Theorem ccatsymb 11290

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 539	. . . . . . . 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 9589	. . . . . . . . . . 11 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> 0 e. ZZ )
6		lencl 11228	. . . . . . . . . . . . 13 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
7	6	nn0zd 9698	. . . . . . . . . . . 12 |- ( A e. Word V -> (♯ ` A ) e. ZZ )
8	7	ad2antrr 488	. . . . . . . . . . 11 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> (♯ ` A ) e. ZZ )
9		elfzo 10483	. . . . . . . . . . 11 |- ( ( I e. ZZ /\ 0 e. ZZ /\ (♯ ` A ) e. ZZ ) -> ( I e. ( 0..^ (♯ ` A ) ) <-> ( 0 <_ I /\ I < (♯ ` A ) ) ) )
10	4, 5, 8, 9	syl3anc 1274	. . . . . . . . . 10 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> ( I e. ( 0..^ (♯ ` A ) ) <-> ( 0 <_ I /\ I < (♯ ` A ) ) ) )
11	10	ad2antrl 490	. . . . . . . . 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 1007	. . . . . . . 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 417	. . . . . . 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 11285	. . . . . . . 8 |- ( ( A e. Word V /\ B e. Word V /\ I e. ( 0..^ (♯ ` A ) ) ) -> ( ( A ++ B ) ` I ) = ( A ` I ) )
16	15	eqcomd 2238	. . . . . . 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 9588	. . . . . . . . . . 11 |- 0 e. ZZ
20		zltnle 9623	. . . . . . . . . . 11 |- ( ( I e. ZZ /\ 0 e. ZZ ) -> ( I < 0 <-> -. 0 <_ I ) )
21	19, 20	mpan2 425	. . . . . . . . . 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 834	. . . . . . . . . . . 12 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < 0 ) -> ( I < 0 \/ (♯ ` A ) <_ I ) )
27		wrdsymb0 11257	. . . . . . . . . . . 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 11281	. . . . . . . . . . . . . 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 834	. . . . . . . . . . . 12 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < 0 ) -> ( I < 0 \/ (♯ ` ( A ++ B ) ) <_ I ) )
33		wrdsymb0 11257	. . . . . . . . . . . 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 2268	. . . . . . . . . 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 529	. . . . . . 7 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < (♯ ` A ) ) -> I e. ZZ )
42		zdcle 9654	. . . . . . 7 |- ( ( 0 e. ZZ /\ I e. ZZ ) -> DECID 0 <_ I )
43	19, 41, 42	sylancr 414	. . . . . 6 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < (♯ ` A ) ) -> DECID 0 <_ I )
44		exmiddc 844	. . . . . 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 806	. . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < (♯ ` A ) ) -> ( A ` I ) = ( ( A ++ B ) ` I ) )
47		simprll 539	. . . . . . . 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 9554	. . . . . . . . . . . . 13 |- ( A e. Word V -> (♯ ` A ) e. RR )
50		zre 9581	. . . . . . . . . . . . 13 |- ( I e. ZZ -> I e. RR )
51		lenlt 8349	. . . . . . . . . . . . 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 477	. . . . . . . . . . 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 11228	. . . . . . . . . . . . . 14 |- ( B e. Word V -> (♯ ` B ) e. NN0 )
57	56	nn0zd 9698	. . . . . . . . . . . . 13 |- ( B e. Word V -> (♯ ` B ) e. ZZ )
58		zaddcl 9617	. . . . . . . . . . . . 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 10483	. . . . . . . . . . 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 1274	. . . . . . . . . 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 490	. . . . . . . . 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 1007	. . . . . . . 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 417	. . . . . . 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 11286	. . . . . . . 8 |- ( ( A e. Word V /\ B e. Word V /\ I e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) -> ( ( A ++ B ) ` I ) = ( B ` ( I - (♯ ` A ) ) ) )
68	67	eqcomd 2238	. . . . . . 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 9554	. . . . . . . . . . 11 |- ( B e. Word V -> (♯ ` B ) e. RR )
72		readdcl 8253	. . . . . . . . . . 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 8349	. . . . . . . . . 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 529	. . . . . . . . . . . . . 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 9705	. . . . . . . . . . . . . . 15 |- ( ( A e. Word V /\ I e. ZZ ) -> ( I - (♯ ` A ) ) e. ZZ )
80	79	adantlr 477	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . . 15 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> (♯ ` A ) e. RR )
84	71	ad2antlr 489	. . . . . . . . . . . . . . 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 8821	. . . . . . . . . . . . . 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 742	. . . . . . . . . . . 12 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ ( (♯ ` A ) + (♯ ` B ) ) <_ I ) -> ( ( I - (♯ ` A ) ) < 0 \/ (♯ ` B ) <_ ( I - (♯ ` A ) ) ) )
89		wrdsymb0 11257	. . . . . . . . . . . 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 11283	. . . . . . . . . . . . . . 15 |- ( ( A e. Word V /\ B e. Word V ) -> (♯ ` ( A ++ B ) ) = ( (♯ ` A ) + (♯ ` B ) ) )
93	92	ad2antrr 488	. . . . . . . . . . . . . 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 4131	. . . . . . . . . . . . 13 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ ( (♯ ` A ) + (♯ ` B ) ) <_ I ) -> (♯ ` ( A ++ B ) ) <_ I )
96	95	olcd 742	. . . . . . . . . . . 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 2268	. . . . . . . . . 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 529	. . . . . . 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 9655	. . . . . . 7 |- ( ( I e. ZZ /\ ( (♯ ` A ) + (♯ ` B ) ) e. ZZ ) -> DECID I < ( (♯ ` A ) + (♯ ` B ) ) )
107	104, 105, 106	syl2anc 411	. . . . . 6 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ -. I < (♯ ` A ) ) -> DECID I < ( (♯ ` A ) + (♯ ` B ) ) )
108		exmiddc 844	. . . . . 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 806	. . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ -. I < (♯ ` A ) ) -> ( B ` ( I - (♯ ` A ) ) ) = ( ( A ++ B ) ` I ) )
111		zdclt 9655	. . . . 5 |- ( ( I e. ZZ /\ (♯ ` A ) e. ZZ ) -> DECID I < (♯ ` A ) )
112	4, 8, 111	syl2anc 411	. . . 4 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> DECID I < (♯ ` A ) )
113	46, 110, 112	ifeqdadc 3655	. . 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 2238	. 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 1221	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 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2203   (/)c0 3508   ifcif 3620   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   RRcr 8126   0cc0 8127   + caddc 8130   < clt 8308   <_ cle 8309   - cmin 8444   ZZcz 9577  ..^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:  swrdccatin2  11421