efgcpbllemb - Metamath Proof Explorer

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Theorem efgcpbllemb 18168

Description: Lemma for efgrelex 18164. Show that

is an equivalence relation containing all direct extensions of a word, so is closed under

. (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 $\$ $/\$ ..^
efgcpbllem.1	$/\$ ++ ++ ++ ++

**Assertion**
Ref	Expression
efgcpbllemb	$/\$

Distinct variable groups:

Allowed substitution hints:

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

Proof of Theorem efgcpbllemb Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgval.w	. . 3 Word
2		efgval.r	. . 3 ~_FG
3		efgval2.m	. . 3 $\$
4		efgval2.t	. . 3 splice
5	1, 2, 3, 4	efgval2 18137	. 2 $/\$
6		efgcpbllem.1	. . . . . . 7 $/\$ ++ ++ ++ ++
7	6	relopabi 5245	. . . . . 6
8	7	a1i 11	. . . . 5 $/\$
9		efgred.d	. . . . . . . . 9 $\$
10		efgred.s	. . . . . . . . 9 Word $\$ $/\$ ..^
11	1, 2, 3, 4, 9, 10, 6	efgcpbllema 18167	. . . . . . . 8 $/\$ $/\$ ++ ++ ++ ++
12	11	simp2bi 1077	. . . . . . 7
13	12	adantl 482	. . . . . 6 $/\$ $/\$
14	11	simp1bi 1076	. . . . . . 7
15	14	adantl 482	. . . . . 6 $/\$ $/\$
16	1, 2	efger 18131	. . . . . . . 8
17	16	a1i 11	. . . . . . 7 $/\$ $/\$
18	11	simp3bi 1078	. . . . . . . 8 ++ ++ ++ ++
19	18	adantl 482	. . . . . . 7 $/\$ $/\$ ++ ++ ++ ++
20	17, 19	ersym 7754	. . . . . 6 $/\$ $/\$ ++ ++ ++ ++
21	1, 2, 3, 4, 9, 10, 6	efgcpbllema 18167	. . . . . 6 $/\$ $/\$ ++ ++ ++ ++
22	13, 15, 20, 21	syl3anbrc 1246	. . . . 5 $/\$ $/\$
23	14	ad2antrl 764	. . . . . 6 $/\$ $/\$ $/\$
24	1, 2, 3, 4, 9, 10, 6	efgcpbllema 18167	. . . . . . . 8 $/\$ $/\$ ++ ++ ++ ++
25	24	simp2bi 1077	. . . . . . 7
26	25	ad2antll 765	. . . . . 6 $/\$ $/\$ $/\$
27	16	a1i 11	. . . . . . 7 $/\$ $/\$ $/\$
28	18	ad2antrl 764	. . . . . . 7 $/\$ $/\$ $/\$ ++ ++ ++ ++
29	24	simp3bi 1078	. . . . . . . 8 ++ ++ ++ ++
30	29	ad2antll 765	. . . . . . 7 $/\$ $/\$ $/\$ ++ ++ ++ ++
31	27, 28, 30	ertrd 7758	. . . . . 6 $/\$ $/\$ $/\$ ++ ++ ++ ++
32	1, 2, 3, 4, 9, 10, 6	efgcpbllema 18167	. . . . . 6 $/\$ $/\$ ++ ++ ++ ++
33	23, 26, 31, 32	syl3anbrc 1246	. . . . 5 $/\$ $/\$ $/\$
34	16	a1i 11	. . . . . . . . 9 $/\$ $/\$
35		fviss 6256	. . . . . . . . . . . . . 14 Word Word
36	1, 35	eqsstri 3635	. . . . . . . . . . . . 13 Word
37		simpll 790	. . . . . . . . . . . . 13 $/\$ $/\$
38	36, 37	sseldi 3601	. . . . . . . . . . . 12 $/\$ $/\$ Word
39		simpr 477	. . . . . . . . . . . . 13 $/\$ $/\$
40	36, 39	sseldi 3601	. . . . . . . . . . . 12 $/\$ $/\$ Word
41		ccatcl 13359	. . . . . . . . . . . 12 Word $/\$ Word ++ Word
42	38, 40, 41	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ ++ Word
43		simplr 792	. . . . . . . . . . . 12 $/\$ $/\$
44	36, 43	sseldi 3601	. . . . . . . . . . 11 $/\$ $/\$ Word
45		ccatcl 13359	. . . . . . . . . . 11 ++ Word $/\$ Word ++ ++ Word
46	42, 44, 45	syl2anc 693	. . . . . . . . . 10 $/\$ $/\$ ++ ++ Word
47	1	efgrcl 18128	. . . . . . . . . . . 12 $/\$ Word
48	47	simprd 479	. . . . . . . . . . 11 Word
49	48	ad2antrr 762	. . . . . . . . . 10 $/\$ $/\$ Word
50	46, 49	eleqtrrd 2704	. . . . . . . . 9 $/\$ $/\$ ++ ++
51	34, 50	erref 7762	. . . . . . . 8 $/\$ $/\$ ++ ++ ++ ++
52	51	ex 450	. . . . . . 7 $/\$ ++ ++ ++ ++
53	52	pm4.71d 666	. . . . . 6 $/\$ $/\$ ++ ++ ++ ++
54	1, 2, 3, 4, 9, 10, 6	efgcpbllema 18167	. . . . . . 7 $/\$ $/\$ ++ ++ ++ ++
55		df-3an 1039	. . . . . . 7 $/\$ $/\$ ++ ++ ++ ++ $/\$ $/\$ ++ ++ ++ ++
56		anidm 676	. . . . . . . 8 $/\$
57	56	anbi1i 731	. . . . . . 7 $/\$ $/\$ ++ ++ ++ ++ $/\$ ++ ++ ++ ++
58	54, 55, 57	3bitri 286	. . . . . 6 $/\$ ++ ++ ++ ++
59	53, 58	syl6bbr 278	. . . . 5 $/\$
60	8, 22, 33, 59	iserd 7768	. . . 4 $/\$
61	1, 2, 3, 4	efgtf 18135	. . . . . . . . . 10 splice $/\$
62	61	simprd 479	. . . . . . . . 9
63	62	adantl 482	. . . . . . . 8 $/\$ $/\$
64		ffn 6045	. . . . . . . 8
65		ovelrn 6810	. . . . . . . 8
66	63, 64, 65	3syl 18	. . . . . . 7 $/\$ $/\$
67		simplr 792	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
68	62	ad2antlr 763	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
69		simprl 794	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
70		simprr 796	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
71	68, 69, 70	fovrnd 6806	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
72	50	adantr 481	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ ++ ++
73	37	adantr 481	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$
74	36, 73	sseldi 3601	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ Word
75	40	adantr 481	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ Word
76		swrdcl 13419	. . . . . . . . . . . . . . . . 17 Word substr Word
77	75, 76	syl 17	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ substr Word
78		ccatcl 13359	. . . . . . . . . . . . . . . 16 Word $/\$ substr Word ++ substr Word
79	74, 77, 78	syl2anc 693	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ ++ substr Word
80	3	efgmf 18126	. . . . . . . . . . . . . . . . . 18
81	80	ffvelrni 6358	. . . . . . . . . . . . . . . . 17
82	81	ad2antll 765	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
83	70, 82	s2cld 13616	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ Word
84		ccatcl 13359	. . . . . . . . . . . . . . 15 ++ substr Word $/\$ Word ++ substr ++ Word
85	79, 83, 84	syl2anc 693	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ ++ substr ++ Word
86		swrdcl 13419	. . . . . . . . . . . . . . 15 Word substr Word
87	75, 86	syl 17	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ substr Word
88	44	adantr 481	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ Word
89		ccatass 13371	. . . . . . . . . . . . . 14 ++ substr ++ Word $/\$ substr Word $/\$ Word ++ substr ++ ++ substr ++ ++ substr ++ ++ substr ++
90	85, 87, 88, 89	syl3anc 1326	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ ++ substr ++ ++ substr ++ ++ substr ++ ++ substr ++
91		ccatcl 13359	. . . . . . . . . . . . . . . . 17 substr Word $/\$ Word substr ++ Word
92	77, 83, 91	syl2anc 693	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ substr ++ Word
93		ccatass 13371	. . . . . . . . . . . . . . . 16 Word $/\$ substr ++ Word $/\$ substr Word ++ substr ++ ++ substr ++ substr ++ ++ substr
94	74, 92, 87, 93	syl3anc 1326	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ ++ substr ++ ++ substr ++ substr ++ ++ substr
95		ccatass 13371	. . . . . . . . . . . . . . . . 17 Word $/\$ substr Word $/\$ Word ++ substr ++ ++ substr ++
96	74, 77, 83, 95	syl3anc 1326	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ ++ substr ++ ++ substr ++
97	96	oveq1d 6665	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ ++ substr ++ ++ substr ++ substr ++ ++ substr
98	1, 2, 3, 4	efgtval 18136	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ splice
99	67, 69, 70, 98	syl3anc 1326	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ splice
100		splval 13502	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ Word splice substr ++ ++ substr
101	67, 69, 69, 83, 100	syl13anc 1328	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ splice substr ++ ++ substr
102	99, 101	eqtrd 2656	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ substr ++ ++ substr
103	102	oveq2d 6666	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ ++ ++ substr ++ ++ substr
104	94, 97, 103	3eqtr4rd 2667	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ ++ ++ substr ++ ++ substr
105	104	oveq1d 6665	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ ++ ++ ++ substr ++ ++ substr ++
106		lencl 13324	. . . . . . . . . . . . . . . . . . 19 Word
107	74, 106	syl 17	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$
108		nn0uz 11722	. . . . . . . . . . . . . . . . . 18
109	107, 108	syl6eleq 2711	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$
110		elfznn0 12433	. . . . . . . . . . . . . . . . . 18
111	110	ad2antrl 764	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$
112		uzaddcl 11744	. . . . . . . . . . . . . . . . 17 $/\$
113	109, 111, 112	syl2anc 693	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
114	42	adantr 481	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ ++ Word
115		ccatlen 13360	. . . . . . . . . . . . . . . . . 18 ++ Word $/\$ Word ++ ++ ++
116	114, 88, 115	syl2anc 693	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ ++ ++ ++
117		ccatlen 13360	. . . . . . . . . . . . . . . . . . . 20 Word $/\$ Word ++
118	74, 75, 117	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ ++
119		elfzuz3 12339	. . . . . . . . . . . . . . . . . . . . . 22
120	119	ad2antrl 764	. . . . . . . . . . . . . . . . . . . . 21 $/\$ $/\$ $/\$ $/\$
121	107	nn0zd 11480	. . . . . . . . . . . . . . . . . . . . 21 $/\$ $/\$ $/\$ $/\$
122		eluzadd 11716	. . . . . . . . . . . . . . . . . . . . 21 $/\$
123	120, 121, 122	syl2anc 693	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$ $/\$ $/\$
124		lencl 13324	. . . . . . . . . . . . . . . . . . . . . . 23 Word
125	75, 124	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 $/\$ $/\$ $/\$ $/\$
126	125	nn0cnd 11353	. . . . . . . . . . . . . . . . . . . . 21 $/\$ $/\$ $/\$ $/\$
127	107	nn0cnd 11353	. . . . . . . . . . . . . . . . . . . . 21 $/\$ $/\$ $/\$ $/\$
128	126, 127	addcomd 10238	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$ $/\$ $/\$
129	111	nn0cnd 11353	. . . . . . . . . . . . . . . . . . . . . 22 $/\$ $/\$ $/\$ $/\$
130	129, 127	addcomd 10238	. . . . . . . . . . . . . . . . . . . . 21 $/\$ $/\$ $/\$ $/\$
131	130	fveq2d 6195	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$ $/\$ $/\$
132	123, 128, 131	3eltr3d 2715	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$
133	118, 132	eqeltrd 2701	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ ++
134		lencl 13324	. . . . . . . . . . . . . . . . . . 19 Word
135	88, 134	syl 17	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$
136		uzaddcl 11744	. . . . . . . . . . . . . . . . . 18 ++ $/\$ ++
137	133, 135, 136	syl2anc 693	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ ++
138	116, 137	eqeltrd 2701	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ ++ ++
139		elfzuzb 12336	. . . . . . . . . . . . . . . 16 ++ ++ $/\$ ++ ++
140	113, 138, 139	sylanbrc 698	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ ++ ++
141	1, 2, 3, 4	efgtval 18136	. . . . . . . . . . . . . . 15 ++ ++ $/\$ ++ ++ $/\$ ++ ++ ++ ++ splice
142	72, 140, 70, 141	syl3anc 1326	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ ++ ++ ++ ++ splice
143		wrd0 13330	. . . . . . . . . . . . . . . 16 Word
144	143	a1i 11	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ Word
145		ccatcl 13359	. . . . . . . . . . . . . . . 16 substr Word $/\$ Word substr ++ Word
146	87, 88, 145	syl2anc 693	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ substr ++ Word
147		ccatrid 13370	. . . . . . . . . . . . . . . . . 18 ++ substr Word ++ substr ++ ++ substr
148	79, 147	syl 17	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ ++ substr ++ ++ substr
149	148	oveq1d 6665	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ ++ substr ++ ++ substr ++ ++ substr ++ substr ++
150		ccatass 13371	. . . . . . . . . . . . . . . . 17 ++ substr Word $/\$ substr Word $/\$ Word ++ substr ++ substr ++ ++ substr ++ substr ++
151	79, 87, 88, 150	syl3anc 1326	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ ++ substr ++ substr ++ ++ substr ++ substr ++
152		ccatass 13371	. . . . . . . . . . . . . . . . . . 19 Word $/\$ substr Word $/\$ substr Word ++ substr ++ substr ++ substr ++ substr
153	74, 77, 87, 152	syl3anc 1326	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ ++ substr ++ substr ++ substr ++ substr
154	111, 108	syl6eleq 2711	. . . . . . . . . . . . . . . . . . . . . 22 $/\$ $/\$ $/\$ $/\$
155		eluzfz1 12348	. . . . . . . . . . . . . . . . . . . . . 22
156	154, 155	syl 17	. . . . . . . . . . . . . . . . . . . . 21 $/\$ $/\$ $/\$ $/\$
157	125, 108	syl6eleq 2711	. . . . . . . . . . . . . . . . . . . . . 22 $/\$ $/\$ $/\$ $/\$
158		eluzfz2 12349	. . . . . . . . . . . . . . . . . . . . . 22
159	157, 158	syl 17	. . . . . . . . . . . . . . . . . . . . 21 $/\$ $/\$ $/\$ $/\$
160		ccatswrd 13456	. . . . . . . . . . . . . . . . . . . . 21 Word $/\$ $/\$ $/\$ substr ++ substr substr
161	75, 156, 69, 159, 160	syl13anc 1328	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$ $/\$ $/\$ substr ++ substr substr
162		swrdid 13428	. . . . . . . . . . . . . . . . . . . . 21 Word substr
163	75, 162	syl 17	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$ $/\$ $/\$ substr
164	161, 163	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ substr ++ substr
165	164	oveq2d 6666	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ ++ substr ++ substr ++
166	153, 165	eqtrd 2656	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ ++ substr ++ substr ++
167	166	oveq1d 6665	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ ++ substr ++ substr ++ ++ ++
168	149, 151, 167	3eqtr2rd 2663	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ ++ ++ ++ substr ++ ++ substr ++
169		ccatlen 13360	. . . . . . . . . . . . . . . . 17 Word $/\$ substr Word ++ substr substr
170	74, 77, 169	syl2anc 693	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ ++ substr substr
171		swrd0len 13422	. . . . . . . . . . . . . . . . . 18 Word $/\$ substr
172	75, 69, 171	syl2anc 693	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ substr
173	172	oveq2d 6666	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ substr
174	170, 173	eqtr2d 2657	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ ++ substr
175		hash0 13158	. . . . . . . . . . . . . . . . 17
176	175	oveq2i 6661	. . . . . . . . . . . . . . . 16
177	107, 111	nn0addcld 11355	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$
178	177	nn0cnd 11353	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$
179	178	addid1d 10236	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
180	176, 179	syl5req 2669	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
181	79, 144, 146, 83, 168, 174, 180	splval2 13508	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ ++ ++ splice ++ substr ++ ++ substr ++
182	142, 181	eqtrd 2656	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ ++ ++ ++ substr ++ ++ substr ++
183	90, 105, 182	3eqtr4d 2666	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ ++ ++ ++ ++
184	1, 2, 3, 4	efgtf 18135	. . . . . . . . . . . . . . 15 ++ ++ ++ ++ ++ ++ ++ ++ splice $/\$ ++ ++ ++ ++
185	184	simprd 479	. . . . . . . . . . . . . 14 ++ ++ ++ ++ ++ ++
186		ffn 6045	. . . . . . . . . . . . . 14 ++ ++ ++ ++ ++ ++ ++ ++
187	72, 185, 186	3syl 18	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ ++ ++ ++ ++
188		fnovrn 6809	. . . . . . . . . . . . 13 ++ ++ ++ ++ $/\$ ++ ++ $/\$ ++ ++ ++ ++
189	187, 140, 70, 188	syl3anc 1326	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ ++ ++ ++ ++
190	183, 189	eqeltrd 2701	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ ++ ++ ++ ++
191	1, 2, 3, 4	efgi2 18138	. . . . . . . . . . 11 ++ ++ $/\$ ++ ++ ++ ++ ++ ++ ++ ++
192	72, 190, 191	syl2anc 693	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ ++ ++ ++ ++
193	1, 2, 3, 4, 9, 10, 6	efgcpbllema 18167	. . . . . . . . . 10 $/\$ $/\$ ++ ++ ++ ++
194	67, 71, 192, 193	syl3anbrc 1246	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
195		vex 3203	. . . . . . . . . . 11
196		vex 3203	. . . . . . . . . . 11
197	195, 196	elec 7786	. . . . . . . . . 10
198		breq2 4657	. . . . . . . . . 10
199	197, 198	syl5bb 272	. . . . . . . . 9
200	194, 199	syl5ibrcom 237	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
201	200	rexlimdvva 3038	. . . . . . 7 $/\$ $/\$
202	66, 201	sylbid 230	. . . . . 6 $/\$ $/\$
203	202	ssrdv 3609	. . . . 5 $/\$ $/\$
204	203	ralrimiva 2966	. . . 4 $/\$
205		fvex 6201	. . . . . . 7 Word
206	1, 205	eqeltri 2697	. . . . . 6
207		erex 7766	. . . . . 6
208	60, 206, 207	mpisyl 21	. . . . 5 $/\$
209		ereq1 7749	. . . . . . 7
210		eceq2 7784	. . . . . . . . 9
211	210	sseq2d 3633	. . . . . . . 8
212	211	ralbidv 2986	. . . . . . 7
213	209, 212	anbi12d 747	. . . . . 6 $/\$ $/\$
214	213	elabg 3351	. . . . 5 $/\$ $/\$
215	208, 214	syl 17	. . . 4 $/\$ $/\$ $/\$
216	60, 204, 215	mpbir2and 957	. . 3 $/\$ $/\$
217		intss1 4492	. . 3 $/\$ $/\$
218	216, 217	syl 17	. 2 $/\$ $/\$
219	5, 218	syl5eqss 3649	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

cab 2608

wral 2912

wrex 2913

crab 2916

cvv 3200 $\$ cdif 3571

wss 3574

c0 3915

csn 4177

cpr 4179

cop 4183

cotp 4185

cint 4475

ciun 4520 class class class wbr 4653

copab 4712

cmpt 4729

cid 5023

cxp 5112

crn 5115

wrel 5119

wfn 5883

wf 5884

cfv 5888 (class class class)co 6650

cmpt2 6652

c1o 7553

c2o 7554

wer 7739

cec 7740

cc0 9936

c1 9937

caddc 9939

cmin 10266

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-iin 4523 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-ec 7744 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-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 df-efg 18122

This theorem is referenced by: efgcpbl 18169

W3C validator