Theorem ccatrn 11377

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 11369	. . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) Fn ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
2		lencl 11308	. . . . . . . . . . . 12 |- ( S e. Word B -> (♯ ` S ) e. NN0 )
3		nn0uz 9957	. . . . . . . . . . . 12 |- NN0 = ( ZZ>= ` 0 )
4	2, 3	eleqtrdi 2331	. . . . . . . . . . 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 9766	. . . . . . . . . . . 12 |- ( S e. Word B -> (♯ ` S ) e. ZZ )
7	6	uzidd 9937	. . . . . . . . . . 11 |- ( S e. Word B -> (♯ ` S ) e. ( ZZ>= ` (♯ ` S ) ) )
8		lencl 11308	. . . . . . . . . . 11 |- ( T e. Word B -> (♯ ` T ) e. NN0 )
9		uzaddcl 9986	. . . . . . . . . . 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 10422	. . . . . . . . . 10 |- ( (♯ ` S ) e. ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) <-> ( (♯ ` S ) e. ( ZZ>= ` 0 ) /\ ( (♯ ` S ) + (♯ ` T ) ) e. ( ZZ>= ` (♯ ` S ) ) ) )
12	5, 10, 11	sylanbrc 421	. . . . . . . . 9 |- ( ( S e. Word B /\ T e. Word B ) -> (♯ ` S ) e. ( 0 ... ( (♯ ` S ) + (♯ ` T ) ) ) )
13		fzosplit 10586	. . . . . . . . 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 2308	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B ) -> ( x e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) <-> x e. ( ( 0..^ (♯ ` S ) ) u. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) ) )
16		elun 3370	. . . . . . 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 11365	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` x ) = ( S ` x ) )
19	18	3expa 1234	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` x ) = ( S ` x ) )
20		ssun1 3392	. . . . . . . . . 10 |- ran S C_ ( ran S u. ran T )
21		wrdfn 11319	. . . . . . . . . . . 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 5840	. . . . . . . . . . 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 3246	. . . . . . . . 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 2315	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` x ) e. ( ran S u. ran T ) )
27		ccatval2 11366	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( S ++ T ) ` x ) = ( T ` ( x - (♯ ` S ) ) ) )
28	27	3expa 1234	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( S ++ T ) ` x ) = ( T ` ( x - (♯ ` S ) ) ) )
29		ssun2 3393	. . . . . . . . . 10 |- ran T C_ ( ran S u. ran T )
30		wrdfn 11319	. . . . . . . . . . . 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 10558	. . . . . . . . . . . . . . 15 |- ( x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) -> x e. ( ZZ>= ` (♯ ` S ) ) )
33		uznn0sub 9954	. . . . . . . . . . . . . . 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 2331	. . . . . . . . . . . . 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 9766	. . . . . . . . . . . . 13 |- ( T e. Word B -> (♯ ` T ) e. ZZ )
38	37	ad2antlr 493	. . . . . . . . . . . 12 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> (♯ ` T ) e. ZZ )
39		elfzolt2 10564	. . . . . . . . . . . . . 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 10554	. . . . . . . . . . . . . . . 16 |- ( x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) -> x e. ZZ )
42	41	zred 9768	. . . . . . . . . . . . . . 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 9621	. . . . . . . . . . . . . . 15 |- ( S e. Word B -> (♯ ` S ) e. RR )
45	44	ad2antrr 492	. . . . . . . . . . . . . 14 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> (♯ ` S ) e. RR )
46	8	nn0red 9621	. . . . . . . . . . . . . . 15 |- ( T e. Word B -> (♯ ` T ) e. RR )
47	46	ad2antlr 493	. . . . . . . . . . . . . 14 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> (♯ ` T ) e. RR )
48	43, 45, 47	ltsubadd2d 8871	. . . . . . . . . . . . 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 10557	. . . . . . . . . . . 12 |- ( ( x - (♯ ` S ) ) e. ( 0..^ (♯ ` T ) ) <-> ( ( x - (♯ ` S ) ) e. ( ZZ>= ` 0 ) /\ (♯ ` T ) e. ZZ /\ ( x - (♯ ` S ) ) < (♯ ` T ) ) )
51	36, 38, 49, 50	syl3anbrc 1212	. . . . . . . . . . 11 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( x - (♯ ` S ) ) e. ( 0..^ (♯ ` T ) ) )
52		fnfvelrn 5840	. . . . . . . . . . 11 |- ( ( T Fn ( 0..^ (♯ ` T ) ) /\ ( x - (♯ ` S ) ) e. ( 0..^ (♯ ` T ) ) ) -> ( T ` ( x - (♯ ` S ) ) ) e. ran T )
53	31, 51, 52	syl2an2r 603	. . . . . . . . . 10 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( T ` ( x - (♯ ` S ) ) ) e. ran T )
54	29, 53	sselid 3246	. . . . . . . . 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 2315	. . . . . . . 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 809	. . . . . . 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 2622	. . . 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 5866	. . . 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 421	. . 3 |- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) : ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) --> ( ran S u. ran T ) )
62	61	frnd 5543	. 2 |- ( ( S e. Word B /\ T e. Word B ) -> ran ( S ++ T ) C_ ( ran S u. ran T ) )
63		fzoss2 10581	. . . . . . . . . 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 3248	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> x e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
66		fnfvelrn 5840	. . . . . . . 8 |- ( ( ( S ++ T ) Fn ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) /\ x e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( S ++ T ) ` x ) e. ran ( S ++ T ) )
67	1, 65, 66	syl2an2r 603	. . . . . . 7 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` x ) e. ran ( S ++ T ) )
68	19, 67	eqeltrrd 2316	. . . . . 6 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( S ` x ) e. ran ( S ++ T ) )
69	68	ralrimiva 2623	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> A. x e. ( 0..^ (♯ ` S ) ) ( S ` x ) e. ran ( S ++ T ) )
70		ffnfv 5866	. . . . 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 421	. . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> S : ( 0..^ (♯ ` S ) ) --> ran ( S ++ T ) )
72	71	frnd 5543	. . 3 |- ( ( S e. Word B /\ T e. Word B ) -> ran S C_ ran ( S ++ T ) )
73		ccatval3 11367	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( ( S ++ T ) ` ( x + (♯ ` S ) ) ) = ( T ` x ) )
74	73	3expa 1234	. . . . . . 7 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( ( S ++ T ) ` ( x + (♯ ` S ) ) ) = ( T ` x ) )
75		elfzouz 10558	. . . . . . . . . 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 9986	. . . . . . . . . 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 9598	. . . . . . . . . . . 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 9766	. . . . . . . . . 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 10598	. . . . . . . . . . . 12 |- ( x e. ( 0..^ (♯ ` T ) ) -> x e. NN0 )
84	83	nn0cnd 9622	. . . . . . . . . . 11 |- ( x e. ( 0..^ (♯ ` T ) ) -> x e. CC )
85	2	nn0cnd 9622	. . . . . . . . . . . 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 8463	. . . . . . . . . . 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 9621	. . . . . . . . . . . 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 493	. . . . . . . . . . 11 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> (♯ ` T ) e. RR )
92	44	ad2antrr 492	. . . . . . . . . . 11 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> (♯ ` S ) e. RR )
93		elfzolt2 10564	. . . . . . . . . . . 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 8750	. . . . . . . . . 10 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( (♯ ` S ) + x ) < ( (♯ ` S ) + (♯ ` T ) ) )
96	88, 95	eqbrtrd 4152	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( x + (♯ ` S ) ) < ( (♯ ` S ) + (♯ ` T ) ) )
97		elfzo2 10557	. . . . . . . . 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 1212	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( x + (♯ ` S ) ) e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
99		fnfvelrn 5840	. . . . . . . 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 603	. . . . . . 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 2316	. . . . . 6 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( T ` x ) e. ran ( S ++ T ) )
102	101	ralrimiva 2623	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> A. x e. ( 0..^ (♯ ` T ) ) ( T ` x ) e. ran ( S ++ T ) )
103		ffnfv 5866	. . . . 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 421	. . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> T : ( 0..^ (♯ ` T ) ) --> ran ( S ++ T ) )
105	104	frnd 5543	. . 3 |- ( ( S e. Word B /\ T e. Word B ) -> ran T C_ ran ( S ++ T ) )
106	72, 105	unssd 3405	. 2 |- ( ( S e. Word B /\ T e. Word B ) -> ( ran S u. ran T ) C_ ran ( S ++ T ) )
107	62, 106	eqssd 3265	1 |- ( ( S e. Word B /\ T e. Word B ) -> ran ( S ++ T ) = ( ran S u. ran T ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   A.wral 2528   u. cun 3218   C_ wss 3220   class class class wbr 4130   ran crn 4775   Fn wfn 5372   -->wf 5373   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   + caddc 8182   < clt 8360   - cmin 8497   NN0cn0 9563   ZZcz 9644   ZZ>=cuz 9921   ...cfz 10411  ..^cfzo 10549  ♯chash 11214  Word cword 11304   ++ cconcat 11358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-fzo 10550  df-ihash 11215  df-word 11305  df-concat 11359

This theorem is used by: (None)