Theorem ccatalpha 11326

Description: A concatenation of two arbitrary words is a word over an alphabet iff the symbols of both words belong to the alphabet. (Contributed by AV, 28-Feb-2021.)

Assertion

Ref	Expression
ccatalpha	|- ( ( A e. Word _V /\ B e. Word _V ) -> ( ( A ++ B ) e. Word S <-> ( A e. Word S /\ B e. Word S ) ) )

Proof of Theorem ccatalpha

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

Step	Hyp	Ref	Expression
1		wrdfin 11268	. . . . 5 |- ( A e. Word _V -> A e. Fin )
2		wrdfin 11268	. . . . 5 |- ( B e. Word _V -> B e. Fin )
3		ccatfvalfi 11305	. . . . 5 |- ( ( A e. Fin /\ B e. Fin ) -> ( A ++ B ) = ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) )
4	1, 2, 3	syl2an 289	. . . 4 |- ( ( A e. Word _V /\ B e. Word _V ) -> ( A ++ B ) = ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) )
5	4	eleq1d 2303	. . 3 |- ( ( A e. Word _V /\ B e. Word _V ) -> ( ( A ++ B ) e. Word S <-> ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) e. Word S ) )
6		wrdf 11255	. . . 4 |- ( ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) e. Word S -> ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) : ( 0..^ (♯ ` ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) ) ) --> S )
7		funmpt 5395	. . . . . . . . . . 11 |- Fun ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) )
8	7	a1i 9	. . . . . . . . . 10 |- ( ( A e. Word _V /\ B e. Word _V ) -> Fun ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) )
9	8	funfnd 5388	. . . . . . . . 9 |- ( ( A e. Word _V /\ B e. Word _V ) -> ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) Fn dom ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) )
10		eqid 2234	. . . . . . . . . . 11 |- ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) = ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) )
11		fvexg 5694	. . . . . . . . . . . . 13 |- ( ( A e. Word _V /\ x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) -> ( A ` x ) e. _V )
12	11	adantlr 477	. . . . . . . . . . . 12 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) -> ( A ` x ) e. _V )
13		simplr 529	. . . . . . . . . . . . 13 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) -> B e. Word _V )
14		elfzoelz 10503	. . . . . . . . . . . . . . 15 |- ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) -> x e. ZZ )
15	14	adantl 277	. . . . . . . . . . . . . 14 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) -> x e. ZZ )
16		simpll 527	. . . . . . . . . . . . . . 15 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) -> A e. Word _V )
17		lencl 11253	. . . . . . . . . . . . . . . 16 |- ( A e. Word _V -> (♯ ` A ) e. NN0 )
18	17	nn0zd 9716	. . . . . . . . . . . . . . 15 |- ( A e. Word _V -> (♯ ` A ) e. ZZ )
19	16, 18	syl 14	. . . . . . . . . . . . . 14 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) -> (♯ ` A ) e. ZZ )
20	15, 19	zsubcld 9723	. . . . . . . . . . . . 13 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) -> ( x - (♯ ` A ) ) e. ZZ )
21		fvexg 5694	. . . . . . . . . . . . 13 |- ( ( B e. Word _V /\ ( x - (♯ ` A ) ) e. ZZ ) -> ( B ` ( x - (♯ ` A ) ) ) e. _V )
22	13, 20, 21	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) -> ( B ` ( x - (♯ ` A ) ) ) e. _V )
23	12, 22	ifexd 4610	. . . . . . . . . . 11 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) -> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) e. _V )
24	10, 23	dmmptd 5494	. . . . . . . . . 10 |- ( ( A e. Word _V /\ B e. Word _V ) -> dom ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) = ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) )
25		0z 9605	. . . . . . . . . . 11 |- 0 e. ZZ
26	17	adantr 276	. . . . . . . . . . . . 13 |- ( ( A e. Word _V /\ B e. Word _V ) -> (♯ ` A ) e. NN0 )
27		lencl 11253	. . . . . . . . . . . . . 14 |- ( B e. Word _V -> (♯ ` B ) e. NN0 )
28	27	adantl 277	. . . . . . . . . . . . 13 |- ( ( A e. Word _V /\ B e. Word _V ) -> (♯ ` B ) e. NN0 )
29	26, 28	nn0addcld 9574	. . . . . . . . . . . 12 |- ( ( A e. Word _V /\ B e. Word _V ) -> ( (♯ ` A ) + (♯ ` B ) ) e. NN0 )
30	29	nn0zd 9716	. . . . . . . . . . 11 |- ( ( A e. Word _V /\ B e. Word _V ) -> ( (♯ ` A ) + (♯ ` B ) ) e. ZZ )
31		fzofig 10818	. . . . . . . . . . 11 |- ( ( 0 e. ZZ /\ ( (♯ ` A ) + (♯ ` B ) ) e. ZZ ) -> ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) e. Fin )
32	25, 30, 31	sylancr 414	. . . . . . . . . 10 |- ( ( A e. Word _V /\ B e. Word _V ) -> ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) e. Fin )
33	24, 32	eqeltrd 2311	. . . . . . . . 9 |- ( ( A e. Word _V /\ B e. Word _V ) -> dom ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) e. Fin )
34		fihashfn 11189	. . . . . . . . 9 |- ( ( ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) Fn dom ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) /\ dom ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) e. Fin ) -> (♯ ` ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) ) = (♯ ` dom ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) ) )
35	9, 33, 34	syl2anc 411	. . . . . . . 8 |- ( ( A e. Word _V /\ B e. Word _V ) -> (♯ ` ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) ) = (♯ ` dom ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) ) )
36	24	fveq2d 5679	. . . . . . . 8 |- ( ( A e. Word _V /\ B e. Word _V ) -> (♯ ` dom ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) ) = (♯ ` ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) )
37		nn0addcl 9548	. . . . . . . . . 10 |- ( ( (♯ ` A ) e. NN0 /\ (♯ ` B ) e. NN0 ) -> ( (♯ ` A ) + (♯ ` B ) ) e. NN0 )
38	17, 27, 37	syl2an 289	. . . . . . . . 9 |- ( ( A e. Word _V /\ B e. Word _V ) -> ( (♯ ` A ) + (♯ ` B ) ) e. NN0 )
39		hashfzo0 11213	. . . . . . . . 9 |- ( ( (♯ ` A ) + (♯ ` B ) ) e. NN0 -> (♯ ` ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
40	38, 39	syl 14	. . . . . . . 8 |- ( ( A e. Word _V /\ B e. Word _V ) -> (♯ ` ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
41	35, 36, 40	3eqtrd 2271	. . . . . . 7 |- ( ( A e. Word _V /\ B e. Word _V ) -> (♯ ` ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
42	41	oveq2d 6074	. . . . . 6 |- ( ( A e. Word _V /\ B e. Word _V ) -> ( 0..^ (♯ ` ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) ) ) = ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) )
43	42	feq2d 5501	. . . . 5 |- ( ( A e. Word _V /\ B e. Word _V ) -> ( ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) : ( 0..^ (♯ ` ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) ) ) --> S <-> ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) : ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) --> S ) )
44	10	fmpt 5832	. . . . . 6 |- ( A. x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) e. S <-> ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) : ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) --> S )
45		simpl 109	. . . . . . . . 9 |- ( ( A e. Word _V /\ B e. Word _V ) -> A e. Word _V )
46		nn0cn 9523	. . . . . . . . . . . . . . . . . 18 |- ( (♯ ` A ) e. NN0 -> (♯ ` A ) e. CC )
47		nn0cn 9523	. . . . . . . . . . . . . . . . . 18 |- ( (♯ ` B ) e. NN0 -> (♯ ` B ) e. CC )
48		addcom 8426	. . . . . . . . . . . . . . . . . 18 |- ( ( (♯ ` A ) e. CC /\ (♯ ` B ) e. CC ) -> ( (♯ ` A ) + (♯ ` B ) ) = ( (♯ ` B ) + (♯ ` A ) ) )
49	46, 47, 48	syl2an 289	. . . . . . . . . . . . . . . . 17 |- ( ( (♯ ` A ) e. NN0 /\ (♯ ` B ) e. NN0 ) -> ( (♯ ` A ) + (♯ ` B ) ) = ( (♯ ` B ) + (♯ ` A ) ) )
50		nn0z 9614	. . . . . . . . . . . . . . . . . . 19 |- ( (♯ ` A ) e. NN0 -> (♯ ` A ) e. ZZ )
51	50	anim1ci 341	. . . . . . . . . . . . . . . . . 18 |- ( ( (♯ ` A ) e. NN0 /\ (♯ ` B ) e. NN0 ) -> ( (♯ ` B ) e. NN0 /\ (♯ ` A ) e. ZZ ) )
52		nn0pzuz 9937	. . . . . . . . . . . . . . . . . 18 |- ( ( (♯ ` B ) e. NN0 /\ (♯ ` A ) e. ZZ ) -> ( (♯ ` B ) + (♯ ` A ) ) e. ( ZZ>= ` (♯ ` A ) ) )
53	51, 52	syl 14	. . . . . . . . . . . . . . . . 17 |- ( ( (♯ ` A ) e. NN0 /\ (♯ ` B ) e. NN0 ) -> ( (♯ ` B ) + (♯ ` A ) ) e. ( ZZ>= ` (♯ ` A ) ) )
54	49, 53	eqeltrd 2311	. . . . . . . . . . . . . . . 16 |- ( ( (♯ ` A ) e. NN0 /\ (♯ ` B ) e. NN0 ) -> ( (♯ ` A ) + (♯ ` B ) ) e. ( ZZ>= ` (♯ ` A ) ) )
55	17, 27, 54	syl2an 289	. . . . . . . . . . . . . . 15 |- ( ( A e. Word _V /\ B e. Word _V ) -> ( (♯ ` A ) + (♯ ` B ) ) e. ( ZZ>= ` (♯ ` A ) ) )
56		fzoss2 10530	. . . . . . . . . . . . . . 15 |- ( ( (♯ ` A ) + (♯ ` B ) ) e. ( ZZ>= ` (♯ ` A ) ) -> ( 0..^ (♯ ` A ) ) C_ ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) )
57	55, 56	syl 14	. . . . . . . . . . . . . 14 |- ( ( A e. Word _V /\ B e. Word _V ) -> ( 0..^ (♯ ` A ) ) C_ ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) )
58	57	sselda 3242	. . . . . . . . . . . . 13 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` A ) ) ) -> y e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) )
59		eleq1 2297	. . . . . . . . . . . . . . . 16 |- ( x = y -> ( x e. ( 0..^ (♯ ` A ) ) <-> y e. ( 0..^ (♯ ` A ) ) ) )
60		fveq2 5675	. . . . . . . . . . . . . . . 16 |- ( x = y -> ( A ` x ) = ( A ` y ) )
61		fvoveq1 6081	. . . . . . . . . . . . . . . 16 |- ( x = y -> ( B ` ( x - (♯ ` A ) ) ) = ( B ` ( y - (♯ ` A ) ) ) )
62	59, 60, 61	ifbieq12d 3653	. . . . . . . . . . . . . . 15 |- ( x = y -> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) = if ( y e. ( 0..^ (♯ ` A ) ) , ( A ` y ) , ( B ` ( y - (♯ ` A ) ) ) ) )
63	62	eleq1d 2303	. . . . . . . . . . . . . 14 |- ( x = y -> ( if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) e. S <-> if ( y e. ( 0..^ (♯ ` A ) ) , ( A ` y ) , ( B ` ( y - (♯ ` A ) ) ) ) e. S ) )
64	63	rspcv 2919	. . . . . . . . . . . . 13 |- ( y e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) -> ( A. x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) e. S -> if ( y e. ( 0..^ (♯ ` A ) ) , ( A ` y ) , ( B ` ( y - (♯ ` A ) ) ) ) e. S ) )
65	58, 64	syl 14	. . . . . . . . . . . 12 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` A ) ) ) -> ( A. x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) e. S -> if ( y e. ( 0..^ (♯ ` A ) ) , ( A ` y ) , ( B ` ( y - (♯ ` A ) ) ) ) e. S ) )
66		iftrue 3631	. . . . . . . . . . . . . 14 |- ( y e. ( 0..^ (♯ ` A ) ) -> if ( y e. ( 0..^ (♯ ` A ) ) , ( A ` y ) , ( B ` ( y - (♯ ` A ) ) ) ) = ( A ` y ) )
67	66	adantl 277	. . . . . . . . . . . . 13 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` A ) ) ) -> if ( y e. ( 0..^ (♯ ` A ) ) , ( A ` y ) , ( B ` ( y - (♯ ` A ) ) ) ) = ( A ` y ) )
68	67	eleq1d 2303	. . . . . . . . . . . 12 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` A ) ) ) -> ( if ( y e. ( 0..^ (♯ ` A ) ) , ( A ` y ) , ( B ` ( y - (♯ ` A ) ) ) ) e. S <-> ( A ` y ) e. S ) )
69	65, 68	sylibd 149	. . . . . . . . . . 11 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` A ) ) ) -> ( A. x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) e. S -> ( A ` y ) e. S ) )
70	69	impancom 260	. . . . . . . . . 10 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ A. x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) e. S ) -> ( y e. ( 0..^ (♯ ` A ) ) -> ( A ` y ) e. S ) )
71	70	ralrimiv 2616	. . . . . . . . 9 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ A. x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) e. S ) -> A. y e. ( 0..^ (♯ ` A ) ) ( A ` y ) e. S )
72		iswrdsymb 11267	. . . . . . . . 9 |- ( ( A e. Word _V /\ A. y e. ( 0..^ (♯ ` A ) ) ( A ` y ) e. S ) -> A e. Word S )
73	45, 71, 72	syl2an2r 599	. . . . . . . 8 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ A. x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) e. S ) -> A e. Word S )
74		simpr 110	. . . . . . . . 9 |- ( ( A e. Word _V /\ B e. Word _V ) -> B e. Word _V )
75		simpr 110	. . . . . . . . . . . . . . 15 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` B ) ) ) -> y e. ( 0..^ (♯ ` B ) ) )
76	26	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` B ) ) ) -> (♯ ` A ) e. NN0 )
77		elincfzoext 10560	. . . . . . . . . . . . . . 15 |- ( ( y e. ( 0..^ (♯ ` B ) ) /\ (♯ ` A ) e. NN0 ) -> ( y + (♯ ` A ) ) e. ( 0..^ ( (♯ ` B ) + (♯ ` A ) ) ) )
78	75, 76, 77	syl2anc 411	. . . . . . . . . . . . . 14 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` B ) ) ) -> ( y + (♯ ` A ) ) e. ( 0..^ ( (♯ ` B ) + (♯ ` A ) ) ) )
79	17	nn0cnd 9572	. . . . . . . . . . . . . . . . . 18 |- ( A e. Word _V -> (♯ ` A ) e. CC )
80	27	nn0cnd 9572	. . . . . . . . . . . . . . . . . 18 |- ( B e. Word _V -> (♯ ` B ) e. CC )
81	79, 80, 48	syl2an 289	. . . . . . . . . . . . . . . . 17 |- ( ( A e. Word _V /\ B e. Word _V ) -> ( (♯ ` A ) + (♯ ` B ) ) = ( (♯ ` B ) + (♯ ` A ) ) )
82	81	oveq2d 6074	. . . . . . . . . . . . . . . 16 |- ( ( A e. Word _V /\ B e. Word _V ) -> ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) = ( 0..^ ( (♯ ` B ) + (♯ ` A ) ) ) )
83	82	eleq2d 2304	. . . . . . . . . . . . . . 15 |- ( ( A e. Word _V /\ B e. Word _V ) -> ( ( y + (♯ ` A ) ) e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) <-> ( y + (♯ ` A ) ) e. ( 0..^ ( (♯ ` B ) + (♯ ` A ) ) ) ) )
84	83	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` B ) ) ) -> ( ( y + (♯ ` A ) ) e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) <-> ( y + (♯ ` A ) ) e. ( 0..^ ( (♯ ` B ) + (♯ ` A ) ) ) ) )
85	78, 84	mpbird 167	. . . . . . . . . . . . 13 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` B ) ) ) -> ( y + (♯ ` A ) ) e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) )
86		eleq1 2297	. . . . . . . . . . . . . . . 16 |- ( x = ( y + (♯ ` A ) ) -> ( x e. ( 0..^ (♯ ` A ) ) <-> ( y + (♯ ` A ) ) e. ( 0..^ (♯ ` A ) ) ) )
87		fveq2 5675	. . . . . . . . . . . . . . . 16 |- ( x = ( y + (♯ ` A ) ) -> ( A ` x ) = ( A ` ( y + (♯ ` A ) ) ) )
88		fvoveq1 6081	. . . . . . . . . . . . . . . 16 |- ( x = ( y + (♯ ` A ) ) -> ( B ` ( x - (♯ ` A ) ) ) = ( B ` ( ( y + (♯ ` A ) ) - (♯ ` A ) ) ) )
89	86, 87, 88	ifbieq12d 3653	. . . . . . . . . . . . . . 15 |- ( x = ( y + (♯ ` A ) ) -> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) = if ( ( y + (♯ ` A ) ) e. ( 0..^ (♯ ` A ) ) , ( A ` ( y + (♯ ` A ) ) ) , ( B ` ( ( y + (♯ ` A ) ) - (♯ ` A ) ) ) ) )
90	89	eleq1d 2303	. . . . . . . . . . . . . 14 |- ( x = ( y + (♯ ` A ) ) -> ( if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) e. S <-> if ( ( y + (♯ ` A ) ) e. ( 0..^ (♯ ` A ) ) , ( A ` ( y + (♯ ` A ) ) ) , ( B ` ( ( y + (♯ ` A ) ) - (♯ ` A ) ) ) ) e. S ) )
91	90	rspcv 2919	. . . . . . . . . . . . 13 |- ( ( y + (♯ ` A ) ) e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) -> ( A. x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) e. S -> if ( ( y + (♯ ` A ) ) e. ( 0..^ (♯ ` A ) ) , ( A ` ( y + (♯ ` A ) ) ) , ( B ` ( ( y + (♯ ` A ) ) - (♯ ` A ) ) ) ) e. S ) )
92	85, 91	syl 14	. . . . . . . . . . . 12 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` B ) ) ) -> ( A. x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) e. S -> if ( ( y + (♯ ` A ) ) e. ( 0..^ (♯ ` A ) ) , ( A ` ( y + (♯ ` A ) ) ) , ( B ` ( ( y + (♯ ` A ) ) - (♯ ` A ) ) ) ) e. S ) )
93	17	nn0red 9571	. . . . . . . . . . . . . . . . . . . 20 |- ( A e. Word _V -> (♯ ` A ) e. RR )
94	93	adantr 276	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. Word _V /\ B e. Word _V ) -> (♯ ` A ) e. RR )
95	94	adantr 276	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` B ) ) ) -> (♯ ` A ) e. RR )
96		elfzoelz 10503	. . . . . . . . . . . . . . . . . . . . . 22 |- ( y e. ( 0..^ (♯ ` B ) ) -> y e. ZZ )
97	96	zred 9718	. . . . . . . . . . . . . . . . . . . . 21 |- ( y e. ( 0..^ (♯ ` B ) ) -> y e. RR )
98	97	adantr 276	. . . . . . . . . . . . . . . . . . . 20 |- ( ( y e. ( 0..^ (♯ ` B ) ) /\ ( A e. Word _V /\ B e. Word _V ) ) -> y e. RR )
99	94	adantl 277	. . . . . . . . . . . . . . . . . . . 20 |- ( ( y e. ( 0..^ (♯ ` B ) ) /\ ( A e. Word _V /\ B e. Word _V ) ) -> (♯ ` A ) e. RR )
100	98, 99	readdcld 8319	. . . . . . . . . . . . . . . . . . 19 |- ( ( y e. ( 0..^ (♯ ` B ) ) /\ ( A e. Word _V /\ B e. Word _V ) ) -> ( y + (♯ ` A ) ) e. RR )
101	100	ancoms 268	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` B ) ) ) -> ( y + (♯ ` A ) ) e. RR )
102		elfzole1 10512	. . . . . . . . . . . . . . . . . . . 20 |- ( y e. ( 0..^ (♯ ` B ) ) -> 0 <_ y )
103	102	adantl 277	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` B ) ) ) -> 0 <_ y )
104		addge02 8764	. . . . . . . . . . . . . . . . . . . 20 |- ( ( (♯ ` A ) e. RR /\ y e. RR ) -> ( 0 <_ y <-> (♯ ` A ) <_ ( y + (♯ ` A ) ) ) )
105	94, 97, 104	syl2an 289	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` B ) ) ) -> ( 0 <_ y <-> (♯ ` A ) <_ ( y + (♯ ` A ) ) ) )
106	103, 105	mpbid 147	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` B ) ) ) -> (♯ ` A ) <_ ( y + (♯ ` A ) ) )
107	95, 101, 106	lensymd 8411	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` B ) ) ) -> -. ( y + (♯ ` A ) ) < (♯ ` A ) )
108	107	intn3an3d 1395	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` B ) ) ) -> -. ( ( y + (♯ ` A ) ) e. NN0 /\ (♯ ` A ) e. NN /\ ( y + (♯ ` A ) ) < (♯ ` A ) ) )
109		elfzo0 10542	. . . . . . . . . . . . . . . 16 |- ( ( y + (♯ ` A ) ) e. ( 0..^ (♯ ` A ) ) <-> ( ( y + (♯ ` A ) ) e. NN0 /\ (♯ ` A ) e. NN /\ ( y + (♯ ` A ) ) < (♯ ` A ) ) )
110	108, 109	sylnibr 684	. . . . . . . . . . . . . . 15 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` B ) ) ) -> -. ( y + (♯ ` A ) ) e. ( 0..^ (♯ ` A ) ) )
111	110	iffalsed 3636	. . . . . . . . . . . . . 14 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` B ) ) ) -> if ( ( y + (♯ ` A ) ) e. ( 0..^ (♯ ` A ) ) , ( A ` ( y + (♯ ` A ) ) ) , ( B ` ( ( y + (♯ ` A ) ) - (♯ ` A ) ) ) ) = ( B ` ( ( y + (♯ ` A ) ) - (♯ ` A ) ) ) )
112	111	eleq1d 2303	. . . . . . . . . . . . 13 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` B ) ) ) -> ( if ( ( y + (♯ ` A ) ) e. ( 0..^ (♯ ` A ) ) , ( A ` ( y + (♯ ` A ) ) ) , ( B ` ( ( y + (♯ ` A ) ) - (♯ ` A ) ) ) ) e. S <-> ( B ` ( ( y + (♯ ` A ) ) - (♯ ` A ) ) ) e. S ) )
113	96	zcnd 9719	. . . . . . . . . . . . . . . . 17 |- ( y e. ( 0..^ (♯ ` B ) ) -> y e. CC )
114	79	adantr 276	. . . . . . . . . . . . . . . . 17 |- ( ( A e. Word _V /\ B e. Word _V ) -> (♯ ` A ) e. CC )
115		pncan 8495	. . . . . . . . . . . . . . . . 17 |- ( ( y e. CC /\ (♯ ` A ) e. CC ) -> ( ( y + (♯ ` A ) ) - (♯ ` A ) ) = y )
116	113, 114, 115	syl2anr 290	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` B ) ) ) -> ( ( y + (♯ ` A ) ) - (♯ ` A ) ) = y )
117	116	fveq2d 5679	. . . . . . . . . . . . . . 15 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` B ) ) ) -> ( B ` ( ( y + (♯ ` A ) ) - (♯ ` A ) ) ) = ( B ` y ) )
118	117	eleq1d 2303	. . . . . . . . . . . . . 14 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` B ) ) ) -> ( ( B ` ( ( y + (♯ ` A ) ) - (♯ ` A ) ) ) e. S <-> ( B ` y ) e. S ) )
119	118	biimpd 144	. . . . . . . . . . . . 13 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` B ) ) ) -> ( ( B ` ( ( y + (♯ ` A ) ) - (♯ ` A ) ) ) e. S -> ( B ` y ) e. S ) )
120	112, 119	sylbid 150	. . . . . . . . . . . 12 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` B ) ) ) -> ( if ( ( y + (♯ ` A ) ) e. ( 0..^ (♯ ` A ) ) , ( A ` ( y + (♯ ` A ) ) ) , ( B ` ( ( y + (♯ ` A ) ) - (♯ ` A ) ) ) ) e. S -> ( B ` y ) e. S ) )
121	92, 120	syld 45	. . . . . . . . . . 11 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ y e. ( 0..^ (♯ ` B ) ) ) -> ( A. x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) e. S -> ( B ` y ) e. S ) )
122	121	impancom 260	. . . . . . . . . 10 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ A. x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) e. S ) -> ( y e. ( 0..^ (♯ ` B ) ) -> ( B ` y ) e. S ) )
123	122	ralrimiv 2616	. . . . . . . . 9 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ A. x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) e. S ) -> A. y e. ( 0..^ (♯ ` B ) ) ( B ` y ) e. S )
124		iswrdsymb 11267	. . . . . . . . 9 |- ( ( B e. Word _V /\ A. y e. ( 0..^ (♯ ` B ) ) ( B ` y ) e. S ) -> B e. Word S )
125	74, 123, 124	syl2an2r 599	. . . . . . . 8 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ A. x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) e. S ) -> B e. Word S )
126	73, 125	jca 306	. . . . . . 7 |- ( ( ( A e. Word _V /\ B e. Word _V ) /\ A. x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) e. S ) -> ( A e. Word S /\ B e. Word S ) )
127	126	ex 115	. . . . . 6 |- ( ( A e. Word _V /\ B e. Word _V ) -> ( A. x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) e. S -> ( A e. Word S /\ B e. Word S ) ) )
128	44, 127	biimtrrid 153	. . . . 5 |- ( ( A e. Word _V /\ B e. Word _V ) -> ( ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) : ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) --> S -> ( A e. Word S /\ B e. Word S ) ) )
129	43, 128	sylbid 150	. . . 4 |- ( ( A e. Word _V /\ B e. Word _V ) -> ( ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) : ( 0..^ (♯ ` ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) ) ) --> S -> ( A e. Word S /\ B e. Word S ) ) )
130	6, 129	syl5 32	. . 3 |- ( ( A e. Word _V /\ B e. Word _V ) -> ( ( x e. ( 0..^ ( (♯ ` A ) + (♯ ` B ) ) ) |-> if ( x e. ( 0..^ (♯ ` A ) ) , ( A ` x ) , ( B ` ( x - (♯ ` A ) ) ) ) ) e. Word S -> ( A e. Word S /\ B e. Word S ) ) )
131	5, 130	sylbid 150	. 2 |- ( ( A e. Word _V /\ B e. Word _V ) -> ( ( A ++ B ) e. Word S -> ( A e. Word S /\ B e. Word S ) ) )
132		ccatcl 11306	. 2 |- ( ( A e. Word S /\ B e. Word S ) -> ( A ++ B ) e. Word S )
133	131, 132	impbid1 142	1 |- ( ( A e. Word _V /\ B e. Word _V ) -> ( ( A ++ B ) e. Word S <-> ( A e. Word S /\ B e. Word S ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   A.wral 2522   _Vcvv 2815   C_ wss 3214   ifcif 3624   class class class wbr 4114   |-> cmpt 4176   dom cdm 4754   Fun wfun 5351   Fn wfn 5352   -->wf 5353   ` cfv 5357  (class class class)co 6058   Fincfn 6988   CCcc 8141   RRcr 8142   0cc0 8143   + caddc 8146   < clt 8324   <_ cle 8325   - cmin 8460   NNcn 9254   NN0cn0 9513   ZZcz 9594   ZZ>=cuz 9871  ..^cfzo 10498  ♯chash 11163  Word cword 11249   ++ cconcat 11303

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-concat 11304

This theorem is referenced by:  ccatrcl1  11327