Theorem ccatrn 11297

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 11289	. . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) Fn ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
2		lencl 11228	. . . . . . . . . . . 12 |- ( S e. Word B -> (♯ ` S ) e. NN0 )
3		nn0uz 9889	. . . . . . . . . . . 12 |- NN0 = ( ZZ>= ` 0 )
4	2, 3	eleqtrdi 2325	. . . . . . . . . . 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 9698	. . . . . . . . . . . 12 |- ( S e. Word B -> (♯ ` S ) e. ZZ )
7	6	uzidd 9869	. . . . . . . . . . 11 |- ( S e. Word B -> (♯ ` S ) e. ( ZZ>= ` (♯ ` S ) ) )
8		lencl 11228	. . . . . . . . . . 11 |- ( T e. Word B -> (♯ ` T ) e. NN0 )
9		uzaddcl 9918	. . . . . . . . . . 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 10353	. . . . . . . . . 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 10513	. . . . . . . . 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 2302	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B ) -> ( x e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) <-> x e. ( ( 0..^ (♯ ` S ) ) u. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) ) )
16		elun 3360	. . . . . . 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 11285	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` x ) = ( S ` x ) )
19	18	3expa 1230	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` x ) = ( S ` x ) )
20		ssun1 3382	. . . . . . . . . 10 |- ran S C_ ( ran S u. ran T )
21		wrdfn 11239	. . . . . . . . . . . 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 5809	. . . . . . . . . . 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 3236	. . . . . . . . 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 2309	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` x ) e. ( ran S u. ran T ) )
27		ccatval2 11286	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( S ++ T ) ` x ) = ( T ` ( x - (♯ ` S ) ) ) )
28	27	3expa 1230	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( S ++ T ) ` x ) = ( T ` ( x - (♯ ` S ) ) ) )
29		ssun2 3383	. . . . . . . . . 10 |- ran T C_ ( ran S u. ran T )
30		wrdfn 11239	. . . . . . . . . . . 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 10485	. . . . . . . . . . . . . . 15 |- ( x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) -> x e. ( ZZ>= ` (♯ ` S ) ) )
33		uznn0sub 9886	. . . . . . . . . . . . . . 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 2325	. . . . . . . . . . . . 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 9698	. . . . . . . . . . . . 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 10491	. . . . . . . . . . . . . 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 10481	. . . . . . . . . . . . . . . 16 |- ( x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) -> x e. ZZ )
42	41	zred 9700	. . . . . . . . . . . . . . 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 9554	. . . . . . . . . . . . . . 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 9554	. . . . . . . . . . . . . . 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 8817	. . . . . . . . . . . . 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 10484	. . . . . . . . . . . 12 |- ( ( x - (♯ ` S ) ) e. ( 0..^ (♯ ` T ) ) <-> ( ( x - (♯ ` S ) ) e. ( ZZ>= ` 0 ) /\ (♯ ` T ) e. ZZ /\ ( x - (♯ ` S ) ) < (♯ ` T ) ) )
51	36, 38, 49, 50	syl3anbrc 1208	. . . . . . . . . . 11 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( x - (♯ ` S ) ) e. ( 0..^ (♯ ` T ) ) )
52		fnfvelrn 5809	. . . . . . . . . . 11 |- ( ( T Fn ( 0..^ (♯ ` T ) ) /\ ( x - (♯ ` S ) ) e. ( 0..^ (♯ ` T ) ) ) -> ( T ` ( x - (♯ ` S ) ) ) e. ran T )
53	31, 51, 52	syl2an2r 599	. . . . . . . . . 10 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( T ` ( x - (♯ ` S ) ) ) e. ran T )
54	29, 53	sselid 3236	. . . . . . . . 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 2309	. . . . . . . 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 805	. . . . . . 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 2614	. . . 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 5835	. . . 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 5518	. 2 |- ( ( S e. Word B /\ T e. Word B ) -> ran ( S ++ T ) C_ ( ran S u. ran T ) )
63		fzoss2 10508	. . . . . . . . . 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 3238	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> x e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
66		fnfvelrn 5809	. . . . . . . 8 |- ( ( ( S ++ T ) Fn ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) /\ x e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( S ++ T ) ` x ) e. ran ( S ++ T ) )
67	1, 65, 66	syl2an2r 599	. . . . . . 7 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` x ) e. ran ( S ++ T ) )
68	19, 67	eqeltrrd 2310	. . . . . 6 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( S ` x ) e. ran ( S ++ T ) )
69	68	ralrimiva 2615	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> A. x e. ( 0..^ (♯ ` S ) ) ( S ` x ) e. ran ( S ++ T ) )
70		ffnfv 5835	. . . . 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 5518	. . 3 |- ( ( S e. Word B /\ T e. Word B ) -> ran S C_ ran ( S ++ T ) )
73		ccatval3 11287	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( ( S ++ T ) ` ( x + (♯ ` S ) ) ) = ( T ` x ) )
74	73	3expa 1230	. . . . . . 7 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( ( S ++ T ) ` ( x + (♯ ` S ) ) ) = ( T ` x ) )
75		elfzouz 10485	. . . . . . . . . 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 9918	. . . . . . . . . 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 9531	. . . . . . . . . . . 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 9698	. . . . . . . . . 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 10525	. . . . . . . . . . . 12 |- ( x e. ( 0..^ (♯ ` T ) ) -> x e. NN0 )
84	83	nn0cnd 9555	. . . . . . . . . . 11 |- ( x e. ( 0..^ (♯ ` T ) ) -> x e. CC )
85	2	nn0cnd 9555	. . . . . . . . . . . 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 8410	. . . . . . . . . . 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 9554	. . . . . . . . . . . 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 10491	. . . . . . . . . . . 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 8696	. . . . . . . . . 10 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( (♯ ` S ) + x ) < ( (♯ ` S ) + (♯ ` T ) ) )
96	88, 95	eqbrtrd 4131	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( x + (♯ ` S ) ) < ( (♯ ` S ) + (♯ ` T ) ) )
97		elfzo2 10484	. . . . . . . . 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 1208	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( x + (♯ ` S ) ) e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
99		fnfvelrn 5809	. . . . . . . 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 599	. . . . . . 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 2310	. . . . . 6 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( T ` x ) e. ran ( S ++ T ) )
102	101	ralrimiva 2615	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> A. x e. ( 0..^ (♯ ` T ) ) ( T ` x ) e. ran ( S ++ T ) )
103		ffnfv 5835	. . . . 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 5518	. . 3 |- ( ( S e. Word B /\ T e. Word B ) -> ran T C_ ran ( S ++ T ) )
106	72, 105	unssd 3395	. 2 |- ( ( S e. Word B /\ T e. Word B ) -> ( ran S u. ran T ) C_ ran ( S ++ T ) )
107	62, 106	eqssd 3255	1 |- ( ( S e. Word B /\ T e. Word B ) -> ran ( S ++ T ) = ( ran S u. ran T ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   = wceq 1398   e. wcel 2203   A.wral 2520   u. cun 3209   C_ wss 3211   class class class wbr 4109   ran crn 4750   Fn wfn 5347   -->wf 5348   ` cfv 5352  (class class class)co 6050   CCcc 8125   RRcr 8126   0cc0 8127   + caddc 8130   < clt 8308   - cmin 8444   NN0cn0 9496   ZZcz 9577   ZZ>=cuz 9853   ...cfz 10342  ..^cfzo 10476  ♯chash 11138  Word cword 11224   ++ cconcat 11278

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343  df-fzo 10477  df-ihash 11139  df-word 11225  df-concat 11279

This theorem is referenced by: (None)