Theorem ccatsymb 11315

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 9606	. . . . . . . . . . 11 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> 0 e. ZZ )
6		lencl 11253	. . . . . . . . . . . . 13 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
7	6	nn0zd 9716	. . . . . . . . . . . 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 10505	. . . . . . . . . . 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 11310	. . . . . . . 8 |- ( ( A e. Word V /\ B e. Word V /\ I e. ( 0..^ (♯ ` A ) ) ) -> ( ( A ++ B ) ` I ) = ( A ` I ) )
16	15	eqcomd 2240	. . . . . . 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 9605	. . . . . . . . . . 11 |- 0 e. ZZ
20		zltnle 9640	. . . . . . . . . . 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 11282	. . . . . . . . . . . 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 11306	. . . . . . . . . . . . . 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 11282	. . . . . . . . . . . 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 2270	. . . . . . . . . 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 9671	. . . . . . 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 9571	. . . . . . . . . . . . 13 |- ( A e. Word V -> (♯ ` A ) e. RR )
50		zre 9598	. . . . . . . . . . . . 13 |- ( I e. ZZ -> I e. RR )
51		lenlt 8365	. . . . . . . . . . . . 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 11253	. . . . . . . . . . . . . 14 |- ( B e. Word V -> (♯ ` B ) e. NN0 )
57	56	nn0zd 9716	. . . . . . . . . . . . 13 |- ( B e. Word V -> (♯ ` B ) e. ZZ )
58		zaddcl 9634	. . . . . . . . . . . . 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 10505	. . . . . . . . . . 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 11311	. . . . . . . 8 |- ( ( A e. Word V /\ B e. Word V /\ I e. ( (♯ ` A )..^ ( (♯ ` A ) + (♯ ` B ) ) ) ) -> ( ( A ++ B ) ` I ) = ( B ` ( I - (♯ ` A ) ) ) )
68	67	eqcomd 2240	. . . . . . 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 9571	. . . . . . . . . . 11 |- ( B e. Word V -> (♯ ` B ) e. RR )
72		readdcl 8269	. . . . . . . . . . 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 8365	. . . . . . . . . 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 9723	. . . . . . . . . . . . . . 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 8838	. . . . . . . . . . . . . 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 11282	. . . . . . . . . . . 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 11308	. . . . . . . . . . . . . . 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 4136	. . . . . . . . . . . . 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 2270	. . . . . . . . . 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 9672	. . . . . . 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 9672	. . . . 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 3659	. . 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 2240	. 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 2205   (/)c0 3512   ifcif 3624   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   RRcr 8142   0cc0 8143   + caddc 8146   < clt 8324   <_ cle 8325   - cmin 8460   ZZcz 9594  ..^cfzo 10498  ♯chash 11163  Word cword 11249   ++ cconcat 11303

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-concat 11304

This theorem is referenced by:  swrdccatin2  11446