Theorem ccatsymb 11166

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 537	. . . . . . . 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 9479	. . . . . . . . . . 11 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> 0 e. ZZ )
6		lencl 11104	. . . . . . . . . . . . 13 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
7	6	nn0zd 9588	. . . . . . . . . . . 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 10372	. . . . . . . . . . 11 |- ( ( I e. ZZ /\ 0 e. ZZ /\ (♯ ` A ) e. ZZ ) -> ( I e. ( 0..^ (♯ ` A ) ) <-> ( 0 <_ I /\ I < (♯ ` A ) ) ) )
10	4, 5, 8, 9	syl3anc 1271	. . . . . . . . . 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 1004	. . . . . . . 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 11161	. . . . . . . 8 |- ( ( A e. Word V /\ B e. Word V /\ I e. ( 0..^ (♯ ` A ) ) ) -> ( ( A ++ B ) ` I ) = ( A ` I ) )
16	15	eqcomd 2235	. . . . . . 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 9478	. . . . . . . . . . 11 |- 0 e. ZZ
20		zltnle 9513	. . . . . . . . . . 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 831	. . . . . . . . . . . 12 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < 0 ) -> ( I < 0 \/ (♯ ` A ) <_ I ) )
27		wrdsymb0 11133	. . . . . . . . . . . 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 11157	. . . . . . . . . . . . . 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 831	. . . . . . . . . . . 12 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < 0 ) -> ( I < 0 \/ (♯ ` ( A ++ B ) ) <_ I ) )
33		wrdsymb0 11133	. . . . . . . . . . . 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 2265	. . . . . . . . . 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 528	. . . . . . 7 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < (♯ ` A ) ) -> I e. ZZ )
42		zdcle 9544	. . . . . . 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 841	. . . . . 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 803	. . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ I < (♯ ` A ) ) -> ( A ` I ) = ( ( A ++ B ) ` I ) )
47		simprll 537	. . . . . . . 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 9444	. . . . . . . . . . . . 13 |- ( A e. Word V -> (♯ ` A ) e. RR )
50		zre 9471	. . . . . . . . . . . . 13 |- ( I e. ZZ -> I e. RR )
51		lenlt 8243	. . . . . . . . . . . . 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 11104	. . . . . . . . . . . . . 14 |- ( B e. Word V -> (♯ ` B ) e. NN0 )
57	56	nn0zd 9588	. . . . . . . . . . . . 13 |- ( B e. Word V -> (♯ ` B ) e. ZZ )
58		zaddcl 9507	. . . . . . . . . . . . 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 10372	. . . . . . . . . . 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 1271	. . . . . . . . . 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 1004	. . . . . . . 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 11162	. . . . . . . 8 |- ( ( A e. Word V /\ B e. Word V /\ I e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) -> ( ( A ++ B ) ` I ) = ( B ` ( I - (♯ ` A ) ) ) )
68	67	eqcomd 2235	. . . . . . 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 9444	. . . . . . . . . . 11 |- ( B e. Word V -> (♯ ` B ) e. RR )
72		readdcl 8146	. . . . . . . . . . 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 8243	. . . . . . . . . 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 528	. . . . . . . . . . . . . 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 9595	. . . . . . . . . . . . . . 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 8715	. . . . . . . . . . . . . 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 739	. . . . . . . . . . . 12 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ ( (♯ ` A ) + (♯ ` B ) ) <_ I ) -> ( ( I - (♯ ` A ) ) < 0 \/ (♯ ` B ) <_ ( I - (♯ ` A ) ) ) )
89		wrdsymb0 11133	. . . . . . . . . . . 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 11159	. . . . . . . . . . . . . . 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 4106	. . . . . . . . . . . . 13 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ ( (♯ ` A ) + (♯ ` B ) ) <_ I ) -> (♯ ` ( A ++ B ) ) <_ I )
96	95	olcd 739	. . . . . . . . . . . 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 2265	. . . . . . . . . 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 528	. . . . . . 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 9545	. . . . . . 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 841	. . . . . 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 803	. . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) /\ -. I < (♯ ` A ) ) -> ( B ` ( I - (♯ ` A ) ) ) = ( ( A ++ B ) ` I ) )
111		zdclt 9545	. . . . 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 3636	. . 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 2235	. 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 1218	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 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   (/)c0 3492   ifcif 3603   class class class wbr 4084   ` cfv 5322  (class class class)co 6011   RRcr 8019   0cc0 8020   + caddc 8023   < clt 8202   <_ cle 8203   - cmin 8338   ZZcz 9467  ..^cfzo 10365  ♯chash 11025  Word cword 11100   ++ cconcat 11154

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4200  ax-sep 4203  ax-nul 4211  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-iinf 4682  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-addcom 8120  ax-addass 8122  ax-distr 8124  ax-i2m1 8125  ax-0lt1 8126  ax-0id 8128  ax-rnegex 8129  ax-cnre 8131  ax-pre-ltirr 8132  ax-pre-ltwlin 8133  ax-pre-lttrn 8134  ax-pre-apti 8135  ax-pre-ltadd 8136

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-tr 4184  df-id 4386  df-iord 4459  df-on 4461  df-ilim 4462  df-suc 4464  df-iom 4685  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-recs 6464  df-frec 6550  df-1o 6575  df-er 6695  df-en 6903  df-dom 6904  df-fin 6905  df-pnf 8204  df-mnf 8205  df-xr 8206  df-ltxr 8207  df-le 8208  df-sub 8340  df-neg 8341  df-inn 9132  df-n0 9391  df-z 9468  df-uz 9744  df-fz 10232  df-fzo 10366  df-ihash 11026  df-word 11101  df-concat 11155

This theorem is referenced by:  swrdccatin2  11297