Theorem swrdccat3blem 11319

Description: Lemma for swrdccat3b 11320. (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 11116	. . . . . . . 8 |- ( B e. Word V -> (♯ ` B ) e. NN0 )
2		nn0le0eq0 9429	. . . . . . . . 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 11131	. . . . . . . . . . . 12 |- ( B e. Word V -> B e. Fin )
7		fihasheq0 11054	. . . . . . . . . . . 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 11116	. . . . . . . . . . . . . . . 16 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
13		swrdccatin2.l	. . . . . . . . . . . . . . . . . . 19 |- L = (♯ ` A )
14	13	eqcomi 2235	. . . . . . . . . . . . . . . . . 18 |- (♯ ` A ) = L
15	14	eleq1i 2297	. . . . . . . . . . . . . . . . 17 |- ( (♯ ` A ) e. NN0 <-> L e. NN0 )
16		nn0re 9410	. . . . . . . . . . . . . . . . . 18 |- ( L e. NN0 -> L e. RR )
17		elfz2nn0 10346	. . . . . . . . . . . . . . . . . . 19 |- ( M e. ( 0 ... ( L + 0 ) ) <-> ( M e. NN0 /\ ( L + 0 ) e. NN0 /\ M <_ ( L + 0 ) ) )
18		recn 8164	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( L e. RR -> L e. CC )
19	18	addridd 8327	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( L e. RR -> ( L + 0 ) = L )
20	19	breq2d 4100	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( L e. RR -> ( M <_ ( L + 0 ) <-> M <_ L ) )
21		nn0re 9410	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8259	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1042	. . . . . . . . . . . . . . . . . . . 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 4216	. . . . . . . . . . . . . . . . . . 19 |- (/) e. _V
40		0z 9489	. . . . . . . . . . . . . . . . . . 19 |- 0 e. ZZ
41		swrd00g 11229	. . . . . . . . . . . . . . . . . . 19 |- ( ( (/) e. _V /\ 0 e. ZZ ) -> ( (/) substr <. 0 , 0 >. ) = (/) )
42	39, 40, 41	mp2an 426	. . . . . . . . . . . . . . . . . 18 |- ( (/) substr <. 0 , 0 >. ) = (/)
43	13, 12	eqeltrid 2318	. . . . . . . . . . . . . . . . . . . 20 |- ( A e. Word V -> L e. NN0 )
44	43	nn0zd 9599	. . . . . . . . . . . . . . . . . . 19 |- ( A e. Word V -> L e. ZZ )
45		swrd00g 11229	. . . . . . . . . . . . . . . . . . 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 2283	. . . . . . . . . . . . . . . . 17 |- ( A e. Word V -> ( (/) substr <. 0 , 0 >. ) = ( A substr <. L , L >. ) )
48		nn0cn 9411	. . . . . . . . . . . . . . . . . . . . 21 |- ( L e. NN0 -> L e. CC )
49	48	subidd 8477	. . . . . . . . . . . . . . . . . . . 20 |- ( L e. NN0 -> ( L - L ) = 0 )
50	49	opeq1d 3868	. . . . . . . . . . . . . . . . . . 19 |- ( L e. NN0 -> <. ( L - L ) , 0 >. = <. 0 , 0 >. )
51	50	oveq2d 6033	. . . . . . . . . . . . . . . . . 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 8327	. . . . . . . . . . . . . . . . . . . 20 |- ( L e. NN0 -> ( L + 0 ) = L )
54	53	opeq2d 3869	. . . . . . . . . . . . . . . . . . 19 |- ( L e. NN0 -> <. L , ( L + 0 ) >. = <. L , L >. )
55	54	oveq2d 6033	. . . . . . . . . . . . . . . . . 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 2274	. . . . . . . . . . . . . . . 16 |- ( A e. Word V -> ( (/) substr <. ( L - L ) , 0 >. ) = ( A substr <. L , ( L + 0 ) >. ) )
58		oveq1 6024	. . . . . . . . . . . . . . . . . . 19 |- ( M = L -> ( M - L ) = ( L - L ) )
59	58	opeq1d 3868	. . . . . . . . . . . . . . . . . 18 |- ( M = L -> <. ( M - L ) , 0 >. = <. ( L - L ) , 0 >. )
60	59	oveq2d 6033	. . . . . . . . . . . . . . . . 17 |- ( M = L -> ( (/) substr <. ( M - L ) , 0 >. ) = ( (/) substr <. ( L - L ) , 0 >. ) )
61		opeq1 3862	. . . . . . . . . . . . . . . . . 18 |- ( M = L -> <. M , ( L + 0 ) >. = <. L , ( L + 0 ) >. )
62	61	oveq2d 6033	. . . . . . . . . . . . . . . . 17 |- ( M = L -> ( A substr <. M , ( L + 0 ) >. ) = ( A substr <. L , ( L + 0 ) >. ) )
63	60, 62	eqeq12d 2246	. . . . . . . . . . . . . . . 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 10259	. . . . . . . . . . . . . . . . 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 11230	. . . . . . . . . . . . . . . 16 |- ( ( A e. Word V /\ M e. ZZ /\ L e. ZZ ) -> ( A substr <. M , L >. ) e. Word V )
73	68, 70, 71, 72	syl3anc 1273	. . . . . . . . . . . . . . 15 |- ( ( A e. Word V /\ M e. ( 0 ... ( L + 0 ) ) ) -> ( A substr <. M , L >. ) e. Word V )
74		ccatrid 11183	. . . . . . . . . . . . . . 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 8316	. . . . . . . . . . . . . . . . . . 19 |- ( L e. CC -> ( L + 0 ) = L )
79	78	eqcomd 2237	. . . . . . . . . . . . . . . . . 18 |- ( L e. CC -> L = ( L + 0 ) )
80	77, 79	syl 14	. . . . . . . . . . . . . . . . 17 |- ( A e. Word V -> L = ( L + 0 ) )
81	80	opeq2d 3869	. . . . . . . . . . . . . . . 16 |- ( A e. Word V -> <. M , L >. = <. M , ( L + 0 ) >. )
82	81	oveq2d 6033	. . . . . . . . . . . . . . 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 2264	. . . . . . . . . . . . 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 9555	. . . . . . . . . . . . 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 3638	. . . . . . . . . . 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 6025	. . . . . . . . . . . . . 14 |- ( (♯ ` B ) = 0 -> ( L + (♯ ` B ) ) = ( L + 0 ) )
92	91	oveq2d 6033	. . . . . . . . . . . . 13 |- ( (♯ ` B ) = 0 -> ( 0 ... ( L + (♯ ` B ) ) ) = ( 0 ... ( L + 0 ) ) )
93	92	eleq2d 2301	. . . . . . . . . . . 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 3863	. . . . . . . . . . . . . . 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 6035	. . . . . . . . . . . . 13 |- ( ( (♯ ` B ) = 0 /\ B = (/) ) -> ( B substr <. ( M - L ) , (♯ ` B ) >. ) = ( (/) substr <. ( M - L ) , 0 >. ) )
99		oveq2 6025	. . . . . . . . . . . . . 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 3625	. . . . . . . . . . . 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 3869	. . . . . . . . . . . . . 14 |- ( (♯ ` B ) = 0 -> <. M , ( L + (♯ ` B ) ) >. = <. M , ( L + 0 ) >. )
103	102	oveq2d 6033	. . . . . . . . . . . . 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 2246	. . . . . . . . . . 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 2297	. . . . . . . 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 9455	. . . . . 6 |- ( B e. Word V -> (♯ ` B ) e. RR )
119		leaddle0 8656	. . . . . 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 9599	. . . 4 |- ( ( ( ( A e. Word V /\ B e. Word V ) /\ M e. ( 0 ... ( L + (♯ ` B ) ) ) ) /\ ( L + (♯ ` B ) ) <_ L ) -> (♯ ` B ) e. ZZ )
127		zdcle 9555	. . . 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 843	. . 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 725	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 715  DECID wdc 841   /\ w3a 1004   = wceq 1397   e. wcel 2202   _Vcvv 2802   (/)c0 3494   ifcif 3605   <.cop 3672   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   Fincfn 6908   CCcc 8029   RRcr 8030   0cc0 8031   + caddc 8034   <_ cle 8214   - cmin 8349   NN0cn0 9401   ZZcz 9478   ...cfz 10242  ♯chash 11036  Word cword 11112   ++ cconcat 11166   substr csubstr 11225

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-n0 9402  df-z 9479  df-uz 9755  df-fz 10243  df-fzo 10377  df-ihash 11037  df-word 11113  df-concat 11167  df-substr 11226

This theorem is referenced by:  swrdccat3b  11320