Theorem ccatsymb 11370

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 9656	. . . . . . . . . . 11 |- ( ( ( A e. Word V /\ B e. Word V ) /\ I e. ZZ ) -> 0 e. ZZ )
6		lencl 11308	. . . . . . . . . . . . 13 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
7	6	nn0zd 9766	. . . . . . . . . . . 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 10556	. . . . . . . . . . 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 11365	. . . . . . . 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 9655	. . . . . . . . . . 11 |- 0 e. ZZ
20		zltnle 9690	. . . . . . . . . . 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 11337	. . . . . . . . . . . 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 11361	. . . . . . . . . . . . . 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 11337	. . . . . . . . . . . 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 9721	. . . . . . 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 9621	. . . . . . . . . . . . 13 |- ( A e. Word V -> (♯ ` A ) e. RR )
50		zre 9648	. . . . . . . . . . . . 13 |- ( I e. ZZ -> I e. RR )
51		lenlt 8401	. . . . . . . . . . . . 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 11308	. . . . . . . . . . . . . 14 |- ( B e. Word V -> (♯ ` B ) e. NN0 )
57	56	nn0zd 9766	. . . . . . . . . . . . 13 |- ( B e. Word V -> (♯ ` B ) e. ZZ )
58		zaddcl 9684	. . . . . . . . . . . . 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 10556	. . . . . . . . . . 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 11366	. . . . . . . 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 9621	. . . . . . . . . . 11 |- ( B e. Word V -> (♯ ` B ) e. RR )
72		readdcl 8305	. . . . . . . . . . 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 8401	. . . . . . . . . 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 9773	. . . . . . . . . . . . . . 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 8875	. . . . . . . . . . . . . 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 11337	. . . . . . . . . . . 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 11363	. . . . . . . . . . . . . . 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 4152	. . . . . . . . . . . . 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 9722	. . . . . . 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 9722	. . . . 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 3673	. . 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 ) ) ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   (/)c0 3520   ifcif 3638   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   RRcr 8178   0cc0 8179   + caddc 8182   < clt 8360   <_ cle 8361   - cmin 8497   ZZcz 9644  ..^cfzo 10549  ♯chash 11214  Word cword 11304   ++ cconcat 11358

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-fzo 10550  df-ihash 11215  df-word 11305  df-concat 11359

This theorem is used by:  swrdccatin2  11501