efgredleme - Metamath Proof Explorer

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Theorem efgredleme 18156

Description: Lemma for efgred 18161. (Contributed by Mario Carneiro, 1-Oct-2015.)

**Hypotheses**
Ref	Expression
efgval.w	Word
efgval.r	~_FG
efgval2.m	$\$
efgval2.t	splice
efgred.d	$\$
efgred.s	Word $\$ $/\$ ..^
efgredlem.1
efgredlem.2
efgredlem.3
efgredlem.4
efgredlem.5
efgredlemb.k
efgredlemb.l
efgredlemb.p
efgredlemb.q
efgredlemb.u
efgredlemb.v
efgredlemb.6
efgredlemb.7
efgredlemb.8
efgredlemd.9
efgredlemd.c
efgredlemd.sc	substr ++ substr

**Assertion**
Ref	Expression
efgredleme	$/\$

Distinct variable groups:

Allowed substitution hints:

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Proof of Theorem efgredleme Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgredlemd.c	. . . . . 6
2		efgval.w	. . . . . . . . 9 Word
3		efgval.r	. . . . . . . . 9 ~_FG
4		efgval2.m	. . . . . . . . 9 $\$
5		efgval2.t	. . . . . . . . 9 splice
6		efgred.d	. . . . . . . . 9 $\$
7		efgred.s	. . . . . . . . 9 Word $\$ $/\$ ..^
8	2, 3, 4, 5, 6, 7	efgsf 18142	. . . . . . . 8 Word $\$ $/\$ ..^
9	8	fdmi 6052	. . . . . . . . 9 Word $\$ $/\$ ..^
10	9	feq2i 6037	. . . . . . . 8 Word $\$ $/\$ ..^
11	8, 10	mpbir 221	. . . . . . 7
12	11	ffvelrni 6358	. . . . . 6
13	1, 12	syl 17	. . . . 5
14		efgredlemb.q	. . . . . . 7
15		elfzuz 12338	. . . . . . 7
16	14, 15	syl 17	. . . . . 6
17		efgredlemd.sc	. . . . . . . . . 10 substr ++ substr
18	17	fveq2d 6195	. . . . . . . . 9 substr ++ substr
19		fviss 6256	. . . . . . . . . . . . 13 Word Word
20	2, 19	eqsstri 3635	. . . . . . . . . . . 12 Word
21		efgredlem.1	. . . . . . . . . . . . . 14
22		efgredlem.2	. . . . . . . . . . . . . 14
23		efgredlem.3	. . . . . . . . . . . . . 14
24		efgredlem.4	. . . . . . . . . . . . . 14
25		efgredlem.5	. . . . . . . . . . . . . 14
26		efgredlemb.k	. . . . . . . . . . . . . 14
27		efgredlemb.l	. . . . . . . . . . . . . 14
28	2, 3, 4, 5, 6, 7, 21, 22, 23, 24, 25, 26, 27	efgredlemf 18154	. . . . . . . . . . . . 13 $/\$
29	28	simprd 479	. . . . . . . . . . . 12
30	20, 29	sseldi 3601	. . . . . . . . . . 11 Word
31		swrdcl 13419	. . . . . . . . . . 11 Word substr Word
32	30, 31	syl 17	. . . . . . . . . 10 substr Word
33	28	simpld 475	. . . . . . . . . . . 12
34	20, 33	sseldi 3601	. . . . . . . . . . 11 Word
35		swrdcl 13419	. . . . . . . . . . 11 Word substr Word
36	34, 35	syl 17	. . . . . . . . . 10 substr Word
37		ccatlen 13360	. . . . . . . . . 10 substr Word $/\$ substr Word substr ++ substr substr substr
38	32, 36, 37	syl2anc 693	. . . . . . . . 9 substr ++ substr substr substr
39		swrd0len 13422	. . . . . . . . . . . 12 Word $/\$ substr
40	30, 14, 39	syl2anc 693	. . . . . . . . . . 11 substr
41		2nn0 11309	. . . . . . . . . . . . . 14
42		uzaddcl 11744	. . . . . . . . . . . . . 14 $/\$
43	16, 41, 42	sylancl 694	. . . . . . . . . . . . 13
44		efgredlemb.p	. . . . . . . . . . . . . . 15
45		elfzuz3 12339	. . . . . . . . . . . . . . 15
46	44, 45	syl 17	. . . . . . . . . . . . . 14
47		efgredlemd.9	. . . . . . . . . . . . . 14
48		uztrn 11704	. . . . . . . . . . . . . 14 $/\$
49	46, 47, 48	syl2anc 693	. . . . . . . . . . . . 13
50		elfzuzb 12336	. . . . . . . . . . . . 13 $/\$
51	43, 49, 50	sylanbrc 698	. . . . . . . . . . . 12
52		lencl 13324	. . . . . . . . . . . . . . 15 Word
53	34, 52	syl 17	. . . . . . . . . . . . . 14
54		nn0uz 11722	. . . . . . . . . . . . . 14
55	53, 54	syl6eleq 2711	. . . . . . . . . . . . 13
56		eluzfz2 12349	. . . . . . . . . . . . 13
57	55, 56	syl 17	. . . . . . . . . . . 12
58		swrdlen 13423	. . . . . . . . . . . 12 Word $/\$ $/\$ substr
59	34, 51, 57, 58	syl3anc 1326	. . . . . . . . . . 11 substr
60	40, 59	oveq12d 6668	. . . . . . . . . 10 substr substr
61		elfzelz 12342	. . . . . . . . . . . . 13
62	14, 61	syl 17	. . . . . . . . . . . 12
63	62	zcnd 11483	. . . . . . . . . . 11
64	53	nn0cnd 11353	. . . . . . . . . . 11
65		2z 11409	. . . . . . . . . . . . 13
66		zaddcl 11417	. . . . . . . . . . . . 13 $/\$
67	62, 65, 66	sylancl 694	. . . . . . . . . . . 12
68	67	zcnd 11483	. . . . . . . . . . 11
69	63, 64, 68	addsubassd 10412	. . . . . . . . . 10
70		2cn 11091	. . . . . . . . . . . 12
71	70	a1i 11	. . . . . . . . . . 11
72	63, 64, 71	pnpcand 10429	. . . . . . . . . 10
73	60, 69, 72	3eqtr2d 2662	. . . . . . . . 9 substr substr
74	18, 38, 73	3eqtrd 2660	. . . . . . . 8
75		elfzelz 12342	. . . . . . . . . . 11
76	44, 75	syl 17	. . . . . . . . . 10
77		zsubcl 11419	. . . . . . . . . 10 $/\$
78	76, 65, 77	sylancl 694	. . . . . . . . 9
79	65	a1i 11	. . . . . . . . 9
80	76	zcnd 11483	. . . . . . . . . . . 12
81		npcan 10290	. . . . . . . . . . . 12 $/\$
82	80, 70, 81	sylancl 694	. . . . . . . . . . 11
83	82	fveq2d 6195	. . . . . . . . . 10
84	46, 83	eleqtrrd 2704	. . . . . . . . 9
85		eluzsub 11717	. . . . . . . . 9 $/\$ $/\$
86	78, 79, 84, 85	syl3anc 1326	. . . . . . . 8
87	74, 86	eqeltrd 2701	. . . . . . 7
88		eluzsub 11717	. . . . . . . 8 $/\$ $/\$
89	62, 79, 47, 88	syl3anc 1326	. . . . . . 7
90		uztrn 11704	. . . . . . 7 $/\$
91	87, 89, 90	syl2anc 693	. . . . . 6
92		elfzuzb 12336	. . . . . 6 $/\$
93	16, 91, 92	sylanbrc 698	. . . . 5
94		efgredlemb.v	. . . . 5
95	2, 3, 4, 5	efgtval 18136	. . . . 5 $/\$ $/\$ splice
96	13, 93, 94, 95	syl3anc 1326	. . . 4 splice
97		swrdcl 13419	. . . . . 6 Word substr Word
98	34, 97	syl 17	. . . . 5 substr Word
99		wrd0 13330	. . . . . 6 Word
100	99	a1i 11	. . . . 5 Word
101	4	efgmf 18126	. . . . . . . 8
102	101	ffvelrni 6358	. . . . . . 7
103	94, 102	syl 17	. . . . . 6
104	94, 103	s2cld 13616	. . . . 5 Word
105		eluzfz1 12348	. . . . . . . . . . . . . 14
106	16, 105	syl 17	. . . . . . . . . . . . 13
107	62	zred 11482	. . . . . . . . . . . . . . . . 17
108		nn0addge1 11339	. . . . . . . . . . . . . . . . 17 $/\$
109	107, 41, 108	sylancl 694	. . . . . . . . . . . . . . . 16
110		eluz2 11693	. . . . . . . . . . . . . . . 16 $/\$ $/\$
111	62, 67, 109, 110	syl3anbrc 1246	. . . . . . . . . . . . . . 15
112		uztrn 11704	. . . . . . . . . . . . . . 15 $/\$
113	47, 111, 112	syl2anc 693	. . . . . . . . . . . . . 14
114		elfzuzb 12336	. . . . . . . . . . . . . 14 $/\$
115	16, 113, 114	sylanbrc 698	. . . . . . . . . . . . 13
116		ccatswrd 13456	. . . . . . . . . . . . 13 Word $/\$ $/\$ $/\$ substr ++ substr substr
117	34, 106, 115, 44, 116	syl13anc 1328	. . . . . . . . . . . 12 substr ++ substr substr
118	117	oveq1d 6665	. . . . . . . . . . 11 substr ++ substr ++ ++ substr substr ++ ++ substr
119		swrdcl 13419	. . . . . . . . . . . . 13 Word substr Word
120	34, 119	syl 17	. . . . . . . . . . . 12 substr Word
121		efgredlemb.u	. . . . . . . . . . . . 13
122	101	ffvelrni 6358	. . . . . . . . . . . . . 14
123	121, 122	syl 17	. . . . . . . . . . . . 13
124	121, 123	s2cld 13616	. . . . . . . . . . . 12 Word
125		swrdcl 13419	. . . . . . . . . . . . 13 Word substr Word
126	34, 125	syl 17	. . . . . . . . . . . 12 substr Word
127		ccatass 13371	. . . . . . . . . . . 12 substr Word $/\$ Word $/\$ substr Word substr ++ ++ substr substr ++ ++ substr
128	120, 124, 126, 127	syl3anc 1326	. . . . . . . . . . 11 substr ++ ++ substr substr ++ ++ substr
129		efgredlemb.6	. . . . . . . . . . . . 13
130	2, 3, 4, 5	efgtval 18136	. . . . . . . . . . . . . 14 $/\$ $/\$ splice
131	33, 44, 121, 130	syl3anc 1326	. . . . . . . . . . . . 13 splice
132		splval 13502	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ Word splice substr ++ ++ substr
133	33, 44, 44, 124, 132	syl13anc 1328	. . . . . . . . . . . . 13 splice substr ++ ++ substr
134	129, 131, 133	3eqtrd 2660	. . . . . . . . . . . 12 substr ++ ++ substr
135		efgredlemb.7	. . . . . . . . . . . . 13
136	2, 3, 4, 5	efgtval 18136	. . . . . . . . . . . . . 14 $/\$ $/\$ splice
137	29, 14, 94, 136	syl3anc 1326	. . . . . . . . . . . . 13 splice
138		splval 13502	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ Word splice substr ++ ++ substr
139	29, 14, 14, 104, 138	syl13anc 1328	. . . . . . . . . . . . 13 splice substr ++ ++ substr
140	135, 137, 139	3eqtrd 2660	. . . . . . . . . . . 12 substr ++ ++ substr
141	24, 134, 140	3eqtr3d 2664	. . . . . . . . . . 11 substr ++ ++ substr substr ++ ++ substr
142	118, 128, 141	3eqtr2d 2662	. . . . . . . . . 10 substr ++ substr ++ ++ substr substr ++ ++ substr
143		swrdcl 13419	. . . . . . . . . . . 12 Word substr Word
144	34, 143	syl 17	. . . . . . . . . . 11 substr Word
145		ccatcl 13359	. . . . . . . . . . . 12 Word $/\$ substr Word ++ substr Word
146	124, 126, 145	syl2anc 693	. . . . . . . . . . 11 ++ substr Word
147		ccatass 13371	. . . . . . . . . . 11 substr Word $/\$ substr Word $/\$ ++ substr Word substr ++ substr ++ ++ substr substr ++ substr ++ ++ substr
148	98, 144, 146, 147	syl3anc 1326	. . . . . . . . . 10 substr ++ substr ++ ++ substr substr ++ substr ++ ++ substr
149		swrdcl 13419	. . . . . . . . . . . 12 Word substr Word
150	30, 149	syl 17	. . . . . . . . . . 11 substr Word
151		ccatass 13371	. . . . . . . . . . 11 substr Word $/\$ Word $/\$ substr Word substr ++ ++ substr substr ++ ++ substr
152	32, 104, 150, 151	syl3anc 1326	. . . . . . . . . 10 substr ++ ++ substr substr ++ ++ substr
153	142, 148, 152	3eqtr3d 2664	. . . . . . . . 9 substr ++ substr ++ ++ substr substr ++ ++ substr
154		ccatcl 13359	. . . . . . . . . . 11 substr Word $/\$ ++ substr Word substr ++ ++ substr Word
155	144, 146, 154	syl2anc 693	. . . . . . . . . 10 substr ++ ++ substr Word
156		ccatcl 13359	. . . . . . . . . . 11 Word $/\$ substr Word ++ substr Word
157	104, 150, 156	syl2anc 693	. . . . . . . . . 10 ++ substr Word
158		uztrn 11704	. . . . . . . . . . . . . 14 $/\$
159	46, 113, 158	syl2anc 693	. . . . . . . . . . . . 13
160		elfzuzb 12336	. . . . . . . . . . . . 13 $/\$
161	16, 159, 160	sylanbrc 698	. . . . . . . . . . . 12
162		swrd0len 13422	. . . . . . . . . . . 12 Word $/\$ substr
163	34, 161, 162	syl2anc 693	. . . . . . . . . . 11 substr
164	163, 40	eqtr4d 2659	. . . . . . . . . 10 substr substr
165		ccatopth 13470	. . . . . . . . . 10 substr Word $/\$ substr ++ ++ substr Word $/\$ substr Word $/\$ ++ substr Word $/\$ substr substr substr ++ substr ++ ++ substr substr ++ ++ substr substr substr $/\$ substr ++ ++ substr ++ substr
166	98, 155, 32, 157, 164, 165	syl221anc 1337	. . . . . . . . 9 substr ++ substr ++ ++ substr substr ++ ++ substr substr substr $/\$ substr ++ ++ substr ++ substr
167	153, 166	mpbid 222	. . . . . . . 8 substr substr $/\$ substr ++ ++ substr ++ substr
168	167	simpld 475	. . . . . . 7 substr substr
169	168	oveq1d 6665	. . . . . 6 substr ++ substr substr ++ substr
170		ccatrid 13370	. . . . . . . 8 substr Word substr ++ substr
171	98, 170	syl 17	. . . . . . 7 substr ++ substr
172	171	oveq1d 6665	. . . . . 6 substr ++ ++ substr substr ++ substr
173	169, 172, 17	3eqtr4rd 2667	. . . . 5 substr ++ ++ substr
174	163	eqcomd 2628	. . . . 5 substr
175		hash0 13158	. . . . . . 7
176	175	oveq2i 6661	. . . . . 6
177	63	addid1d 10236	. . . . . 6
178	176, 177	syl5req 2669	. . . . 5
179	98, 100, 36, 104, 173, 174, 178	splval2 13508	. . . 4 splice substr ++ ++ substr
180		elfzuzb 12336	. . . . . . . . . . . . . 14 $/\$
181	16, 111, 180	sylanbrc 698	. . . . . . . . . . . . 13
182		elfzuzb 12336	. . . . . . . . . . . . . 14 $/\$
183	43, 47, 182	sylanbrc 698	. . . . . . . . . . . . 13
184		ccatswrd 13456	. . . . . . . . . . . . 13 Word $/\$ $/\$ $/\$ substr ++ substr substr
185	34, 181, 183, 44, 184	syl13anc 1328	. . . . . . . . . . . 12 substr ++ substr substr
186	185	oveq1d 6665	. . . . . . . . . . 11 substr ++ substr ++ ++ substr substr ++ ++ substr
187		swrdcl 13419	. . . . . . . . . . . . 13 Word substr Word
188	34, 187	syl 17	. . . . . . . . . . . 12 substr Word
189		swrdcl 13419	. . . . . . . . . . . . 13 Word substr Word
190	34, 189	syl 17	. . . . . . . . . . . 12 substr Word
191		ccatass 13371	. . . . . . . . . . . 12 substr Word $/\$ substr Word $/\$ ++ substr Word substr ++ substr ++ ++ substr substr ++ substr ++ ++ substr
192	188, 190, 146, 191	syl3anc 1326	. . . . . . . . . . 11 substr ++ substr ++ ++ substr substr ++ substr ++ ++ substr
193	167	simprd 479	. . . . . . . . . . 11 substr ++ ++ substr ++ substr
194	186, 192, 193	3eqtr3d 2664	. . . . . . . . . 10 substr ++ substr ++ ++ substr ++ substr
195		ccatcl 13359	. . . . . . . . . . . 12 substr Word $/\$ ++ substr Word substr ++ ++ substr Word
196	190, 146, 195	syl2anc 693	. . . . . . . . . . 11 substr ++ ++ substr Word
197		swrdlen 13423	. . . . . . . . . . . . . 14 Word $/\$ $/\$ substr
198	34, 181, 51, 197	syl3anc 1326	. . . . . . . . . . . . 13 substr
199		pncan2 10288	. . . . . . . . . . . . . 14 $/\$
200	63, 70, 199	sylancl 694	. . . . . . . . . . . . 13
201	198, 200	eqtrd 2656	. . . . . . . . . . . 12 substr
202		s2len 13634	. . . . . . . . . . . 12
203	201, 202	syl6eqr 2674	. . . . . . . . . . 11 substr
204		ccatopth 13470	. . . . . . . . . . 11 substr Word $/\$ substr ++ ++ substr Word $/\$ Word $/\$ substr Word $/\$ substr substr ++ substr ++ ++ substr ++ substr substr $/\$ substr ++ ++ substr substr
205	188, 196, 104, 150, 203, 204	syl221anc 1337	. . . . . . . . . 10 substr ++ substr ++ ++ substr ++ substr substr $/\$ substr ++ ++ substr substr
206	194, 205	mpbid 222	. . . . . . . . 9 substr $/\$ substr ++ ++ substr substr
207	206	simpld 475	. . . . . . . 8 substr
208	207	oveq2d 6666	. . . . . . 7 substr ++ substr substr ++
209		ccatswrd 13456	. . . . . . . 8 Word $/\$ $/\$ $/\$ substr ++ substr substr
210	34, 106, 181, 51, 209	syl13anc 1328	. . . . . . 7 substr ++ substr substr
211	208, 210	eqtr3d 2658	. . . . . 6 substr ++ substr
212	211	oveq1d 6665	. . . . 5 substr ++ ++ substr substr ++ substr
213		eluzfz1 12348	. . . . . . 7
214	43, 213	syl 17	. . . . . 6
215		ccatswrd 13456	. . . . . 6 Word $/\$ $/\$ $/\$ substr ++ substr substr
216	34, 214, 51, 57, 215	syl13anc 1328	. . . . 5 substr ++ substr substr
217		swrdid 13428	. . . . . 6 Word substr
218	34, 217	syl 17	. . . . 5 substr
219	212, 216, 218	3eqtrd 2660	. . . 4 substr ++ ++ substr
220	96, 179, 219	3eqtrd 2660	. . 3
221	2, 3, 4, 5	efgtf 18135	. . . . . . 7 splice $/\$
222	13, 221	syl 17	. . . . . 6 splice $/\$
223	222	simprd 479	. . . . 5
224		ffn 6045	. . . . 5
225	223, 224	syl 17	. . . 4
226		fnovrn 6809	. . . 4 $/\$ $/\$
227	225, 93, 94, 226	syl3anc 1326	. . 3
228	220, 227	eqeltrrd 2702	. 2
229		uztrn 11704	. . . . . . 7 $/\$
230	89, 16, 229	syl2anc 693	. . . . . 6
231		elfzuzb 12336	. . . . . 6 $/\$
232	230, 87, 231	sylanbrc 698	. . . . 5
233	2, 3, 4, 5	efgtval 18136	. . . . 5 $/\$ $/\$ splice
234	13, 232, 121, 233	syl3anc 1326	. . . 4 splice
235		swrdcl 13419	. . . . . 6 Word substr Word
236	30, 235	syl 17	. . . . 5 substr Word
237		swrdcl 13419	. . . . . 6 Word substr Word
238	30, 237	syl 17	. . . . 5 substr Word
239		ccatswrd 13456	. . . . . . . . . . 11 Word $/\$ $/\$ $/\$ substr ++ substr substr
240	34, 183, 44, 57, 239	syl13anc 1328	. . . . . . . . . 10 substr ++ substr substr
241	206	simprd 479	. . . . . . . . . . . . . 14 substr ++ ++ substr substr
242		elfzuzb 12336	. . . . . . . . . . . . . . . 16 $/\$
243	16, 89, 242	sylanbrc 698	. . . . . . . . . . . . . . 15
244	2, 3, 4, 5, 6, 7, 21, 22, 23, 24, 25, 26, 27, 44, 14, 121, 94, 129, 135	efgredlemg 18155	. . . . . . . . . . . . . . . . . 18
245	244, 46	eqeltrrd 2702	. . . . . . . . . . . . . . . . 17
246		0le2 11111	. . . . . . . . . . . . . . . . . . . 20
247	246	a1i 11	. . . . . . . . . . . . . . . . . . 19
248	76	zred 11482	. . . . . . . . . . . . . . . . . . . 20
249		2re 11090	. . . . . . . . . . . . . . . . . . . 20
250		subge02 10544	. . . . . . . . . . . . . . . . . . . 20 $/\$
251	248, 249, 250	sylancl 694	. . . . . . . . . . . . . . . . . . 19
252	247, 251	mpbid 222	. . . . . . . . . . . . . . . . . 18
253		eluz2 11693	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
254	78, 76, 252, 253	syl3anbrc 1246	. . . . . . . . . . . . . . . . 17
255		uztrn 11704	. . . . . . . . . . . . . . . . 17 $/\$
256	245, 254, 255	syl2anc 693	. . . . . . . . . . . . . . . 16
257		elfzuzb 12336	. . . . . . . . . . . . . . . 16 $/\$
258	230, 256, 257	sylanbrc 698	. . . . . . . . . . . . . . 15
259		lencl 13324	. . . . . . . . . . . . . . . . . 18 Word
260	30, 259	syl 17	. . . . . . . . . . . . . . . . 17
261	260, 54	syl6eleq 2711	. . . . . . . . . . . . . . . 16
262		eluzfz2 12349	. . . . . . . . . . . . . . . 16
263	261, 262	syl 17	. . . . . . . . . . . . . . 15
264		ccatswrd 13456	. . . . . . . . . . . . . . 15 Word $/\$ $/\$ $/\$ substr ++ substr substr
265	30, 243, 258, 263, 264	syl13anc 1328	. . . . . . . . . . . . . 14 substr ++ substr substr
266	241, 265	eqtr4d 2659	. . . . . . . . . . . . 13 substr ++ ++ substr substr ++ substr
267		swrdcl 13419	. . . . . . . . . . . . . . 15 Word substr Word
268	30, 267	syl 17	. . . . . . . . . . . . . 14 substr Word
269		swrdcl 13419	. . . . . . . . . . . . . . 15 Word substr Word
270	30, 269	syl 17	. . . . . . . . . . . . . 14 substr Word
271		swrdlen 13423	. . . . . . . . . . . . . . . 16 Word $/\$ $/\$ substr
272	34, 183, 44, 271	syl3anc 1326	. . . . . . . . . . . . . . 15 substr
273		swrdlen 13423	. . . . . . . . . . . . . . . . 17 Word $/\$ $/\$ substr
274	30, 243, 258, 273	syl3anc 1326	. . . . . . . . . . . . . . . 16 substr
275	80, 63, 71	sub32d 10424	. . . . . . . . . . . . . . . 16
276	80, 63, 71	subsub4d 10423	. . . . . . . . . . . . . . . 16
277	274, 275, 276	3eqtr2d 2662	. . . . . . . . . . . . . . 15 substr
278	272, 277	eqtr4d 2659	. . . . . . . . . . . . . 14 substr substr
279		ccatopth 13470	. . . . . . . . . . . . . 14 substr Word $/\$ ++ substr Word $/\$ substr Word $/\$ substr Word $/\$ substr substr substr ++ ++ substr substr ++ substr substr substr $/\$ ++ substr substr
280	190, 146, 268, 270, 278, 279	syl221anc 1337	. . . . . . . . . . . . 13 substr ++ ++ substr substr ++ substr substr substr $/\$ ++ substr substr
281	266, 280	mpbid 222	. . . . . . . . . . . 12 substr substr $/\$ ++ substr substr
282	281	simpld 475	. . . . . . . . . . 11 substr substr
283	281	simprd 479	. . . . . . . . . . . . . 14 ++ substr substr
284		elfzuzb 12336	. . . . . . . . . . . . . . . 16 $/\$
285	230, 254, 284	sylanbrc 698	. . . . . . . . . . . . . . 15
286		elfzuz 12338	. . . . . . . . . . . . . . . . 17
287	44, 286	syl 17	. . . . . . . . . . . . . . . 16
288		elfzuzb 12336	. . . . . . . . . . . . . . . 16 $/\$
289	287, 245, 288	sylanbrc 698	. . . . . . . . . . . . . . 15
290		ccatswrd 13456	. . . . . . . . . . . . . . 15 Word $/\$ $/\$ $/\$ substr ++ substr substr
291	30, 285, 289, 263, 290	syl13anc 1328	. . . . . . . . . . . . . 14 substr ++ substr substr
292	283, 291	eqtr4d 2659	. . . . . . . . . . . . 13 ++ substr substr ++ substr
293		swrdcl 13419	. . . . . . . . . . . . . . 15 Word substr Word
294	30, 293	syl 17	. . . . . . . . . . . . . 14 substr Word
295		s2len 13634	. . . . . . . . . . . . . . 15
296		swrdlen 13423	. . . . . . . . . . . . . . . . 17 Word $/\$ $/\$ substr
297	30, 285, 289, 296	syl3anc 1326	. . . . . . . . . . . . . . . 16 substr
298		nncan 10310	. . . . . . . . . . . . . . . . 17 $/\$
299	80, 70, 298	sylancl 694	. . . . . . . . . . . . . . . 16
300	297, 299	eqtr2d 2657	. . . . . . . . . . . . . . 15 substr
301	295, 300	syl5eq 2668	. . . . . . . . . . . . . 14 substr
302		ccatopth 13470	. . . . . . . . . . . . . 14 Word $/\$ substr Word $/\$ substr Word $/\$ substr Word $/\$ substr ++ substr substr ++ substr substr $/\$ substr substr
303	124, 126, 294, 238, 301, 302	syl221anc 1337	. . . . . . . . . . . . 13 ++ substr substr ++ substr substr $/\$ substr substr
304	292, 303	mpbid 222	. . . . . . . . . . . 12 substr $/\$ substr substr
305	304	simprd 479	. . . . . . . . . . 11 substr substr
306	282, 305	oveq12d 6668	. . . . . . . . . 10 substr ++ substr substr ++ substr
307	240, 306	eqtr3d 2658	. . . . . . . . 9 substr substr ++ substr
308	307	oveq2d 6666	. . . . . . . 8 substr ++ substr substr ++ substr ++ substr
309		ccatass 13371	. . . . . . . . 9 substr Word $/\$ substr Word $/\$ substr Word substr ++ substr ++ substr substr ++ substr ++ substr
310	32, 268, 238, 309	syl3anc 1326	. . . . . . . 8 substr ++ substr ++ substr substr ++ substr ++ substr
311	308, 310	eqtr4d 2659	. . . . . . 7 substr ++ substr substr ++ substr ++ substr
312		ccatswrd 13456	. . . . . . . . 9 Word $/\$ $/\$ $/\$ substr ++ substr substr
313	30, 106, 243, 258, 312	syl13anc 1328	. . . . . . . 8 substr ++ substr substr
314	313	oveq1d 6665	. . . . . . 7 substr ++ substr ++ substr substr ++ substr
315	17, 311, 314	3eqtrd 2660	. . . . . 6 substr ++ substr
316		ccatrid 13370	. . . . . . . 8 substr Word substr ++ substr
317	236, 316	syl 17	. . . . . . 7 substr ++ substr
318	317	oveq1d 6665	. . . . . 6 substr ++ ++ substr substr ++ substr
319	315, 318	eqtr4d 2659	. . . . 5 substr ++ ++ substr
320		swrd0len 13422	. . . . . . 7 Word $/\$ substr
321	30, 258, 320	syl2anc 693	. . . . . 6 substr
322	321	eqcomd 2628	. . . . 5 substr
323	175	oveq2i 6661	. . . . . 6
324	78	zcnd 11483	. . . . . . 7
325	324	addid1d 10236	. . . . . 6
326	323, 325	syl5req 2669	. . . . 5
327	236, 100, 238, 124, 319, 322, 326	splval2 13508	. . . 4 splice substr ++ ++ substr
328	304	simpld 475	. . . . . . . 8 substr
329	328	oveq2d 6666	. . . . . . 7 substr ++ substr ++ substr
330		eluzfz1 12348	. . . . . . . . 9
331	230, 330	syl 17	. . . . . . . 8
332		ccatswrd 13456	. . . . . . . 8 Word $/\$ $/\$ $/\$ substr ++ substr substr
333	30, 331, 285, 289, 332	syl13anc 1328	. . . . . . 7 substr ++ substr substr
334	329, 333	eqtrd 2656	. . . . . 6 substr ++ substr
335	334	oveq1d 6665	. . . . 5 substr ++ ++ substr substr ++ substr
336		eluzfz1 12348	. . . . . . 7
337	287, 336	syl 17	. . . . . 6
338		ccatswrd 13456	. . . . . 6 Word $/\$ $/\$ $/\$ substr ++ substr substr
339	30, 337, 289, 263, 338	syl13anc 1328	. . . . 5 substr ++ substr substr
340		swrdid 13428	. . . . . 6 Word substr
341	30, 340	syl 17	. . . . 5 substr
342	335, 339, 341	3eqtrd 2660	. . . 4 substr ++ ++ substr
343	234, 327, 342	3eqtrd 2660	. . 3
344		fnovrn 6809	. . . 4 $/\$ $/\$
345	225, 232, 121, 344	syl3anc 1326	. . 3
346	343, 345	eqeltrrd 2702	. 2
347	228, 346	jca 554	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990

wral 2912

crab 2916 $\$ cdif 3571

c0 3915

csn 4177

cop 4183

cotp 4185

ciun 4520 class class class wbr 4653

cmpt 4729

cid 5023

cxp 5112

cdm 5114

crn 5115

wfn 5883

wf 5884

cfv 5888 (class class class)co 6650

cmpt2 6652

c1o 7553

c2o 7554

cc 9934

cr 9935

cc0 9936

c1 9937

caddc 9939

clt 10074

cle 10075

cmin 10266

c2 11070

cn0 11292

cz 11377

cuz 11687

cfz 12326 ..^cfzo 12465

chash 13117 Word cword 13291 ++ cconcat 13293 substr csubstr 13295 splice csplice 13296

cs2 13586 ~_FG cefg 18119

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-substr 13303 df-splice 13304 df-s2 13593

This theorem is referenced by: efgredlemd 18157

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