Theorem swrdccat3blem 11511

Description: Lemma for swrdccat3b 11512. (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 11308	. . . . . . . 8 |- ( B e. Word V -> (♯ ` B ) e. NN0 )
2		nn0le0eq0 9591	. . . . . . . . 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 11323	. . . . . . . . . . . 12 |- ( B e. Word V -> B e. Fin )
7		fihasheq0 11232	. . . . . . . . . . . 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 11308	. . . . . . . . . . . . . . . 16 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
13		swrdccatin2.l	. . . . . . . . . . . . . . . . . . 19 |- L = (♯ ` A )
14	13	eqcomi 2242	. . . . . . . . . . . . . . . . . 18 |- (♯ ` A ) = L
15	14	eleq1i 2304	. . . . . . . . . . . . . . . . 17 |- ( (♯ ` A ) e. NN0 <-> L e. NN0 )
16		nn0re 9572	. . . . . . . . . . . . . . . . . 18 |- ( L e. NN0 -> L e. RR )
17		elfz2nn0 10519	. . . . . . . . . . . . . . . . . . 19 |- ( M e. ( 0 ... ( L + 0 ) ) <-> ( M e. NN0 /\ ( L + 0 ) e. NN0 /\ M <_ ( L + 0 ) ) )
18		recn 8312	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( L e. RR -> L e. CC )
19	18	addridd 8475	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( L e. RR -> ( L + 0 ) = L )
20	19	breq2d 4142	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( L e. RR -> ( M <_ ( L + 0 ) <-> M <_ L ) )
21		nn0re 9572	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8406	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1047	. . . . . . . . . . . . . . . . . . . 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 4260	. . . . . . . . . . . . . . . . . . 19 |- (/) e. _V
40		0z 9655	. . . . . . . . . . . . . . . . . . 19 |- 0 e. ZZ
41		swrd00g 11421	. . . . . . . . . . . . . . . . . . 19 |- ( ( (/) e. _V /\ 0 e. ZZ ) -> ( (/) substr <. 0 , 0 >. ) = (/) )
42	39, 40, 41	mp2an 430	. . . . . . . . . . . . . . . . . 18 |- ( (/) substr <. 0 , 0 >. ) = (/)
43	13, 12	eqeltrid 2325	. . . . . . . . . . . . . . . . . . . 20 |- ( A e. Word V -> L e. NN0 )
44	43	nn0zd 9766	. . . . . . . . . . . . . . . . . . 19 |- ( A e. Word V -> L e. ZZ )
45		swrd00g 11421	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. Word V /\ L e. ZZ ) -> ( A substr <. L , L >. ) = (/) )
46	44, 45	mpdan 425	. . . . . . . . . . . . . . . . . 18 |- ( A e. Word V -> ( A substr <. L , L >. ) = (/) )
47	42, 46	eqtr4id 2290	. . . . . . . . . . . . . . . . 17 |- ( A e. Word V -> ( (/) substr <. 0 , 0 >. ) = ( A substr <. L , L >. ) )
48		nn0cn 9573	. . . . . . . . . . . . . . . . . . . . 21 |- ( L e. NN0 -> L e. CC )
49	48	subidd 8625	. . . . . . . . . . . . . . . . . . . 20 |- ( L e. NN0 -> ( L - L ) = 0 )
50	49	opeq1d 3910	. . . . . . . . . . . . . . . . . . 19 |- ( L e. NN0 -> <. ( L - L ) , 0 >. = <. 0 , 0 >. )
51	50	oveq2d 6101	. . . . . . . . . . . . . . . . . 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 8475	. . . . . . . . . . . . . . . . . . . 20 |- ( L e. NN0 -> ( L + 0 ) = L )
54	53	opeq2d 3911	. . . . . . . . . . . . . . . . . . 19 |- ( L e. NN0 -> <. L , ( L + 0 ) >. = <. L , L >. )
55	54	oveq2d 6101	. . . . . . . . . . . . . . . . . 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 2281	. . . . . . . . . . . . . . . 16 |- ( A e. Word V -> ( (/) substr <. ( L - L ) , 0 >. ) = ( A substr <. L , ( L + 0 ) >. ) )
58		oveq1 6092	. . . . . . . . . . . . . . . . . . 19 |- ( M = L -> ( M - L ) = ( L - L ) )
59	58	opeq1d 3910	. . . . . . . . . . . . . . . . . 18 |- ( M = L -> <. ( M - L ) , 0 >. = <. ( L - L ) , 0 >. )
60	59	oveq2d 6101	. . . . . . . . . . . . . . . . 17 |- ( M = L -> ( (/) substr <. ( M - L ) , 0 >. ) = ( (/) substr <. ( L - L ) , 0 >. ) )
61		opeq1 3904	. . . . . . . . . . . . . . . . . 18 |- ( M = L -> <. M , ( L + 0 ) >. = <. L , ( L + 0 ) >. )
62	61	oveq2d 6101	. . . . . . . . . . . . . . . . 17 |- ( M = L -> ( A substr <. M , ( L + 0 ) >. ) = ( A substr <. L , ( L + 0 ) >. ) )
63	60, 62	eqeq12d 2253	. . . . . . . . . . . . . . . 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 10428	. . . . . . . . . . . . . . . . 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 11422	. . . . . . . . . . . . . . . 16 |- ( ( A e. Word V /\ M e. ZZ /\ L e. ZZ ) -> ( A substr <. M , L >. ) e. Word V )
73	68, 70, 71, 72	syl3anc 1278	. . . . . . . . . . . . . . 15 |- ( ( A e. Word V /\ M e. ( 0 ... ( L + 0 ) ) ) -> ( A substr <. M , L >. ) e. Word V )
74		ccatrid 11375	. . . . . . . . . . . . . . 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 8464	. . . . . . . . . . . . . . . . . . 19 |- ( L e. CC -> ( L + 0 ) = L )
79	78	eqcomd 2244	. . . . . . . . . . . . . . . . . 18 |- ( L e. CC -> L = ( L + 0 ) )
80	77, 79	syl 14	. . . . . . . . . . . . . . . . 17 |- ( A e. Word V -> L = ( L + 0 ) )
81	80	opeq2d 3911	. . . . . . . . . . . . . . . 16 |- ( A e. Word V -> <. M , L >. = <. M , ( L + 0 ) >. )
82	81	oveq2d 6101	. . . . . . . . . . . . . . 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 2271	. . . . . . . . . . . . 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 9721	. . . . . . . . . . . . 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 3673	. . . . . . . . . . 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 496	. . . . . . . . 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 6093	. . . . . . . . . . . . . 14 |- ( (♯ ` B ) = 0 -> ( L + (♯ ` B ) ) = ( L + 0 ) )
92	91	oveq2d 6101	. . . . . . . . . . . . 13 |- ( (♯ ` B ) = 0 -> ( 0 ... ( L + (♯ ` B ) ) ) = ( 0 ... ( L + 0 ) ) )
93	92	eleq2d 2308	. . . . . . . . . . . 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 3905	. . . . . . . . . . . . . . 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 6103	. . . . . . . . . . . . 13 |- ( ( (♯ ` B ) = 0 /\ B = (/) ) -> ( B substr <. ( M - L ) , (♯ ` B ) >. ) = ( (/) substr <. ( M - L ) , 0 >. ) )
99		oveq2 6093	. . . . . . . . . . . . . 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 3660	. . . . . . . . . . . 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 3911	. . . . . . . . . . . . . 14 |- ( (♯ ` B ) = 0 -> <. M , ( L + (♯ ` B ) ) >. = <. M , ( L + 0 ) >. )
103	102	oveq2d 6101	. . . . . . . . . . . . 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 2253	. . . . . . . . . . 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 480	. . . . . . . . 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 425	. . . . . . 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 2304	. . . . . . . 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 9621	. . . . . 6 |- ( B e. Word V -> (♯ ` B ) e. RR )
119		leaddle0 8805	. . . . . 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		pm2.24 630	. . . . 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 497	. . . . 5 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ ( L + (♯ ` B ) ) <_ L ) -> (♯ ` B ) e. NN0 )
126	125	nn0zd 9766	. . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ ( L + (♯ ` B ) ) <_ L ) -> (♯ ` B ) e. ZZ )
127		zdcle 9721	. . . 4 |- ( ( (♯ ` B ) e. ZZ /\ 0 e. ZZ ) -> DECID (♯ ` B ) <_ 0 )
128	126, 40, 127	sylancl 417	. . 3 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ ( L + (♯ ` B ) ) <_ L ) -> DECID (♯ ` B ) <_ 0 )
129		exmiddc 848	. . 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 730	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 ) ) >. ) )

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   _Vcvv 2821   (/)c0 3520   ifcif 3638   <.cop 3712   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   Fincfn 7022   CCcc 8177   RRcr 8178   0cc0 8179   + caddc 8182   <_ cle 8361   - cmin 8497   NN0cn0 9563   ZZcz 9644   ...cfz 10411  ♯chash 11214  Word cword 11304   ++ cconcat 11358   substr csubstr 11417

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  df-substr 11418

This theorem is used by:  swrdccat3b  11512