Theorem ccatsymb 11348

Description: The symbol at a given position in a concatenated word. (Contributed by AV, 26-May-2018.) (Proof shortened by AV, 24-Nov-2018.)

Assertion

Ref	Expression
ccatsymb	|- ( ( A e. Word V /\ B e. Word V /\ I e. ZZ ) -> ( ( A ++ B ) ` I ) = if ( I < (♯ ` A ) , ( A ` I ) , ( B ` ( I - (♯ ` A ) ) ) ) )

Proof of Theorem ccatsymb

Step	Hyp	Ref	Expression
1		simprll 543	. . . . . . . 8 |- ( ( 0 <_ I /\ ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < (♯ ` A ) ) ) -> ( A e. Word V /\ B e. Word V ) )
2		simpr 110	. . . . . . . . . 10 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < (♯ ` A ) ) -> I < (♯ ` A ) )
3	2	anim2i 342	. . . . . . . . 9 |- ( ( 0 <_ I /\ ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < (♯ ` A ) ) ) -> ( 0 <_ I /\ I < (♯ ` A ) ) )
4		simpr 110	. . . . . . . . . . 11 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> I e. ZZ )
5		0zd 9635	. . . . . . . . . . 11 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> 0 e. ZZ )
6		lencl 11286	. . . . . . . . . . . . 13 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
7	6	nn0zd 9745	. . . . . . . . . . . 12 |- ( A e. Word V -> (♯ ` A ) e. ZZ )
8	7	ad2antrr 492	. . . . . . . . . . 11 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> (♯ ` A ) e. ZZ )
9		elfzo 10534	. . . . . . . . . . 11 |- ( ( I e. ZZ /\ 0 e. ZZ /\ (♯ ` A ) e. ZZ ) -> ( I e. ( 0..^ (♯ ` A ) ) <-> ( 0 <_ I /\ I < (♯ ` A ) ) ) )
10	4, 5, 8, 9	syl3anc 1278	. . . . . . . . . 10 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> ( I e. ( 0..^ (♯ ` A ) ) <-> ( 0 <_ I /\ I < (♯ ` A ) ) ) )
11	10	ad2antrl 494	. . . . . . . . 9 |- ( ( 0 <_ I /\ ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < (♯ ` A ) ) ) -> ( I e. ( 0..^ (♯ ` A ) ) <-> ( 0 <_ I /\ I < (♯ ` A ) ) ) )
12	3, 11	mpbird 167	. . . . . . . 8 |- ( ( 0 <_ I /\ ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < (♯ ` A ) ) ) -> I e. ( 0..^ (♯ ` A ) ) )
13		df-3an 1011	. . . . . . . 8 |- ( ( A e. Word V /\ B e. Word V /\ I e. ( 0..^ (♯ ` A ) ) ) <-> ( ( A e. Word V /\ B e. Word V ) /\ I e. ( 0..^ (♯ ` A ) ) ) )
14	1, 12, 13	sylanbrc 421	. . . . . . 7 |- ( ( 0 <_ I /\ ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < (♯ ` A ) ) ) -> ( A e. Word V /\ B e. Word V /\ I e. ( 0..^ (♯ ` A ) ) ) )
15		ccatval1 11343	. . . . . . . 8 |- ( ( A e. Word V /\ B e. Word V /\ I e. ( 0..^ (♯ ` A ) ) ) -> ( ( A ++ B ) ` I ) = ( A ` I ) )
16	15	eqcomd 2244	. . . . . . 7 |- ( ( A e. Word V /\ B e. Word V /\ I e. ( 0..^ (♯ ` A ) ) ) -> ( A ` I ) = ( ( A ++ B ) ` I ) )
17	14, 16	syl 14	. . . . . 6 |- ( ( 0 <_ I /\ ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < (♯ ` A ) ) ) -> ( A ` I ) = ( ( A ++ B ) ` I ) )
18	17	ancoms 268	. . . . 5 |- ( ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < (♯ ` A ) ) /\ 0 <_ I ) -> ( A ` I ) = ( ( A ++ B ) ` I ) )
19		0z 9634	. . . . . . . . . . 11 |- 0 e. ZZ
20		zltnle 9669	. . . . . . . . . . 11 |- ( ( I e. ZZ /\ 0 e. ZZ ) -> ( I < 0 <-> -. 0 <_ I ) )
21	19, 20	mpan2 429	. . . . . . . . . 10 |- ( I e. ZZ -> ( I < 0 <-> -. 0 <_ I ) )
22	21	adantl 277	. . . . . . . . 9 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> ( I < 0 <-> -. 0 <_ I ) )
23		simpl 109	. . . . . . . . . . . . . 14 |- ( ( A e. Word V /\ B e. Word V ) -> A e. Word V )
24	23	anim1i 340	. . . . . . . . . . . . 13 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> ( A e. Word V /\ I e. ZZ ) )
25	24	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < 0 ) -> ( A e. Word V /\ I e. ZZ ) )
26		animorrl 838	. . . . . . . . . . . 12 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < 0 ) -> ( I < 0 \/ (♯ ` A ) <_ I ) )
27		wrdsymb0 11315	. . . . . . . . . . . 12 |- ( ( A e. Word V /\ I e. ZZ ) -> ( ( I < 0 \/ (♯ ` A ) <_ I ) -> ( A ` I ) = (/) ) )
28	25, 26, 27	sylc 62	. . . . . . . . . . 11 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < 0 ) -> ( A ` I ) = (/) )
29		ccatcl 11339	. . . . . . . . . . . . . 14 |- ( ( A e. Word V /\ B e. Word V ) -> ( A ++ B ) e. Word V )
30	29	anim1i 340	. . . . . . . . . . . . 13 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> ( ( A ++ B ) e. Word V /\ I e. ZZ ) )
31	30	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < 0 ) -> ( ( A ++ B ) e. Word V /\ I e. ZZ ) )
32		animorrl 838	. . . . . . . . . . . 12 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < 0 ) -> ( I < 0 \/ (♯ ` ( A ++ B ) ) <_ I ) )
33		wrdsymb0 11315	. . . . . . . . . . . 12 |- ( ( ( A ++ B ) e. Word V /\ I e. ZZ ) -> ( ( I < 0 \/ (♯ ` ( A ++ B ) ) <_ I ) -> ( ( A ++ B ) ` I ) = (/) ) )
34	31, 32, 33	sylc 62	. . . . . . . . . . 11 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < 0 ) -> ( ( A ++ B ) ` I ) = (/) )
35	28, 34	eqtr4d 2274	. . . . . . . . . 10 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < 0 ) -> ( A ` I ) = ( ( A ++ B ) ` I ) )
36	35	ex 115	. . . . . . . . 9 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> ( I < 0 -> ( A ` I ) = ( ( A ++ B ) ` I ) ) )
37	22, 36	sylbird 170	. . . . . . . 8 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> ( -. 0 <_ I -> ( A ` I ) = ( ( A ++ B ) ` I ) ) )
38	37	com12 30	. . . . . . 7 |- ( -. 0 <_ I -> ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> ( A ` I ) = ( ( A ++ B ) ` I ) ) )
39	38	adantrd 279	. . . . . 6 |- ( -. 0 <_ I -> ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < (♯ ` A ) ) -> ( A ` I ) = ( ( A ++ B ) ` I ) ) )
40	39	impcom 125	. . . . 5 |- ( ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < (♯ ` A ) ) /\ -. 0 <_ I ) -> ( A ` I ) = ( ( A ++ B ) ` I ) )
41		simplr 533	. . . . . . 7 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < (♯ ` A ) ) -> I e. ZZ )
42		zdcle 9700	. . . . . . 7 |- ( ( 0 e. ZZ /\ I e. ZZ ) -> DECID 0 <_ I )
43	19, 41, 42	sylancr 418	. . . . . 6 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < (♯ ` A ) ) -> DECID 0 <_ I )
44		exmiddc 848	. . . . . 6 |- (DECID 0 <_ I -> ( 0 <_ I \/ -. 0 <_ I ) )
45	43, 44	syl 14	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < (♯ ` A ) ) -> ( 0 <_ I \/ -. 0 <_ I ) )
46	18, 40, 45	mpjaodan 810	. . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < (♯ ` A ) ) -> ( A ` I ) = ( ( A ++ B ) ` I ) )
47		simprll 543	. . . . . . . 8 |- ( ( I < ( (♯ ` A ) + (♯ ` B ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ -. I < (♯ ` A ) ) ) -> ( A e. Word V /\ B e. Word V ) )
48		id 19	. . . . . . . . . 10 |- ( I < ( (♯ ` A ) + (♯ ` B ) ) -> I < ( (♯ ` A ) + (♯ ` B ) ) )
49	6	nn0red 9600	. . . . . . . . . . . . 13 |- ( A e. Word V -> (♯ ` A ) e. RR )
50		zre 9627	. . . . . . . . . . . . 13 |- ( I e. ZZ -> I e. RR )
51		lenlt 8391	. . . . . . . . . . . . 13 |- ( ( (♯ ` A ) e. RR /\ I e. RR ) -> ( (♯ ` A ) <_ I <-> -. I < (♯ ` A ) ) )
52	49, 50, 51	syl2an 289	. . . . . . . . . . . 12 |- ( ( A e. Word V /\ I e. ZZ ) -> ( (♯ ` A ) <_ I <-> -. I < (♯ ` A ) ) )
53	52	adantlr 481	. . . . . . . . . . 11 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> ( (♯ ` A ) <_ I <-> -. I < (♯ ` A ) ) )
54	53	biimpar 297	. . . . . . . . . 10 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ -. I < (♯ ` A ) ) -> (♯ ` A ) <_ I )
55	48, 54	anim12ci 339	. . . . . . . . 9 |- ( ( I < ( (♯ ` A ) + (♯ ` B ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ -. I < (♯ ` A ) ) ) -> ( (♯ ` A ) <_ I /\ I < ( (♯ ` A ) + (♯ ` B ) ) ) )
56		lencl 11286	. . . . . . . . . . . . . 14 |- ( B e. Word V -> (♯ ` B ) e. NN0 )
57	56	nn0zd 9745	. . . . . . . . . . . . 13 |- ( B e. Word V -> (♯ ` B ) e. ZZ )
58		zaddcl 9663	. . . . . . . . . . . . 13 |- ( ( (♯ ` A ) e. ZZ /\ (♯ ` B ) e. ZZ ) -> ( (♯ ` A ) + (♯ ` B ) ) e. ZZ )
59	7, 57, 58	syl2an 289	. . . . . . . . . . . 12 |- ( ( A e. Word V /\ B e. Word V ) -> ( (♯ ` A ) + (♯ ` B ) ) e. ZZ )
60	59	adantr 276	. . . . . . . . . . 11 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> ( (♯ ` A ) + (♯ ` B ) ) e. ZZ )
61		elfzo 10534	. . . . . . . . . . 11 |- ( ( I e. ZZ /\ (♯ ` A ) e. ZZ /\ ( (♯ ` A ) + (♯ ` B ) ) e. ZZ ) -> ( I e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) <-> ( (♯ ` A ) <_ I /\ I < ( (♯ ` A ) + (♯ ` B ) ) ) ) )
62	4, 8, 60, 61	syl3anc 1278	. . . . . . . . . 10 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> ( I e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) <-> ( (♯ ` A ) <_ I /\ I < ( (♯ ` A ) + (♯ ` B ) ) ) ) )
63	62	ad2antrl 494	. . . . . . . . 9 |- ( ( I < ( (♯ ` A ) + (♯ ` B ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ -. I < (♯ ` A ) ) ) -> ( I e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) <-> ( (♯ ` A ) <_ I /\ I < ( (♯ ` A ) + (♯ ` B ) ) ) ) )
64	55, 63	mpbird 167	. . . . . . . 8 |- ( ( I < ( (♯ ` A ) + (♯ ` B ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ -. I < (♯ ` A ) ) ) -> I e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) )
65		df-3an 1011	. . . . . . . 8 |- ( ( A e. Word V /\ B e. Word V /\ I e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) <-> ( ( A e. Word V /\ B e. Word V ) /\ I e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) )
66	47, 64, 65	sylanbrc 421	. . . . . . 7 |- ( ( I < ( (♯ ` A ) + (♯ ` B ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ -. I < (♯ ` A ) ) ) -> ( A e. Word V /\ B e. Word V /\ I e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) )
67		ccatval2 11344	. . . . . . . 8 |- ( ( A e. Word V /\ B e. Word V /\ I e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) -> ( ( A ++ B ) ` I ) = ( B ` ( I - (♯ ` A ) ) ) )
68	67	eqcomd 2244	. . . . . . 7 |- ( ( A e. Word V /\ B e. Word V /\ I e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) -> ( B ` ( I - (♯ ` A ) ) ) = ( ( A ++ B ) ` I ) )
69	66, 68	syl 14	. . . . . 6 |- ( ( I < ( (♯ ` A ) + (♯ ` B ) ) /\ ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ -. I < (♯ ` A ) ) ) -> ( B ` ( I - (♯ ` A ) ) ) = ( ( A ++ B ) ` I ) )
70	69	ancoms 268	. . . . 5 |- ( ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ -. I < (♯ ` A ) ) /\ I < ( (♯ ` A ) + (♯ ` B ) ) ) -> ( B ` ( I - (♯ ` A ) ) ) = ( ( A ++ B ) ` I ) )
71	56	nn0red 9600	. . . . . . . . . . 11 |- ( B e. Word V -> (♯ ` B ) e. RR )
72		readdcl 8295	. . . . . . . . . . 11 |- ( ( (♯ ` A ) e. RR /\ (♯ ` B ) e. RR ) -> ( (♯ ` A ) + (♯ ` B ) ) e. RR )
73	49, 71, 72	syl2an 289	. . . . . . . . . 10 |- ( ( A e. Word V /\ B e. Word V ) -> ( (♯ ` A ) + (♯ ` B ) ) e. RR )
74		lenlt 8391	. . . . . . . . . 10 |- ( ( ( (♯ ` A ) + (♯ ` B ) ) e. RR /\ I e. RR ) -> ( ( (♯ ` A ) + (♯ ` B ) ) <_ I <-> -. I < ( (♯ ` A ) + (♯ ` B ) ) ) )
75	73, 50, 74	syl2an 289	. . . . . . . . 9 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> ( ( (♯ ` A ) + (♯ ` B ) ) <_ I <-> -. I < ( (♯ ` A ) + (♯ ` B ) ) ) )
76		simplr 533	. . . . . . . . . . . . . 14 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> B e. Word V )
77		simpr 110	. . . . . . . . . . . . . . . 16 |- ( ( A e. Word V /\ I e. ZZ ) -> I e. ZZ )
78	7	adantr 276	. . . . . . . . . . . . . . . 16 |- ( ( A e. Word V /\ I e. ZZ ) -> (♯ ` A ) e. ZZ )
79	77, 78	zsubcld 9752	. . . . . . . . . . . . . . 15 |- ( ( A e. Word V /\ I e. ZZ ) -> ( I - (♯ ` A ) ) e. ZZ )
80	79	adantlr 481	. . . . . . . . . . . . . 14 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> ( I - (♯ ` A ) ) e. ZZ )
81	76, 80	jca 306	. . . . . . . . . . . . 13 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> ( B e. Word V /\ ( I - (♯ ` A ) ) e. ZZ ) )
82	81	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ ( (♯ ` A ) + (♯ ` B ) ) <_ I ) -> ( B e. Word V /\ ( I - (♯ ` A ) ) e. ZZ ) )
83	49	ad2antrr 492	. . . . . . . . . . . . . . 15 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> (♯ ` A ) e. RR )
84	71	ad2antlr 493	. . . . . . . . . . . . . . 15 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> (♯ ` B ) e. RR )
85	50	adantl 277	. . . . . . . . . . . . . . 15 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> I e. RR )
86	83, 84, 85	leaddsub2d 8865	. . . . . . . . . . . . . 14 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> ( ( (♯ ` A ) + (♯ ` B ) ) <_ I <-> (♯ ` B ) <_ ( I - (♯ ` A ) ) ) )
87	86	biimpa 296	. . . . . . . . . . . . 13 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ ( (♯ ` A ) + (♯ ` B ) ) <_ I ) -> (♯ ` B ) <_ ( I - (♯ ` A ) ) )
88	87	olcd 746	. . . . . . . . . . . 12 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ ( (♯ ` A ) + (♯ ` B ) ) <_ I ) -> ( ( I - (♯ ` A ) ) < 0 \/ (♯ ` B ) <_ ( I - (♯ ` A ) ) ) )
89		wrdsymb0 11315	. . . . . . . . . . . 12 |- ( ( B e. Word V /\ ( I - (♯ ` A ) ) e. ZZ ) -> ( ( ( I - (♯ ` A ) ) < 0 \/ (♯ ` B ) <_ ( I - (♯ ` A ) ) ) -> ( B ` ( I - (♯ ` A ) ) ) = (/) ) )
90	82, 88, 89	sylc 62	. . . . . . . . . . 11 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ ( (♯ ` A ) + (♯ ` B ) ) <_ I ) -> ( B ` ( I - (♯ ` A ) ) ) = (/) )
91	30	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ ( (♯ ` A ) + (♯ ` B ) ) <_ I ) -> ( ( A ++ B ) e. Word V /\ I e. ZZ ) )
92		ccatlen 11341	. . . . . . . . . . . . . . 15 |- ( ( A e. Word V /\ B e. Word V ) -> (♯ ` ( A ++ B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
93	92	ad2antrr 492	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ ( (♯ ` A ) + (♯ ` B ) ) <_ I ) -> (♯ ` ( A ++ B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
94		simpr 110	. . . . . . . . . . . . . 14 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ ( (♯ ` A ) + (♯ ` B ) ) <_ I ) -> ( (♯ ` A ) + (♯ ` B ) ) <_ I )
95	93, 94	eqbrtrd 4147	. . . . . . . . . . . . 13 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ ( (♯ ` A ) + (♯ ` B ) ) <_ I ) -> (♯ ` ( A ++ B ) ) <_ I )
96	95	olcd 746	. . . . . . . . . . . 12 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ ( (♯ ` A ) + (♯ ` B ) ) <_ I ) -> ( I < 0 \/ (♯ ` ( A ++ B ) ) <_ I ) )
97	91, 96, 33	sylc 62	. . . . . . . . . . 11 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ ( (♯ ` A ) + (♯ ` B ) ) <_ I ) -> ( ( A ++ B ) ` I ) = (/) )
98	90, 97	eqtr4d 2274	. . . . . . . . . 10 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ ( (♯ ` A ) + (♯ ` B ) ) <_ I ) -> ( B ` ( I - (♯ ` A ) ) ) = ( ( A ++ B ) ` I ) )
99	98	ex 115	. . . . . . . . 9 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> ( ( (♯ ` A ) + (♯ ` B ) ) <_ I -> ( B ` ( I - (♯ ` A ) ) ) = ( ( A ++ B ) ` I ) ) )
100	75, 99	sylbird 170	. . . . . . . 8 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> ( -. I < ( (♯ ` A ) + (♯ ` B ) ) -> ( B ` ( I - (♯ ` A ) ) ) = ( ( A ++ B ) ` I ) ) )
101	100	com12 30	. . . . . . 7 |- ( -. I < ( (♯ ` A ) + (♯ ` B ) ) -> ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> ( B ` ( I - (♯ ` A ) ) ) = ( ( A ++ B ) ` I ) ) )
102	101	adantrd 279	. . . . . 6 |- ( -. I < ( (♯ ` A ) + (♯ ` B ) ) -> ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ -. I < (♯ ` A ) ) -> ( B ` ( I - (♯ ` A ) ) ) = ( ( A ++ B ) ` I ) ) )
103	102	impcom 125	. . . . 5 |- ( ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ -. I < (♯ ` A ) ) /\ -. I < ( (♯ ` A ) + (♯ ` B ) ) ) -> ( B ` ( I - (♯ ` A ) ) ) = ( ( A ++ B ) ` I ) )
104		simplr 533	. . . . . . 7 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ -. I < (♯ ` A ) ) -> I e. ZZ )
105	60	adantr 276	. . . . . . 7 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ -. I < (♯ ` A ) ) -> ( (♯ ` A ) + (♯ ` B ) ) e. ZZ )
106		zdclt 9701	. . . . . . 7 |- ( ( I e. ZZ /\ ( (♯ ` A ) + (♯ ` B ) ) e. ZZ ) -> DECID I < ( (♯ ` A ) + (♯ ` B ) ) )
107	104, 105, 106	syl2anc 415	. . . . . 6 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ -. I < (♯ ` A ) ) -> DECID I < ( (♯ ` A ) + (♯ ` B ) ) )
108		exmiddc 848	. . . . . 6 |- (DECID I < ( (♯ ` A ) + (♯ ` B ) ) -> ( I < ( (♯ ` A ) + (♯ ` B ) ) \/ -. I < ( (♯ ` A ) + (♯ ` B ) ) ) )
109	107, 108	syl 14	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ -. I < (♯ ` A ) ) -> ( I < ( (♯ ` A ) + (♯ ` B ) ) \/ -. I < ( (♯ ` A ) + (♯ ` B ) ) ) )
110	70, 103, 109	mpjaodan 810	. . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ -. I < (♯ ` A ) ) -> ( B ` ( I - (♯ ` A ) ) ) = ( ( A ++ B ) ` I ) )
111		zdclt 9701	. . . . 5 |- ( ( I e. ZZ /\ (♯ ` A ) e. ZZ ) -> DECID I < (♯ ` A ) )
112	4, 8, 111	syl2anc 415	. . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> DECID I < (♯ ` A ) )
113	46, 110, 112	ifeqdadc 3670	. . 3 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> if ( I < (♯ ` A ) , ( A ` I ) , ( B ` ( I - (♯ ` A ) ) ) ) = ( ( A ++ B ) ` I ) )
114	113	eqcomd 2244	. 2 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> ( ( A ++ B ) ` I ) = if ( I < (♯ ` A ) , ( A ` I ) , ( B ` ( I - (♯ ` A ) ) ) ) )
115	114	3impa 1225	1 |- ( ( A e. Word V /\ B e. Word V /\ I e. ZZ ) -> ( ( A ++ B ) ` I ) = if ( I < (♯ ` A ) , ( A ` I ) , ( B ` ( I - (♯ ` A ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   (/)c0 3520   ifcif 3635   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   RRcr 8168   0cc0 8169   + caddc 8172   < clt 8350   <_ cle 8351   - cmin 8487   ZZcz 9623  ..^cfzo 10527  ♯chash 11192  Word cword 11282   ++ cconcat 11336

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-ihash 11193  df-word 11283  df-concat 11337

This theorem is referenced by:  swrdccatin2  11479