Theorem ccatrn 11088

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 11080	. . . 4 |- ( ( S e. Word B /\ T e. Word B ) -> ( S ++ T ) Fn ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
2		lencl 11020	. . . . . . . . . . . 12 |- ( S e. Word B -> (♯ ` S ) e. NN0 )
3		nn0uz 9703	. . . . . . . . . . . 12 |- NN0 = ( ZZ>= ` 0 )
4	2, 3	eleqtrdi 2299	. . . . . . . . . . 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 9513	. . . . . . . . . . . 12 |- ( S e. Word B -> (♯ ` S ) e. ZZ )
7	6	uzidd 9683	. . . . . . . . . . 11 |- ( S e. Word B -> (♯ ` S ) e. ( ZZ>= ` (♯ ` S ) ) )
8		lencl 11020	. . . . . . . . . . 11 |- ( T e. Word B -> (♯ ` T ) e. NN0 )
9		uzaddcl 9727	. . . . . . . . . . 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 10161	. . . . . . . . . 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 10321	. . . . . . . . 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 2276	. . . . . . 7 |- ( ( S e. Word B /\ T e. Word B ) -> ( x e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) <-> x e. ( ( 0..^ (♯ ` S ) ) u. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) ) )
16		elun 3318	. . . . . . 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 11076	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` x ) = ( S ` x ) )
19	18	3expa 1206	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` x ) = ( S ` x ) )
20		ssun1 3340	. . . . . . . . . 10 |- ran S C_ ( ran S u. ran T )
21		wrdfn 11031	. . . . . . . . . . . 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 5725	. . . . . . . . . . 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 3195	. . . . . . . . 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 2283	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` x ) e. ( ran S u. ran T ) )
27		ccatval2 11077	. . . . . . . . . 10 |- ( ( S e. Word B /\ T e. Word B /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( S ++ T ) ` x ) = ( T ` ( x - (♯ ` S ) ) ) )
28	27	3expa 1206	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( S ++ T ) ` x ) = ( T ` ( x - (♯ ` S ) ) ) )
29		ssun2 3341	. . . . . . . . . 10 |- ran T C_ ( ran S u. ran T )
30		wrdfn 11031	. . . . . . . . . . . 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 10293	. . . . . . . . . . . . . . 15 |- ( x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) -> x e. ( ZZ>= ` (♯ ` S ) ) )
33		uznn0sub 9700	. . . . . . . . . . . . . . 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 2299	. . . . . . . . . . . . 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 9513	. . . . . . . . . . . . 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 10299	. . . . . . . . . . . . . 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 10289	. . . . . . . . . . . . . . . 16 |- ( x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) -> x e. ZZ )
42	41	zred 9515	. . . . . . . . . . . . . . 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 9369	. . . . . . . . . . . . . . 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 9369	. . . . . . . . . . . . . . 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 8636	. . . . . . . . . . . . 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 10292	. . . . . . . . . . . 12 |- ( ( x - (♯ ` S ) ) e. ( 0..^ (♯ ` T ) ) <-> ( ( x - (♯ ` S ) ) e. ( ZZ>= ` 0 ) /\ (♯ ` T ) e. ZZ /\ ( x - (♯ ` S ) ) < (♯ ` T ) ) )
51	36, 38, 49, 50	syl3anbrc 1184	. . . . . . . . . . 11 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( x - (♯ ` S ) ) e. ( 0..^ (♯ ` T ) ) )
52		fnfvelrn 5725	. . . . . . . . . . 11 |- ( ( T Fn ( 0..^ (♯ ` T ) ) /\ ( x - (♯ ` S ) ) e. ( 0..^ (♯ ` T ) ) ) -> ( T ` ( x - (♯ ` S ) ) ) e. ran T )
53	31, 51, 52	syl2an2r 595	. . . . . . . . . 10 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( (♯ ` S )..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( T ` ( x - (♯ ` S ) ) ) e. ran T )
54	29, 53	sselid 3195	. . . . . . . . 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 2283	. . . . . . . 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 799	. . . . . . 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 2579	. . . 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 5751	. . . 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 5445	. 2 |- ( ( S e. Word B /\ T e. Word B ) -> ran ( S ++ T ) C_ ( ran S u. ran T ) )
63		fzoss2 10316	. . . . . . . . . 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 3197	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> x e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
66		fnfvelrn 5725	. . . . . . . 8 |- ( ( ( S ++ T ) Fn ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) /\ x e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) ) -> ( ( S ++ T ) ` x ) e. ran ( S ++ T ) )
67	1, 65, 66	syl2an2r 595	. . . . . . 7 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( ( S ++ T ) ` x ) e. ran ( S ++ T ) )
68	19, 67	eqeltrrd 2284	. . . . . 6 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` S ) ) ) -> ( S ` x ) e. ran ( S ++ T ) )
69	68	ralrimiva 2580	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> A. x e. ( 0..^ (♯ ` S ) ) ( S ` x ) e. ran ( S ++ T ) )
70		ffnfv 5751	. . . . 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 5445	. . 3 |- ( ( S e. Word B /\ T e. Word B ) -> ran S C_ ran ( S ++ T ) )
73		ccatval3 11078	. . . . . . . 8 |- ( ( S e. Word B /\ T e. Word B /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( ( S ++ T ) ` ( x + (♯ ` S ) ) ) = ( T ` x ) )
74	73	3expa 1206	. . . . . . 7 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( ( S ++ T ) ` ( x + (♯ ` S ) ) ) = ( T ` x ) )
75		elfzouz 10293	. . . . . . . . . 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 9727	. . . . . . . . . 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 9350	. . . . . . . . . . . 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 9513	. . . . . . . . . 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 10332	. . . . . . . . . . . 12 |- ( x e. ( 0..^ (♯ ` T ) ) -> x e. NN0 )
84	83	nn0cnd 9370	. . . . . . . . . . 11 |- ( x e. ( 0..^ (♯ ` T ) ) -> x e. CC )
85	2	nn0cnd 9370	. . . . . . . . . . . 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 8229	. . . . . . . . . . 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 9369	. . . . . . . . . . . 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 10299	. . . . . . . . . . . 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 8515	. . . . . . . . . 10 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( (♯ ` S ) + x ) < ( (♯ ` S ) + (♯ ` T ) ) )
96	88, 95	eqbrtrd 4073	. . . . . . . . 9 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( x + (♯ ` S ) ) < ( (♯ ` S ) + (♯ ` T ) ) )
97		elfzo2 10292	. . . . . . . . 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 1184	. . . . . . . 8 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( x + (♯ ` S ) ) e. ( 0..^ ( (♯ ` S ) + (♯ ` T ) ) ) )
99		fnfvelrn 5725	. . . . . . . 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 595	. . . . . . 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 2284	. . . . . 6 |- ( ( ( S e. Word B /\ T e. Word B ) /\ x e. ( 0..^ (♯ ` T ) ) ) -> ( T ` x ) e. ran ( S ++ T ) )
102	101	ralrimiva 2580	. . . . 5 |- ( ( S e. Word B /\ T e. Word B ) -> A. x e. ( 0..^ (♯ ` T ) ) ( T ` x ) e. ran ( S ++ T ) )
103		ffnfv 5751	. . . . 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 5445	. . 3 |- ( ( S e. Word B /\ T e. Word B ) -> ran T C_ ran ( S ++ T ) )
106	72, 105	unssd 3353	. 2 |- ( ( S e. Word B /\ T e. Word B ) -> ( ran S u. ran T ) C_ ran ( S ++ T ) )
107	62, 106	eqssd 3214	1 |- ( ( S e. Word B /\ T e. Word B ) -> ran ( S ++ T ) = ( ran S u. ran T ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 710   = wceq 1373   e. wcel 2177   A.wral 2485   u. cun 3168   C_ wss 3170   class class class wbr 4051   ran crn 4684   Fn wfn 5275   -->wf 5276   ` cfv 5280  (class class class)co 5957   CCcc 7943   RRcr 7944   0cc0 7945   + caddc 7948   < clt 8127   - cmin 8263   NN0cn0 9315   ZZcz 9392   ZZ>=cuz 9668   ...cfz 10150  ..^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)