pfxccatin12 - Mathbox for Alexander van der Vekens

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Theorem pfxccatin12 41425

Description: The subword of a concatenation of two words within both of the concatenated words. Could replace swrdccatin12 13491. (Contributed by AV, 9-May-2020.)

**Hypothesis**
Ref	Expression
pfxccatin12.l

**Assertion**
Ref	Expression
pfxccatin12	Word $/\$ Word $/\$ ++ substr substr ++ prefix

Proof of Theorem pfxccatin12 Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ccatcl 13359	. . . . 5 Word $/\$ Word ++ Word
2	1	adantr 481	. . . 4 Word $/\$ Word $/\$ $/\$ ++ Word
3		elfz0fzfz0 12444	. . . . 5 $/\$
4	3	adantl 482	. . . 4 Word $/\$ Word $/\$ $/\$
5		elfzuz2 12346	. . . . . . . . 9
6	5	adantl 482	. . . . . . . 8 Word $/\$ Word $/\$
7		fzss1 12380	. . . . . . . 8
8	6, 7	syl 17	. . . . . . 7 Word $/\$ Word $/\$
9		ccatlen 13360	. . . . . . . . . 10 Word $/\$ Word ++
10		pfxccatin12.l	. . . . . . . . . . . 12
11	10	eqcomi 2631	. . . . . . . . . . 11
12	11	oveq1i 6660	. . . . . . . . . 10
13	9, 12	syl6eq 2672	. . . . . . . . 9 Word $/\$ Word ++
14	13	adantr 481	. . . . . . . 8 Word $/\$ Word $/\$ ++
15	14	oveq2d 6666	. . . . . . 7 Word $/\$ Word $/\$ ++
16	8, 15	sseqtr4d 3642	. . . . . 6 Word $/\$ Word $/\$ ++
17	16	sseld 3602	. . . . 5 Word $/\$ Word $/\$ ++
18	17	impr 649	. . . 4 Word $/\$ Word $/\$ $/\$ ++
19		swrdvalfn 13426	. . . 4 ++ Word $/\$ $/\$ ++ ++ substr ..^
20	2, 4, 18, 19	syl3anc 1326	. . 3 Word $/\$ Word $/\$ $/\$ ++ substr ..^
21		swrdcl 13419	. . . . . . 7 Word substr Word
22		pfxcl 41386	. . . . . . 7 Word prefix Word
23	21, 22	anim12i 590	. . . . . 6 Word $/\$ Word substr Word $/\$ prefix Word
24	23	adantr 481	. . . . 5 Word $/\$ Word $/\$ $/\$ substr Word $/\$ prefix Word
25		ccatvalfn 13365	. . . . 5 substr Word $/\$ prefix Word substr ++ prefix ..^ substr prefix
26	24, 25	syl 17	. . . 4 Word $/\$ Word $/\$ $/\$ substr ++ prefix ..^ substr prefix
27		simpll 790	. . . . . . . . 9 Word $/\$ Word $/\$ $/\$ Word
28		simprl 794	. . . . . . . . 9 Word $/\$ Word $/\$ $/\$
29		lencl 13324	. . . . . . . . . . 11 Word
30		elnn0uz 11725	. . . . . . . . . . . . 13
31		eluzfz2 12349	. . . . . . . . . . . . 13
32	30, 31	sylbi 207	. . . . . . . . . . . 12
33	10, 32	syl5eqel 2705	. . . . . . . . . . 11
34	29, 33	syl 17	. . . . . . . . . 10 Word
35	34	ad2antrr 762	. . . . . . . . 9 Word $/\$ Word $/\$ $/\$
36		swrdlen 13423	. . . . . . . . 9 Word $/\$ $/\$ substr
37	27, 28, 35, 36	syl3anc 1326	. . . . . . . 8 Word $/\$ Word $/\$ $/\$ substr
38		simplr 792	. . . . . . . . 9 Word $/\$ Word $/\$ $/\$ Word
39		lencl 13324	. . . . . . . . . . . . 13 Word
40	39	nn0zd 11480	. . . . . . . . . . . 12 Word
41	40	adantl 482	. . . . . . . . . . 11 Word $/\$ Word
42		simpr 477	. . . . . . . . . . 11 $/\$
43	41, 42	anim12i 590	. . . . . . . . . 10 Word $/\$ Word $/\$ $/\$ $/\$
44		elfzmlbp 12450	. . . . . . . . . 10 $/\$
45	43, 44	syl 17	. . . . . . . . 9 Word $/\$ Word $/\$ $/\$
46		pfxlen 41391	. . . . . . . . 9 Word $/\$ prefix
47	38, 45, 46	syl2anc 693	. . . . . . . 8 Word $/\$ Word $/\$ $/\$ prefix
48	37, 47	oveq12d 6668	. . . . . . 7 Word $/\$ Word $/\$ $/\$ substr prefix
49		elfz2nn0 12431	. . . . . . . . . . 11 $/\$ $/\$
50		nn0cn 11302	. . . . . . . . . . . . . . . . 17
51	50	adantl 482	. . . . . . . . . . . . . . . 16 $/\$
52	51	adantl 482	. . . . . . . . . . . . . . 15 $/\$ $/\$
53		nn0cn 11302	. . . . . . . . . . . . . . . 16
54	53	ad2antrl 764	. . . . . . . . . . . . . . 15 $/\$ $/\$
55		zcn 11382	. . . . . . . . . . . . . . . 16
56	55	adantr 481	. . . . . . . . . . . . . . 15 $/\$ $/\$
57	52, 54, 56	3jca 1242	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
58	57	ex 450	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
59		elfzelz 12342	. . . . . . . . . . . . 13
60	58, 59	syl11 33	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
61	60	3adant3 1081	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
62	49, 61	sylbi 207	. . . . . . . . . 10 $/\$ $/\$
63	62	imp 445	. . . . . . . . 9 $/\$ $/\$ $/\$
64	63	adantl 482	. . . . . . . 8 Word $/\$ Word $/\$ $/\$ $/\$ $/\$
65		npncan3 10319	. . . . . . . 8 $/\$ $/\$
66	64, 65	syl 17	. . . . . . 7 Word $/\$ Word $/\$ $/\$
67	48, 66	eqtr2d 2657	. . . . . 6 Word $/\$ Word $/\$ $/\$ substr prefix
68	67	oveq2d 6666	. . . . 5 Word $/\$ Word $/\$ $/\$ ..^ ..^ substr prefix
69	68	fneq2d 5982	. . . 4 Word $/\$ Word $/\$ $/\$ substr ++ prefix ..^ substr ++ prefix ..^ substr prefix
70	26, 69	mpbird 247	. . 3 Word $/\$ Word $/\$ $/\$ substr ++ prefix ..^
71		simprl 794	. . . . . 6 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ Word $/\$ Word $/\$ $/\$
72		simpr 477	. . . . . . . 8 Word $/\$ Word $/\$ $/\$ $/\$ ..^ ..^
73	72	anim2i 593	. . . . . . 7 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ..^ $/\$ ..^
74	73	ancomd 467	. . . . . 6 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ..^ $/\$ ..^
75	10	swrdccatin12lem3 13490	. . . . . 6 Word $/\$ Word $/\$ $/\$ ..^ $/\$ ..^ ++ substr substr
76	71, 74, 75	sylc 65	. . . . 5 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ++ substr substr
77	24	ad2antrl 764	. . . . . . 7 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ substr Word $/\$ prefix Word
78		simpl 473	. . . . . . . 8 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ..^
79		nn0fz0 12437	. . . . . . . . . . . . . . 15
80	29, 79	sylib 208	. . . . . . . . . . . . . 14 Word
81	10, 80	syl5eqel 2705	. . . . . . . . . . . . 13 Word
82	81	ad2antrr 762	. . . . . . . . . . . 12 Word $/\$ Word $/\$ $/\$
83	27, 28, 82	3jca 1242	. . . . . . . . . . 11 Word $/\$ Word $/\$ $/\$ Word $/\$ $/\$
84	83	ad2antrl 764	. . . . . . . . . 10 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ Word $/\$ $/\$
85	84, 36	syl 17	. . . . . . . . 9 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ substr
86	85	oveq2d 6666	. . . . . . . 8 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ..^ substr ..^
87	78, 86	eleqtrrd 2704	. . . . . . 7 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ..^ substr
88		df-3an 1039	. . . . . . 7 substr Word $/\$ prefix Word $/\$ ..^ substr substr Word $/\$ prefix Word $/\$ ..^ substr
89	77, 87, 88	sylanbrc 698	. . . . . 6 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ substr Word $/\$ prefix Word $/\$ ..^ substr
90		ccatval1 13361	. . . . . 6 substr Word $/\$ prefix Word $/\$ ..^ substr substr ++ prefix substr
91	89, 90	syl 17	. . . . 5 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ substr ++ prefix substr
92	76, 91	eqtr4d 2659	. . . 4 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ++ substr substr ++ prefix
93		simprl 794	. . . . . 6 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ Word $/\$ Word $/\$ $/\$
94	72	anim2i 593	. . . . . . 7 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ..^ $/\$ ..^
95	94	ancomd 467	. . . . . 6 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ..^ $/\$ ..^
96	10	pfxccatin12lem2 41424	. . . . . 6 Word $/\$ Word $/\$ $/\$ ..^ $/\$ ..^ ++ substr prefix substr
97	93, 95, 96	sylc 65	. . . . 5 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ++ substr prefix substr
98	24	ad2antrl 764	. . . . . . 7 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ substr Word $/\$ prefix Word
99		elfzuz 12338	. . . . . . . . . . . . 13
100		eluzelz 11697	. . . . . . . . . . . . . 14
101		simpll 790	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
102		simplr 792	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
103		simpr 477	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
104	101, 102, 103	3jca 1242	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$
105	104	ex 450	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
106	105	ancoms 469	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$
107	106	3adant3 1081	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
108	49, 107	sylbi 207	. . . . . . . . . . . . . 14 $/\$ $/\$
109	100, 108	syl5com 31	. . . . . . . . . . . . 13 $/\$ $/\$
110	99, 109	syl 17	. . . . . . . . . . . 12 $/\$ $/\$
111	110	impcom 446	. . . . . . . . . . 11 $/\$ $/\$ $/\$
112	111	adantl 482	. . . . . . . . . 10 Word $/\$ Word $/\$ $/\$ $/\$ $/\$
113	112	ad2antrl 764	. . . . . . . . 9 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ $/\$ $/\$
114		swrdccatin12lem1 13484	. . . . . . . . 9 $/\$ $/\$ ..^ $/\$ ..^ ..^
115	113, 95, 114	sylc 65	. . . . . . . 8 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ..^
116	27, 28, 82, 36	syl3anc 1326	. . . . . . . . . 10 Word $/\$ Word $/\$ $/\$ substr
117		simpr 477	. . . . . . . . . . . . . . . . 17 Word $/\$ Word Word
118	117	adantl 482	. . . . . . . . . . . . . . . 16 $/\$ Word $/\$ Word Word
119	41	adantl 482	. . . . . . . . . . . . . . . . 17 $/\$ Word $/\$ Word
120		simpl 473	. . . . . . . . . . . . . . . . 17 $/\$ Word $/\$ Word
121	119, 120, 44	syl2anc 693	. . . . . . . . . . . . . . . 16 $/\$ Word $/\$ Word
122	118, 121	jca 554	. . . . . . . . . . . . . . 15 $/\$ Word $/\$ Word Word $/\$
123	122	ex 450	. . . . . . . . . . . . . 14 Word $/\$ Word Word $/\$
124	123	adantl 482	. . . . . . . . . . . . 13 $/\$ Word $/\$ Word Word $/\$
125	124	impcom 446	. . . . . . . . . . . 12 Word $/\$ Word $/\$ $/\$ Word $/\$
126	125, 46	syl 17	. . . . . . . . . . 11 Word $/\$ Word $/\$ $/\$ prefix
127	116, 126	oveq12d 6668	. . . . . . . . . 10 Word $/\$ Word $/\$ $/\$ substr prefix
128	116, 127	oveq12d 6668	. . . . . . . . 9 Word $/\$ Word $/\$ $/\$ substr ..^ substr prefix ..^
129	128	ad2antrl 764	. . . . . . . 8 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ substr ..^ substr prefix ..^
130	115, 129	eleqtrrd 2704	. . . . . . 7 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ substr ..^ substr prefix
131		df-3an 1039	. . . . . . 7 substr Word $/\$ prefix Word $/\$ substr ..^ substr prefix substr Word $/\$ prefix Word $/\$ substr ..^ substr prefix
132	98, 130, 131	sylanbrc 698	. . . . . 6 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ substr Word $/\$ prefix Word $/\$ substr ..^ substr prefix
133		ccatval2 13362	. . . . . 6 substr Word $/\$ prefix Word $/\$ substr ..^ substr prefix substr ++ prefix prefix substr
134	132, 133	syl 17	. . . . 5 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ substr ++ prefix prefix substr
135	97, 134	eqtr4d 2659	. . . 4 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ++ substr substr ++ prefix
136	92, 135	pm2.61ian 831	. . 3 Word $/\$ Word $/\$ $/\$ $/\$ ..^ ++ substr substr ++ prefix
137	20, 70, 136	eqfnfvd 6314	. 2 Word $/\$ Word $/\$ $/\$ ++ substr substr ++ prefix
138	137	ex 450	1 Word $/\$ Word $/\$ ++ substr substr ++ prefix

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wss 3574

cop 4183 class class class wbr 4653

wfn 5883

cfv 5888 (class class class)co 6650

cc 9934

cc0 9936

caddc 9939

cle 10075

cmin 10266

cn0 11292

cz 11377

cuz 11687

cfz 12326 ..^cfzo 12465

chash 13117 Word cword 13291 ++ cconcat 13293 substr csubstr 13295 prefix cpfx 41381

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-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-oadd 7564 df-er 7742 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-substr 13303 df-pfx 41382

This theorem is referenced by: pfxccat3 41426 pfxccatpfx2 41428 pfxccatin12d 41432

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