Theorem swrdccat3blem 11431

Description: Lemma for swrdccat3b 11432. (Contributed by AV, 30-May-2018.)

Hypothesis

Ref	Expression
swrdccatin2.l	|- L = (♯ ` A )

Assertion

Ref	Expression
swrdccat3blem	|- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ ( L + (♯ ` B ) ) <_ L ) -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( A substr <. M , ( L + (♯ ` B ) ) >. ) )

Proof of Theorem swrdccat3blem

Step	Hyp	Ref	Expression
1		lencl 11228	. . . . . . . 8 |- ( B e. Word V -> (♯ ` B ) e. NN0 )
2		nn0le0eq0 9524	. . . . . . . . 9 |- ( (♯ ` B ) e. NN0 -> ( (♯ ` B ) <_ 0 <-> (♯ ` B ) = 0 ) )
3	2	biimpd 144	. . . . . . . 8 |- ( (♯ ` B ) e. NN0 -> ( (♯ ` B ) <_ 0 -> (♯ ` B ) = 0 ) )
4	1, 3	syl 14	. . . . . . 7 |- ( B e. Word V -> ( (♯ ` B ) <_ 0 -> (♯ ` B ) = 0 ) )
5	4	adantl 277	. . . . . 6 |- ( ( A e. Word V /\ B e. Word V ) -> ( (♯ ` B ) <_ 0 -> (♯ ` B ) = 0 ) )
6		wrdfin 11243	. . . . . . . . . . . 12 |- ( B e. Word V -> B e. Fin )
7		fihasheq0 11156	. . . . . . . . . . . 12 |- ( B e. Fin -> ( (♯ ` B ) = 0 <-> B = (/) ) )
8	6, 7	syl 14	. . . . . . . . . . 11 |- ( B e. Word V -> ( (♯ ` B ) = 0 <-> B = (/) ) )
9	8	biimpd 144	. . . . . . . . . 10 |- ( B e. Word V -> ( (♯ ` B ) = 0 -> B = (/) ) )
10	9	adantl 277	. . . . . . . . 9 |- ( ( A e. Word V /\ B e. Word V ) -> ( (♯ ` B ) = 0 -> B = (/) ) )
11	10	imp 124	. . . . . . . 8 |- ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` B ) = 0 ) -> B = (/) )
12		lencl 11228	. . . . . . . . . . . . . . . 16 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
13		swrdccatin2.l	. . . . . . . . . . . . . . . . . . 19 |- L = (♯ ` A )
14	13	eqcomi 2236	. . . . . . . . . . . . . . . . . 18 |- (♯ ` A ) = L
15	14	eleq1i 2298	. . . . . . . . . . . . . . . . 17 |- ( (♯ ` A ) e. NN0 <-> L e. NN0 )
16		nn0re 9505	. . . . . . . . . . . . . . . . . 18 |- ( L e. NN0 -> L e. RR )
17		elfz2nn0 10446	. . . . . . . . . . . . . . . . . . 19 |- ( M e. ( 0 ... ( L + 0 ) ) <-> ( M e. NN0 /\ ( L + 0 ) e. NN0 /\ M <_ ( L + 0 ) ) )
18		recn 8260	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( L e. RR -> L e. CC )
19	18	addridd 8422	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( L e. RR -> ( L + 0 ) = L )
20	19	breq2d 4121	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( L e. RR -> ( M <_ ( L + 0 ) <-> M <_ L ) )
21		nn0re 9505	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ( M e. NN0 -> M e. RR )
22	21	anim1i 340	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( ( M e. NN0 /\ L e. RR ) -> ( M e. RR /\ L e. RR ) )
23	22	ancoms 268	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( L e. RR /\ M e. NN0 ) -> ( M e. RR /\ L e. RR ) )
24		letri3 8354	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( ( M e. RR /\ L e. RR ) -> ( M = L <-> ( M <_ L /\ L <_ M ) ) )
25	23, 24	syl 14	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( L e. RR /\ M e. NN0 ) -> ( M = L <-> ( M <_ L /\ L <_ M ) ) )
26	25	biimprd 158	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( L e. RR /\ M e. NN0 ) -> ( ( M <_ L /\ L <_ M ) -> M = L ) )
27	26	exp4b 367	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( L e. RR -> ( M e. NN0 -> ( M <_ L -> ( L <_ M -> M = L ) ) ) )
28	27	com23 78	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( L e. RR -> ( M <_ L -> ( M e. NN0 -> ( L <_ M -> M = L ) ) ) )
29	20, 28	sylbid 150	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( L e. RR -> ( M <_ ( L + 0 ) -> ( M e. NN0 -> ( L <_ M -> M = L ) ) ) )
30	29	com3l 81	. . . . . . . . . . . . . . . . . . . . . 22 |- ( M <_ ( L + 0 ) -> ( M e. NN0 -> ( L e. RR -> ( L <_ M -> M = L ) ) ) )
31	30	impcom 125	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( M e. NN0 /\ M <_ ( L + 0 ) ) -> ( L e. RR -> ( L <_ M -> M = L ) ) )
32	31	3adant2 1043	. . . . . . . . . . . . . . . . . . . 20 |- ( ( M e. NN0 /\ ( L + 0 ) e. NN0 /\ M <_ ( L + 0 ) ) -> ( L e. RR -> ( L <_ M -> M = L ) ) )
33	32	com12 30	. . . . . . . . . . . . . . . . . . 19 |- ( L e. RR -> ( ( M e. NN0 /\ ( L + 0 ) e. NN0 /\ M <_ ( L + 0 ) ) -> ( L <_ M -> M = L ) ) )
34	17, 33	biimtrid 152	. . . . . . . . . . . . . . . . . 18 |- ( L e. RR -> ( M e. ( 0 ... ( L + 0 ) ) -> ( L <_ M -> M = L ) ) )
35	16, 34	syl 14	. . . . . . . . . . . . . . . . 17 |- ( L e. NN0 -> ( M e. ( 0 ... ( L + 0 ) ) -> ( L <_ M -> M = L ) ) )
36	15, 35	sylbi 121	. . . . . . . . . . . . . . . 16 |- ( (♯ ` A ) e. NN0 -> ( M e. ( 0 ... ( L + 0 ) ) -> ( L <_ M -> M = L ) ) )
37	12, 36	syl 14	. . . . . . . . . . . . . . 15 |- ( A e. Word V -> ( M e. ( 0 ... ( L + 0 ) ) -> ( L <_ M -> M = L ) ) )
38	37	imp 124	. . . . . . . . . . . . . 14 |- ( ( A e. Word V /\ M e. ( 0 ... ( L + 0 ) ) ) -> ( L <_ M -> M = L ) )
39		0ex 4237	. . . . . . . . . . . . . . . . . . 19 |- (/) e. _V
40		0z 9588	. . . . . . . . . . . . . . . . . . 19 |- 0 e. ZZ
41		swrd00g 11341	. . . . . . . . . . . . . . . . . . 19 |- ( ( (/) e. _V /\ 0 e. ZZ ) -> ( (/) substr <. 0 , 0 >. ) = (/) )
42	39, 40, 41	mp2an 426	. . . . . . . . . . . . . . . . . 18 |- ( (/) substr <. 0 , 0 >. ) = (/)
43	13, 12	eqeltrid 2319	. . . . . . . . . . . . . . . . . . . 20 |- ( A e. Word V -> L e. NN0 )
44	43	nn0zd 9698	. . . . . . . . . . . . . . . . . . 19 |- ( A e. Word V -> L e. ZZ )
45		swrd00g 11341	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. Word V /\ L e. ZZ ) -> ( A substr <. L , L >. ) = (/) )
46	44, 45	mpdan 421	. . . . . . . . . . . . . . . . . 18 |- ( A e. Word V -> ( A substr <. L , L >. ) = (/) )
47	42, 46	eqtr4id 2284	. . . . . . . . . . . . . . . . 17 |- ( A e. Word V -> ( (/) substr <. 0 , 0 >. ) = ( A substr <. L , L >. ) )
48		nn0cn 9506	. . . . . . . . . . . . . . . . . . . . 21 |- ( L e. NN0 -> L e. CC )
49	48	subidd 8572	. . . . . . . . . . . . . . . . . . . 20 |- ( L e. NN0 -> ( L - L ) = 0 )
50	49	opeq1d 3889	. . . . . . . . . . . . . . . . . . 19 |- ( L e. NN0 -> <. ( L - L ) , 0 >. = <. 0 , 0 >. )
51	50	oveq2d 6066	. . . . . . . . . . . . . . . . . 18 |- ( L e. NN0 -> ( (/) substr <. ( L - L ) , 0 >. ) = ( (/) substr <. 0 , 0 >. ) )
52	43, 51	syl 14	. . . . . . . . . . . . . . . . 17 |- ( A e. Word V -> ( (/) substr <. ( L - L ) , 0 >. ) = ( (/) substr <. 0 , 0 >. ) )
53	48	addridd 8422	. . . . . . . . . . . . . . . . . . . 20 |- ( L e. NN0 -> ( L + 0 ) = L )
54	53	opeq2d 3890	. . . . . . . . . . . . . . . . . . 19 |- ( L e. NN0 -> <. L , ( L + 0 ) >. = <. L , L >. )
55	54	oveq2d 6066	. . . . . . . . . . . . . . . . . 18 |- ( L e. NN0 -> ( A substr <. L , ( L + 0 ) >. ) = ( A substr <. L , L >. ) )
56	43, 55	syl 14	. . . . . . . . . . . . . . . . 17 |- ( A e. Word V -> ( A substr <. L , ( L + 0 ) >. ) = ( A substr <. L , L >. ) )
57	47, 52, 56	3eqtr4d 2275	. . . . . . . . . . . . . . . 16 |- ( A e. Word V -> ( (/) substr <. ( L - L ) , 0 >. ) = ( A substr <. L , ( L + 0 ) >. ) )
58		oveq1 6057	. . . . . . . . . . . . . . . . . . 19 |- ( M = L -> ( M - L ) = ( L - L ) )
59	58	opeq1d 3889	. . . . . . . . . . . . . . . . . 18 |- ( M = L -> <. ( M - L ) , 0 >. = <. ( L - L ) , 0 >. )
60	59	oveq2d 6066	. . . . . . . . . . . . . . . . 17 |- ( M = L -> ( (/) substr <. ( M - L ) , 0 >. ) = ( (/) substr <. ( L - L ) , 0 >. ) )
61		opeq1 3883	. . . . . . . . . . . . . . . . . 18 |- ( M = L -> <. M , ( L + 0 ) >. = <. L , ( L + 0 ) >. )
62	61	oveq2d 6066	. . . . . . . . . . . . . . . . 17 |- ( M = L -> ( A substr <. M , ( L + 0 ) >. ) = ( A substr <. L , ( L + 0 ) >. ) )
63	60, 62	eqeq12d 2247	. . . . . . . . . . . . . . . 16 |- ( M = L -> ( ( (/) substr <. ( M - L ) , 0 >. ) = ( A substr <. M , ( L + 0 ) >. ) <-> ( (/) substr <. ( L - L ) , 0 >. ) = ( A substr <. L , ( L + 0 ) >. ) ) )
64	57, 63	syl5ibrcom 157	. . . . . . . . . . . . . . 15 |- ( A e. Word V -> ( M = L -> ( (/) substr <. ( M - L ) , 0 >. ) = ( A substr <. M , ( L + 0 ) >. ) ) )
65	64	adantr 276	. . . . . . . . . . . . . 14 |- ( ( A e. Word V /\ M e. ( 0 ... ( L + 0 ) ) ) -> ( M = L -> ( (/) substr <. ( M - L ) , 0 >. ) = ( A substr <. M , ( L + 0 ) >. ) ) )
66	38, 65	syld 45	. . . . . . . . . . . . 13 |- ( ( A e. Word V /\ M e. ( 0 ... ( L + 0 ) ) ) -> ( L <_ M -> ( (/) substr <. ( M - L ) , 0 >. ) = ( A substr <. M , ( L + 0 ) >. ) ) )
67	66	imp 124	. . . . . . . . . . . 12 |- ( ( ( A e. Word V /\ M e. ( 0 ... ( L + 0 ) ) ) /\ L <_ M ) -> ( (/) substr <. ( M - L ) , 0 >. ) = ( A substr <. M , ( L + 0 ) >. ) )
68		simpl 109	. . . . . . . . . . . . . . . 16 |- ( ( A e. Word V /\ M e. ( 0 ... ( L + 0 ) ) ) -> A e. Word V )
69		elfzelz 10359	. . . . . . . . . . . . . . . . 17 |- ( M e. ( 0 ... ( L + 0 ) ) -> M e. ZZ )
70	69	adantl 277	. . . . . . . . . . . . . . . 16 |- ( ( A e. Word V /\ M e. ( 0 ... ( L + 0 ) ) ) -> M e. ZZ )
71	44	adantr 276	. . . . . . . . . . . . . . . 16 |- ( ( A e. Word V /\ M e. ( 0 ... ( L + 0 ) ) ) -> L e. ZZ )
72		swrdclg 11342	. . . . . . . . . . . . . . . 16 |- ( ( A e. Word V /\ M e. ZZ /\ L e. ZZ ) -> ( A substr <. M , L >. ) e. Word V )
73	68, 70, 71, 72	syl3anc 1274	. . . . . . . . . . . . . . 15 |- ( ( A e. Word V /\ M e. ( 0 ... ( L + 0 ) ) ) -> ( A substr <. M , L >. ) e. Word V )
74		ccatrid 11295	. . . . . . . . . . . . . . 15 |- ( ( A substr <. M , L >. ) e. Word V -> ( ( A substr <. M , L >. ) ++ (/) ) = ( A substr <. M , L >. ) )
75	73, 74	syl 14	. . . . . . . . . . . . . 14 |- ( ( A e. Word V /\ M e. ( 0 ... ( L + 0 ) ) ) -> ( ( A substr <. M , L >. ) ++ (/) ) = ( A substr <. M , L >. ) )
76	15, 48	sylbi 121	. . . . . . . . . . . . . . . . . . 19 |- ( (♯ ` A ) e. NN0 -> L e. CC )
77	12, 76	syl 14	. . . . . . . . . . . . . . . . . 18 |- ( A e. Word V -> L e. CC )
78		addrid 8411	. . . . . . . . . . . . . . . . . . 19 |- ( L e. CC -> ( L + 0 ) = L )
79	78	eqcomd 2238	. . . . . . . . . . . . . . . . . 18 |- ( L e. CC -> L = ( L + 0 ) )
80	77, 79	syl 14	. . . . . . . . . . . . . . . . 17 |- ( A e. Word V -> L = ( L + 0 ) )
81	80	opeq2d 3890	. . . . . . . . . . . . . . . 16 |- ( A e. Word V -> <. M , L >. = <. M , ( L + 0 ) >. )
82	81	oveq2d 6066	. . . . . . . . . . . . . . 15 |- ( A e. Word V -> ( A substr <. M , L >. ) = ( A substr <. M , ( L + 0 ) >. ) )
83	82	adantr 276	. . . . . . . . . . . . . 14 |- ( ( A e. Word V /\ M e. ( 0 ... ( L + 0 ) ) ) -> ( A substr <. M , L >. ) = ( A substr <. M , ( L + 0 ) >. ) )
84	75, 83	eqtrd 2265	. . . . . . . . . . . . 13 |- ( ( A e. Word V /\ M e. ( 0 ... ( L + 0 ) ) ) -> ( ( A substr <. M , L >. ) ++ (/) ) = ( A substr <. M , ( L + 0 ) >. ) )
85	84	adantr 276	. . . . . . . . . . . 12 |- ( ( ( A e. Word V /\ M e. ( 0 ... ( L + 0 ) ) ) /\ -. L <_ M ) -> ( ( A substr <. M , L >. ) ++ (/) ) = ( A substr <. M , ( L + 0 ) >. ) )
86		zdcle 9654	. . . . . . . . . . . . 13 |- ( ( L e. ZZ /\ M e. ZZ ) -> DECID L <_ M )
87	44, 69, 86	syl2an 289	. . . . . . . . . . . 12 |- ( ( A e. Word V /\ M e. ( 0 ... ( L + 0 ) ) ) -> DECID L <_ M )
88	67, 85, 87	ifeqdadc 3655	. . . . . . . . . . 11 |- ( ( A e. Word V /\ M e. ( 0 ... ( L + 0 ) ) ) -> if ( L <_ M , ( (/) substr <. ( M - L ) , 0 >. ) , ( ( A substr <. M , L >. ) ++ (/) ) ) = ( A substr <. M , ( L + 0 ) >. ) )
89	88	ex 115	. . . . . . . . . 10 |- ( A e. Word V -> ( M e. ( 0 ... ( L + 0 ) ) -> if ( L <_ M , ( (/) substr <. ( M - L ) , 0 >. ) , ( ( A substr <. M , L >. ) ++ (/) ) ) = ( A substr <. M , ( L + 0 ) >. ) ) )
90	89	ad3antrrr 492	. . . . . . . . 9 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` B ) = 0 ) /\ B = (/) ) -> ( M e. ( 0 ... ( L + 0 ) ) -> if ( L <_ M , ( (/) substr <. ( M - L ) , 0 >. ) , ( ( A substr <. M , L >. ) ++ (/) ) ) = ( A substr <. M , ( L + 0 ) >. ) ) )
91		oveq2 6058	. . . . . . . . . . . . . 14 |- ( (♯ ` B ) = 0 -> ( L + (♯ ` B ) ) = ( L + 0 ) )
92	91	oveq2d 6066	. . . . . . . . . . . . 13 |- ( (♯ ` B ) = 0 -> ( 0 ... ( L + (♯ ` B ) ) ) = ( 0 ... ( L + 0 ) ) )
93	92	eleq2d 2302	. . . . . . . . . . . 12 |- ( (♯ ` B ) = 0 -> ( M e. ( 0 ... ( L + (♯ ` B ) ) ) <-> M e. ( 0 ... ( L + 0 ) ) ) )
94	93	adantr 276	. . . . . . . . . . 11 |- ( ( (♯ ` B ) = 0 /\ B = (/) ) -> ( M e. ( 0 ... ( L + (♯ ` B ) ) ) <-> M e. ( 0 ... ( L + 0 ) ) ) )
95		simpr 110	. . . . . . . . . . . . . 14 |- ( ( (♯ ` B ) = 0 /\ B = (/) ) -> B = (/) )
96		opeq2 3884	. . . . . . . . . . . . . . 15 |- ( (♯ ` B ) = 0 -> <. ( M - L ) , (♯ ` B ) >. = <. ( M - L ) , 0 >. )
97	96	adantr 276	. . . . . . . . . . . . . 14 |- ( ( (♯ ` B ) = 0 /\ B = (/) ) -> <. ( M - L ) , (♯ ` B ) >. = <. ( M - L ) , 0 >. )
98	95, 97	oveq12d 6068	. . . . . . . . . . . . 13 |- ( ( (♯ ` B ) = 0 /\ B = (/) ) -> ( B substr <. ( M - L ) , (♯ ` B ) >. ) = ( (/) substr <. ( M - L ) , 0 >. ) )
99		oveq2 6058	. . . . . . . . . . . . . 14 |- ( B = (/) -> ( ( A substr <. M , L >. ) ++ B ) = ( ( A substr <. M , L >. ) ++ (/) ) )
100	99	adantl 277	. . . . . . . . . . . . 13 |- ( ( (♯ ` B ) = 0 /\ B = (/) ) -> ( ( A substr <. M , L >. ) ++ B ) = ( ( A substr <. M , L >. ) ++ (/) ) )
101	98, 100	ifeq12d 3642	. . . . . . . . . . . 12 |- ( ( (♯ ` B ) = 0 /\ B = (/) ) -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = if ( L <_ M , ( (/) substr <. ( M - L ) , 0 >. ) , ( ( A substr <. M , L >. ) ++ (/) ) ) )
102	91	opeq2d 3890	. . . . . . . . . . . . . 14 |- ( (♯ ` B ) = 0 -> <. M , ( L + (♯ ` B ) ) >. = <. M , ( L + 0 ) >. )
103	102	oveq2d 6066	. . . . . . . . . . . . 13 |- ( (♯ ` B ) = 0 -> ( A substr <. M , ( L + (♯ ` B ) ) >. ) = ( A substr <. M , ( L + 0 ) >. ) )
104	103	adantr 276	. . . . . . . . . . . 12 |- ( ( (♯ ` B ) = 0 /\ B = (/) ) -> ( A substr <. M , ( L + (♯ ` B ) ) >. ) = ( A substr <. M , ( L + 0 ) >. ) )
105	101, 104	eqeq12d 2247	. . . . . . . . . . 11 |- ( ( (♯ ` B ) = 0 /\ B = (/) ) -> ( if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( A substr <. M , ( L + (♯ ` B ) ) >. ) <-> if ( L <_ M , ( (/) substr <. ( M - L ) , 0 >. ) , ( ( A substr <. M , L >. ) ++ (/) ) ) = ( A substr <. M , ( L + 0 ) >. ) ) )
106	94, 105	imbi12d 234	. . . . . . . . . 10 |- ( ( (♯ ` B ) = 0 /\ B = (/) ) -> ( ( M e. ( 0 ... ( L + (♯ ` B ) ) ) -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( A substr <. M , ( L + (♯ ` B ) ) >. ) ) <-> ( M e. ( 0 ... ( L + 0 ) ) -> if ( L <_ M , ( (/) substr <. ( M - L ) , 0 >. ) , ( ( A substr <. M , L >. ) ++ (/) ) ) = ( A substr <. M , ( L + 0 ) >. ) ) ) )
107	106	adantll 476	. . . . . . . . 9 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` B ) = 0 ) /\ B = (/) ) -> ( ( M e. ( 0 ... ( L + (♯ ` B ) ) ) -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( A substr <. M , ( L + (♯ ` B ) ) >. ) ) <-> ( M e. ( 0 ... ( L + 0 ) ) -> if ( L <_ M , ( (/) substr <. ( M - L ) , 0 >. ) , ( ( A substr <. M , L >. ) ++ (/) ) ) = ( A substr <. M , ( L + 0 ) >. ) ) ) )
108	90, 107	mpbird 167	. . . . . . . 8 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` B ) = 0 ) /\ B = (/) ) -> ( M e. ( 0 ... ( L + (♯ ` B ) ) ) -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( A substr <. M , ( L + (♯ ` B ) ) >. ) ) )
109	11, 108	mpdan 421	. . . . . . 7 |- ( ( ( A e. Word V /\ B e. Word V ) /\ (♯ ` B ) = 0 ) -> ( M e. ( 0 ... ( L + (♯ ` B ) ) ) -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( A substr <. M , ( L + (♯ ` B ) ) >. ) ) )
110	109	ex 115	. . . . . 6 |- ( ( A e. Word V /\ B e. Word V ) -> ( (♯ ` B ) = 0 -> ( M e. ( 0 ... ( L + (♯ ` B ) ) ) -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( A substr <. M , ( L + (♯ ` B ) ) >. ) ) ) )
111	5, 110	syld 45	. . . . 5 |- ( ( A e. Word V /\ B e. Word V ) -> ( (♯ ` B ) <_ 0 -> ( M e. ( 0 ... ( L + (♯ ` B ) ) ) -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( A substr <. M , ( L + (♯ ` B ) ) >. ) ) ) )
112	111	com23 78	. . . 4 |- ( ( A e. Word V /\ B e. Word V ) -> ( M e. ( 0 ... ( L + (♯ ` B ) ) ) -> ( (♯ ` B ) <_ 0 -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( A substr <. M , ( L + (♯ ` B ) ) >. ) ) ) )
113	112	imp 124	. . 3 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> ( (♯ ` B ) <_ 0 -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( A substr <. M , ( L + (♯ ` B ) ) >. ) ) )
114	113	adantr 276	. 2 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ ( L + (♯ ` B ) ) <_ L ) -> ( (♯ ` B ) <_ 0 -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( A substr <. M , ( L + (♯ ` B ) ) >. ) ) )
115	13	eleq1i 2298	. . . . . . . 8 |- ( L e. NN0 <-> (♯ ` A ) e. NN0 )
116	115, 16	sylbir 135	. . . . . . 7 |- ( (♯ ` A ) e. NN0 -> L e. RR )
117	12, 116	syl 14	. . . . . 6 |- ( A e. Word V -> L e. RR )
118	1	nn0red 9554	. . . . . 6 |- ( B e. Word V -> (♯ ` B ) e. RR )
119		leaddle0 8751	. . . . . 6 |- ( ( L e. RR /\ (♯ ` B ) e. RR ) -> ( ( L + (♯ ` B ) ) <_ L <-> (♯ ` B ) <_ 0 ) )
120	117, 118, 119	syl2an 289	. . . . 5 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( L + (♯ ` B ) ) <_ L <-> (♯ ` B ) <_ 0 ) )
121		pm2.24 626	. . . . 5 |- ( (♯ ` B ) <_ 0 -> ( -. (♯ ` B ) <_ 0 -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( A substr <. M , ( L + (♯ ` B ) ) >. ) ) )
122	120, 121	biimtrdi 163	. . . 4 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( L + (♯ ` B ) ) <_ L -> ( -. (♯ ` B ) <_ 0 -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( A substr <. M , ( L + (♯ ` B ) ) >. ) ) ) )
123	122	adantr 276	. . 3 |- ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) -> ( ( L + (♯ ` B ) ) <_ L -> ( -. (♯ ` B ) <_ 0 -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( A substr <. M , ( L + (♯ ` B ) ) >. ) ) ) )
124	123	imp 124	. 2 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ ( L + (♯ ` B ) ) <_ L ) -> ( -. (♯ ` B ) <_ 0 -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( A substr <. M , ( L + (♯ ` B ) ) >. ) ) )
125	1	ad3antlr 493	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ ( L + (♯ ` B ) ) <_ L ) -> (♯ ` B ) e. NN0 )
126	125	nn0zd 9698	. . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ ( L + (♯ ` B ) ) <_ L ) -> (♯ ` B ) e. ZZ )
127		zdcle 9654	. . . 4 |- ( ( (♯ ` B ) e. ZZ /\ 0 e. ZZ ) -> DECID (♯ ` B ) <_ 0 )
128	126, 40, 127	sylancl 413	. . 3 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ ( L + (♯ ` B ) ) <_ L ) -> DECID (♯ ` B ) <_ 0 )
129		exmiddc 844	. . 3 |- (DECID (♯ ` B ) <_ 0 -> ( (♯ ` B ) <_ 0 \/ -. (♯ ` B ) <_ 0 ) )
130	128, 129	syl 14	. 2 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ ( L + (♯ ` B ) ) <_ L ) -> ( (♯ ` B ) <_ 0 \/ -. (♯ ` B ) <_ 0 ) )
131	114, 124, 130	mpjaod 726	1 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ ( L + (♯ ` B ) ) <_ L ) -> if ( L <_ M , ( B substr <. ( M - L ) , (♯ ` B ) >. ) , ( ( A substr <. M , L >. ) ++ B ) ) = ( A substr <. M , ( L + (♯ ` B ) ) >. ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2203   _Vcvv 2813   (/)c0 3508   ifcif 3620   <.cop 3692   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   Fincfn 6975   CCcc 8125   RRcr 8126   0cc0 8127   + caddc 8130   <_ cle 8309   - cmin 8444   NN0cn0 9496   ZZcz 9577   ...cfz 10342  ♯chash 11138  Word cword 11224   ++ cconcat 11278   substr csubstr 11337

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  df-substr 11338

This theorem is referenced by:  swrdccat3b  11432