Theorem ccatrn 11139

Description: The range of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.)

Assertion

Ref	Expression
ccatrn	|- ( ( S e. Word B /\ T e. Word B ) -> ran ( S ++ T ) = ( ran S u. ran T ) )

Proof of Theorem ccatrn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ccatvalfn 11131	. . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) Fn ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
2		lencl 11070	. . . . . . . . . . . 12 |- ( S e. Word B -> (♯ ` S ) e. NN0 )
3		nn0uz 9753	. . . . . . . . . . . 12 |- NN0 = ( ZZ>= ` 0 )
4	2, 3	eleqtrdi 2322	. . . . . . . . . . 11 |- ( S e. Word B -> (♯ ` S ) e. ( ZZ>= ` 0 ) )
5	4	adantr 276	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` S ) e. ( ZZ>= ` 0 ) )
6	2	nn0zd 9563	. . . . . . . . . . . 12 |- ( S e. Word B -> (♯ ` S ) e. ZZ )
7	6	uzidd 9733	. . . . . . . . . . 11 |- ( S e. Word B -> (♯ ` S ) e. ( ZZ>= ` (♯ ` S ) ) )
8		lencl 11070	. . . . . . . . . . 11 |- ( T e. Word B -> (♯ ` T ) e. NN0 )
9		uzaddcl 9777	. . . . . . . . . . 11 |- ( ( (♯ ` S ) e. ( ZZ>= ` (♯ ` S ) ) /\ (♯ ` T ) e. NN0 ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` (♯ ` S ) ) )
10	7, 8, 9	syl2an 289	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` (♯ ` S ) ) )
11		elfzuzb 10211	. . . . . . . . . 10 |- ( (♯ ` S ) e. ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) <-> ( (♯ ` S ) e. ( ZZ>= ` 0 ) /\ ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` (♯ ` S ) ) ) )
12	5, 10, 11	sylanbrc 417	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` S ) e. ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) )
13		fzosplit 10371	. . . . . . . . 9 |- ( (♯ ` S ) e. ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) -> ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) = ( ( 0..^ (♯ ` S ) ) u. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) )
14	12, 13	syl 14	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) = ( ( 0..^ (♯ ` S ) ) u. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) )
15	14	eleq2d 2299	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B ) -> ( x e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) <-> x e. ( ( 0..^ (♯ ` S ) ) u. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) ) )
16		elun 3345	. . . . . . 7 |- ( x e. ( ( 0..^ (♯ ` S ) ) u. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) <-> ( x e. ( 0..^ (♯ ` S ) ) \/ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) )
17	15, 16	bitrdi 196	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B ) -> ( x e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) <-> ( x e. ( 0..^ (♯ ` S ) ) \/ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) ) )
18		ccatval1 11127	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` x ) = ( S ` x ) )
19	18	3expa 1227	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` x ) = ( S ` x ) )
20		ssun1 3367	. . . . . . . . . 10 |- ran S C_ ( ran S u. ran T )
21		wrdfn 11081	. . . . . . . . . . . 12 |- ( S e. Word B -> S Fn ( 0..^ (♯ ` S ) ) )
22	21	adantr 276	. . . . . . . . . . 11 |- ( ( S e. Word B /\ T e. Word B ) -> S Fn ( 0..^ (♯ ` S ) ) )
23		fnfvelrn 5766	. . . . . . . . . . 11 |- ( ( S Fn ( 0..^ (♯ ` S ) ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( S ` x ) e. ran S )
24	22, 23	sylan 283	. . . . . . . . . 10 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( S ` x ) e. ran S )
25	20, 24	sselid 3222	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( S ` x ) e. ( ran S u. ran T ) )
26	19, 25	eqeltrd 2306	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` x ) e. ( ran S u. ran T ) )
27		ccatval2 11128	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( S ++ T ) ` x ) = ( T ` ( x - (♯ ` S ) ) ) )
28	27	3expa 1227	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( S ++ T ) ` x ) = ( T ` ( x - (♯ ` S ) ) ) )
29		ssun2 3368	. . . . . . . . . 10 |- ran T C_ ( ran S u. ran T )
30		wrdfn 11081	. . . . . . . . . . . 12 |- ( T e. Word B -> T Fn ( 0..^ (♯ ` T ) ) )
31	30	adantl 277	. . . . . . . . . . 11 |- ( ( S e. Word B /\ T e. Word B ) -> T Fn ( 0..^ (♯ ` T ) ) )
32		elfzouz 10343	. . . . . . . . . . . . . . 15 |- ( x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) -> x e. ( ZZ>= ` (♯ ` S ) ) )
33		uznn0sub 9750	. . . . . . . . . . . . . . 15 |- ( x e. ( ZZ>= ` (♯ ` S ) ) -> ( x - (♯ ` S ) ) e. NN0 )
34	32, 33	syl 14	. . . . . . . . . . . . . 14 |- ( x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) -> ( x - (♯ ` S ) ) e. NN0 )
35	34, 3	eleqtrdi 2322	. . . . . . . . . . . . 13 |- ( x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) -> ( x - (♯ ` S ) ) e. ( ZZ>= ` 0 ) )
36	35	adantl 277	. . . . . . . . . . . 12 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( x - (♯ ` S ) ) e. ( ZZ>= ` 0 ) )
37	8	nn0zd 9563	. . . . . . . . . . . . 13 |- ( T e. Word B -> (♯ ` T ) e. ZZ )
38	37	ad2antlr 489	. . . . . . . . . . . 12 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> (♯ ` T ) e. ZZ )
39		elfzolt2 10349	. . . . . . . . . . . . . 14 |- ( x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) -> x < ( (♯ ` S ) + (♯ ` T ) ) )
40	39	adantl 277	. . . . . . . . . . . . 13 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> x < ( (♯ ` S ) + (♯ ` T ) ) )
41		elfzoelz 10339	. . . . . . . . . . . . . . . 16 |- ( x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) -> x e. ZZ )
42	41	zred 9565	. . . . . . . . . . . . . . 15 |- ( x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) -> x e. RR )
43	42	adantl 277	. . . . . . . . . . . . . 14 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> x e. RR )
44	2	nn0red 9419	. . . . . . . . . . . . . . 15 |- ( S e. Word B -> (♯ ` S ) e. RR )
45	44	ad2antrr 488	. . . . . . . . . . . . . 14 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> (♯ ` S ) e. RR )
46	8	nn0red 9419	. . . . . . . . . . . . . . 15 |- ( T e. Word B -> (♯ ` T ) e. RR )
47	46	ad2antlr 489	. . . . . . . . . . . . . 14 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> (♯ ` T ) e. RR )
48	43, 45, 47	ltsubadd2d 8686	. . . . . . . . . . . . 13 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( x - (♯ ` S ) ) < (♯ ` T ) <-> x < ( (♯ ` S ) + (♯ ` T ) ) ) )
49	40, 48	mpbird 167	. . . . . . . . . . . 12 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( x - (♯ ` S ) ) < (♯ ` T ) )
50		elfzo2 10342	. . . . . . . . . . . 12 |- ( ( x - (♯ ` S ) ) e. ( 0..^ (♯ ` T ) ) <-> ( ( x - (♯ ` S ) ) e. ( ZZ>= ` 0 ) /\ (♯ ` T ) e. ZZ /\ ( x - (♯ ` S ) ) < (♯ ` T ) ) )
51	36, 38, 49, 50	syl3anbrc 1205	. . . . . . . . . . 11 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( x - (♯ ` S ) ) e. ( 0..^ (♯ ` T ) ) )
52		fnfvelrn 5766	. . . . . . . . . . 11 |- ( ( T Fn ( 0..^ (♯ ` T ) ) /\ ( x - (♯ ` S ) ) e. ( 0..^ (♯ ` T ) ) ) -> ( T ` ( x - (♯ ` S ) ) ) e. ran T )
53	31, 51, 52	syl2an2r 597	. . . . . . . . . 10 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( T ` ( x - (♯ ` S ) ) ) e. ran T )
54	29, 53	sselid 3222	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( T ` ( x - (♯ ` S ) ) ) e. ( ran S u. ran T ) )
55	28, 54	eqeltrd 2306	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( S ++ T ) ` x ) e. ( ran S u. ran T ) )
56	26, 55	jaodan 802	. . . . . . 7 |- ( ( ( S e. Word B /\ T e. Word B ) /\ ( x e. ( 0..^ (♯ ` S ) ) \/ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) ) -> ( ( S ++ T ) ` x ) e. ( ran S u. ran T ) )
57	56	ex 115	. . . . . 6 |- ( ( S e. Word B /\ T e. Word B ) -> ( ( x e. ( 0..^ (♯ ` S ) ) \/ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( S ++ T ) ` x ) e. ( ran S u. ran T ) ) )
58	17, 57	sylbid 150	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> ( x e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) -> ( ( S ++ T ) ` x ) e. ( ran S u. ran T ) ) )
59	58	ralrimiv 2602	. . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> A. x e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) ( ( S ++ T ) ` x ) e. ( ran S u. ran T ) )
60		ffnfv 5792	. . . 4 |- ( ( S ++ T ) : ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) --> ( ran S u. ran T ) <-> ( ( S ++ T ) Fn ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) /\ A. x e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) ( ( S ++ T ) ` x ) e. ( ran S u. ran T ) ) )
61	1, 59, 60	sylanbrc 417	. . 3 |- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) : ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) --> ( ran S u. ran T ) )
62	61	frnd 5482	. 2 |- ( ( S e. Word B /\ T e. Word B ) -> ran ( S ++ T ) C_ ( ran S u. ran T ) )
63		fzoss2 10366	. . . . . . . . . 10 |- ( ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` (♯ ` S ) ) -> ( 0..^ (♯ ` S ) ) C_ ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
64	10, 63	syl 14	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> ( 0..^ (♯ ` S ) ) C_ ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
65	64	sselda 3224	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> x e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
66		fnfvelrn 5766	. . . . . . . 8 |- ( ( ( S ++ T ) Fn ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) /\ x e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( S ++ T ) ` x ) e. ran ( S ++ T ) )
67	1, 65, 66	syl2an2r 597	. . . . . . 7 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` x ) e. ran ( S ++ T ) )
68	19, 67	eqeltrrd 2307	. . . . . 6 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( S ` x ) e. ran ( S ++ T ) )
69	68	ralrimiva 2603	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> A. x e. ( 0..^ (♯ ` S ) ) ( S ` x ) e. ran ( S ++ T ) )
70		ffnfv 5792	. . . . 5 |- ( S : ( 0..^ (♯ ` S ) ) --> ran ( S ++ T ) <-> ( S Fn ( 0..^ (♯ ` S ) ) /\ A. x e. ( 0..^ (♯ ` S ) ) ( S ` x ) e. ran ( S ++ T ) ) )
71	22, 69, 70	sylanbrc 417	. . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> S : ( 0..^ (♯ ` S ) ) --> ran ( S ++ T ) )
72	71	frnd 5482	. . 3 |- ( ( S e. Word B /\ T e. Word B ) -> ran S C_ ran ( S ++ T ) )
73		ccatval3 11129	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( ( S ++ T ) ` ( x + (♯ ` S ) ) ) = ( T ` x ) )
74	73	3expa 1227	. . . . . . 7 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( ( S ++ T ) ` ( x + (♯ ` S ) ) ) = ( T ` x ) )
75		elfzouz 10343	. . . . . . . . . 10 |- ( x e. ( 0..^ (♯ ` T ) ) -> x e. ( ZZ>= ` 0 ) )
76	2	adantr 276	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` S ) e. NN0 )
77		uzaddcl 9777	. . . . . . . . . 10 |- ( ( x e. ( ZZ>= ` 0 ) /\ (♯ ` S ) e. NN0 ) -> ( x + (♯ ` S ) ) e. ( ZZ>= ` 0 ) )
78	75, 76, 77	syl2anr 290	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( x + (♯ ` S ) ) e. ( ZZ>= ` 0 ) )
79		nn0addcl 9400	. . . . . . . . . . . 12 |- ( ( (♯ ` S ) e. NN0 /\ (♯ ` T ) e. NN0 ) -> ( (♯ ` S ) + (♯ ` T ) ) e. NN0 )
80	2, 8, 79	syl2an 289	. . . . . . . . . . 11 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. NN0 )
81	80	nn0zd 9563	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ZZ )
82	81	adantr 276	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( (♯ ` S ) + (♯ ` T ) ) e. ZZ )
83		elfzonn0 10382	. . . . . . . . . . . 12 |- ( x e. ( 0..^ (♯ ` T ) ) -> x e. NN0 )
84	83	nn0cnd 9420	. . . . . . . . . . 11 |- ( x e. ( 0..^ (♯ ` T ) ) -> x e. CC )
85	2	nn0cnd 9420	. . . . . . . . . . . 12 |- ( S e. Word B -> (♯ ` S ) e. CC )
86	85	adantr 276	. . . . . . . . . . 11 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` S ) e. CC )
87		addcom 8279	. . . . . . . . . . 11 |- ( ( x e. CC /\ (♯ ` S ) e. CC ) -> ( x + (♯ ` S ) ) = ( (♯ ` S ) + x ) )
88	84, 86, 87	syl2anr 290	. . . . . . . . . 10 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( x + (♯ ` S ) ) = ( (♯ ` S ) + x ) )
89	83	nn0red 9419	. . . . . . . . . . . 12 |- ( x e. ( 0..^ (♯ ` T ) ) -> x e. RR )
90	89	adantl 277	. . . . . . . . . . 11 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> x e. RR )
91	46	ad2antlr 489	. . . . . . . . . . 11 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> (♯ ` T ) e. RR )
92	44	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> (♯ ` S ) e. RR )
93		elfzolt2 10349	. . . . . . . . . . . 12 |- ( x e. ( 0..^ (♯ ` T ) ) -> x < (♯ ` T ) )
94	93	adantl 277	. . . . . . . . . . 11 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> x < (♯ ` T ) )
95	90, 91, 92, 94	ltadd2dd 8565	. . . . . . . . . 10 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( (♯ ` S ) + x ) < ( (♯ ` S ) + (♯ ` T ) ) )
96	88, 95	eqbrtrd 4104	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( x + (♯ ` S ) ) < ( (♯ ` S ) + (♯ ` T ) ) )
97		elfzo2 10342	. . . . . . . . 9 |- ( ( x + (♯ ` S ) ) e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) <-> ( ( x + (♯ ` S ) ) e. ( ZZ>= ` 0 ) /\ ( (♯ ` S ) + (♯ ` T ) ) e. ZZ /\ ( x + (♯ ` S ) ) < ( (♯ ` S ) + (♯ ` T ) ) ) )
98	78, 82, 96, 97	syl3anbrc 1205	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( x + (♯ ` S ) ) e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
99		fnfvelrn 5766	. . . . . . . 8 |- ( ( ( S ++ T ) Fn ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) /\ ( x + (♯ ` S ) ) e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( S ++ T ) ` ( x + (♯ ` S ) ) ) e. ran ( S ++ T ) )
100	1, 98, 99	syl2an2r 597	. . . . . . 7 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( ( S ++ T ) ` ( x + (♯ ` S ) ) ) e. ran ( S ++ T ) )
101	74, 100	eqeltrrd 2307	. . . . . 6 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( T ` x ) e. ran ( S ++ T ) )
102	101	ralrimiva 2603	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> A. x e. ( 0..^ (♯ ` T ) ) ( T ` x ) e. ran ( S ++ T ) )
103		ffnfv 5792	. . . . 5 |- ( T : ( 0..^ (♯ ` T ) ) --> ran ( S ++ T ) <-> ( T Fn ( 0..^ (♯ ` T ) ) /\ A. x e. ( 0..^ (♯ ` T ) ) ( T ` x ) e. ran ( S ++ T ) ) )
104	31, 102, 103	sylanbrc 417	. . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> T : ( 0..^ (♯ ` T ) ) --> ran ( S ++ T ) )
105	104	frnd 5482	. . 3 |- ( ( S e. Word B /\ T e. Word B ) -> ran T C_ ran ( S ++ T ) )
106	72, 105	unssd 3380	. 2 |- ( ( S e. Word B /\ T e. Word B ) -> ( ran S u. ran T ) C_ ran ( S ++ T ) )
107	62, 106	eqssd 3241	1 |- ( ( S e. Word B /\ T e. Word B ) -> ran ( S ++ T ) = ( ran S u. ran T ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   = wceq 1395   e. wcel 2200   A.wral 2508   u. cun 3195   C_ wss 3197   class class class wbr 4082   ran crn 4719   Fn wfn 5312   -->wf 5313   ` cfv 5317  (class class class)co 6000   CCcc 7993   RRcr 7994   0cc0 7995   + caddc 7998   < clt 8177   - cmin 8313   NN0cn0 9365   ZZcz 9442   ZZ>=cuz 9718   ...cfz 10200  ..^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: (None)