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 3236	. . . . . . . . 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 3197	. . . . . . . 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 1250	. . . . . . 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 2249	. . . . . 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 799	. . . . 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 799	. . 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 5693	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 710   /\ w3a 981   = wceq 1373   e. wcel 2177   C_ wss 3170   Fn wfn 5275   ` cfv 5280  (class class class)co 5957   CCcc 7943   0cc0 7945   + caddc 7948   - cmin 8263   NN0cn0 9315   ZZcz 9392   ZZ>=cuz 9668  ..^cfzo 10284  ♯chash 10942  Word cword 11016   ++ cconcat 11069

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-addass 8047  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-0id 8053  ax-rnegex 8054  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-iord 4421  df-on 4423  df-ilim 4424  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-frec 6490  df-1o 6515  df-er 6633  df-en 6841  df-dom 6842  df-fin 6843  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-inn 9057  df-n0 9316  df-z 9393  df-uz 9669  df-fz 10151  df-fzo 10285  df-ihash 10943  df-word 11017  df-concat 11070

This theorem is referenced by: (None)