swrdccatin12 - Metamath Proof Explorer

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Theorem swrdccatin12 13491

Description: The subword of a concatenation of two words within both of the concatenated words. (Contributed by Alexander van der Vekens, 5-Apr-2018.) (Revised by Alexander van der Vekens, 27-May-2018.)

**Hypothesis**
Ref	Expression
swrdccatin12.l

**Assertion**
Ref	Expression
swrdccatin12	Word $/\$ Word $/\$ ++ substr substr ++ substr

Proof of Theorem swrdccatin12 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		swrdccatin12.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		swrdcl 13419	. . . . . . 7 Word substr Word
23	21, 22	anim12i 590	. . . . . 6 Word $/\$ Word substr Word $/\$ substr Word
24	23	adantr 481	. . . . 5 Word $/\$ Word $/\$ $/\$ substr Word $/\$ substr Word
25		ccatvalfn 13365	. . . . 5 substr Word $/\$ substr Word substr ++ substr ..^ substr substr
26	24, 25	syl 17	. . . 4 Word $/\$ Word $/\$ $/\$ substr ++ substr ..^ substr substr
27		simpll 790	. . . . . . . . 9 Word $/\$ Word $/\$ $/\$ Word
28		simprl 794	. . . . . . . . 9 Word $/\$ Word $/\$ $/\$
29		lencl 13324	. . . . . . . . . . . 12 Word
30		elnn0uz 11725	. . . . . . . . . . . . . 14
31		eluzfz2 12349	. . . . . . . . . . . . . 14
32	30, 31	sylbi 207	. . . . . . . . . . . . 13
33	10, 32	syl5eqel 2705	. . . . . . . . . . . 12
34	29, 33	syl 17	. . . . . . . . . . 11 Word
35	34	adantr 481	. . . . . . . . . 10 Word $/\$ Word
36	35	adantr 481	. . . . . . . . 9 Word $/\$ Word $/\$ $/\$
37		swrdlen 13423	. . . . . . . . 9 Word $/\$ $/\$ substr
38	27, 28, 36, 37	syl3anc 1326	. . . . . . . 8 Word $/\$ Word $/\$ $/\$ substr
39		simpr 477	. . . . . . . . . 10 Word $/\$ Word Word
40	39	adantr 481	. . . . . . . . 9 Word $/\$ Word $/\$ $/\$ Word
41		lencl 13324	. . . . . . . . . . . . 13 Word
42	41	nn0zd 11480	. . . . . . . . . . . 12 Word
43	42	adantl 482	. . . . . . . . . . 11 Word $/\$ Word
44		simpr 477	. . . . . . . . . . 11 $/\$
45	43, 44	anim12i 590	. . . . . . . . . 10 Word $/\$ Word $/\$ $/\$ $/\$
46		elfzmlbp 12450	. . . . . . . . . 10 $/\$
47	45, 46	syl 17	. . . . . . . . 9 Word $/\$ Word $/\$ $/\$
48		swrd0len 13422	. . . . . . . . 9 Word $/\$ substr
49	40, 47, 48	syl2anc 693	. . . . . . . 8 Word $/\$ Word $/\$ $/\$ substr
50	38, 49	oveq12d 6668	. . . . . . 7 Word $/\$ Word $/\$ $/\$ substr substr
51		elfz2nn0 12431	. . . . . . . . . . 11 $/\$ $/\$
52		nn0cn 11302	. . . . . . . . . . . . . . . . 17
53	52	adantl 482	. . . . . . . . . . . . . . . 16 $/\$
54	53	adantl 482	. . . . . . . . . . . . . . 15 $/\$ $/\$
55		nn0cn 11302	. . . . . . . . . . . . . . . 16
56	55	ad2antrl 764	. . . . . . . . . . . . . . 15 $/\$ $/\$
57		zcn 11382	. . . . . . . . . . . . . . . 16
58	57	adantr 481	. . . . . . . . . . . . . . 15 $/\$ $/\$
59	54, 56, 58	3jca 1242	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
60	59	ex 450	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
61		elfzelz 12342	. . . . . . . . . . . . 13
62	60, 61	syl11 33	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
63	62	3adant3 1081	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
64	51, 63	sylbi 207	. . . . . . . . . 10 $/\$ $/\$
65	64	imp 445	. . . . . . . . 9 $/\$ $/\$ $/\$
66	65	adantl 482	. . . . . . . 8 Word $/\$ Word $/\$ $/\$ $/\$ $/\$
67		npncan3 10319	. . . . . . . 8 $/\$ $/\$
68	66, 67	syl 17	. . . . . . 7 Word $/\$ Word $/\$ $/\$
69	50, 68	eqtr2d 2657	. . . . . 6 Word $/\$ Word $/\$ $/\$ substr substr
70	69	oveq2d 6666	. . . . 5 Word $/\$ Word $/\$ $/\$ ..^ ..^ substr substr
71	70	fneq2d 5982	. . . 4 Word $/\$ Word $/\$ $/\$ substr ++ substr ..^ substr ++ substr ..^ substr substr
72	26, 71	mpbird 247	. . 3 Word $/\$ Word $/\$ $/\$ substr ++ substr ..^
73		simprl 794	. . . . . 6 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ Word $/\$ Word $/\$ $/\$
74		simpr 477	. . . . . . . 8 Word $/\$ Word $/\$ $/\$ $/\$ ..^ ..^
75	74	anim2i 593	. . . . . . 7 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ..^ $/\$ ..^
76	75	ancomd 467	. . . . . 6 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ..^ $/\$ ..^
77	10	swrdccatin12lem3 13490	. . . . . 6 Word $/\$ Word $/\$ $/\$ ..^ $/\$ ..^ ++ substr substr
78	73, 76, 77	sylc 65	. . . . 5 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ++ substr substr
79	24	ad2antrl 764	. . . . . . 7 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ substr Word $/\$ substr Word
80		simpl 473	. . . . . . . 8 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ..^
81		nn0fz0 12437	. . . . . . . . . . . . . . . 16
82	29, 81	sylib 208	. . . . . . . . . . . . . . 15 Word
83	10, 82	syl5eqel 2705	. . . . . . . . . . . . . 14 Word
84	83	adantr 481	. . . . . . . . . . . . 13 Word $/\$ Word
85	84	adantr 481	. . . . . . . . . . . 12 Word $/\$ Word $/\$ $/\$
86	27, 28, 85	3jca 1242	. . . . . . . . . . 11 Word $/\$ Word $/\$ $/\$ Word $/\$ $/\$
87	86	ad2antrl 764	. . . . . . . . . 10 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ Word $/\$ $/\$
88	87, 37	syl 17	. . . . . . . . 9 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ substr
89	88	oveq2d 6666	. . . . . . . 8 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ..^ substr ..^
90	80, 89	eleqtrrd 2704	. . . . . . 7 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ..^ substr
91		df-3an 1039	. . . . . . 7 substr Word $/\$ substr Word $/\$ ..^ substr substr Word $/\$ substr Word $/\$ ..^ substr
92	79, 90, 91	sylanbrc 698	. . . . . 6 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ substr Word $/\$ substr Word $/\$ ..^ substr
93		ccatval1 13361	. . . . . 6 substr Word $/\$ substr Word $/\$ ..^ substr substr ++ substr substr
94	92, 93	syl 17	. . . . 5 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ substr ++ substr substr
95	78, 94	eqtr4d 2659	. . . 4 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ++ substr substr ++ substr
96		simprl 794	. . . . . 6 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ Word $/\$ Word $/\$ $/\$
97	74	anim2i 593	. . . . . . 7 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ..^ $/\$ ..^
98	97	ancomd 467	. . . . . 6 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ..^ $/\$ ..^
99	10	swrdccatin12lem2 13489	. . . . . 6 Word $/\$ Word $/\$ $/\$ ..^ $/\$ ..^ ++ substr substr substr
100	96, 98, 99	sylc 65	. . . . 5 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ++ substr substr substr
101	24	ad2antrl 764	. . . . . . 7 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ substr Word $/\$ substr Word
102		elfzuz 12338	. . . . . . . . . . . . 13
103		eluzelz 11697	. . . . . . . . . . . . . 14
104		simpll 790	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
105		simpr 477	. . . . . . . . . . . . . . . . . . . 20 $/\$
106	105	adantr 481	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
107		simpr 477	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
108	104, 106, 107	3jca 1242	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$
109	108	ex 450	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$
110	109	ancoms 469	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$
111	110	3adant3 1081	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
112	51, 111	sylbi 207	. . . . . . . . . . . . . 14 $/\$ $/\$
113	103, 112	syl5com 31	. . . . . . . . . . . . 13 $/\$ $/\$
114	102, 113	syl 17	. . . . . . . . . . . 12 $/\$ $/\$
115	114	impcom 446	. . . . . . . . . . 11 $/\$ $/\$ $/\$
116	115	adantl 482	. . . . . . . . . 10 Word $/\$ Word $/\$ $/\$ $/\$ $/\$
117	116	ad2antrl 764	. . . . . . . . 9 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ $/\$ $/\$
118		swrdccatin12lem1 13484	. . . . . . . . 9 $/\$ $/\$ ..^ $/\$ ..^ ..^
119	117, 98, 118	sylc 65	. . . . . . . 8 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ..^
120	27, 28, 85, 37	syl3anc 1326	. . . . . . . . . 10 Word $/\$ Word $/\$ $/\$ substr
121	39	adantl 482	. . . . . . . . . . . . . . . 16 $/\$ Word $/\$ Word Word
122	43	adantl 482	. . . . . . . . . . . . . . . . 17 $/\$ Word $/\$ Word
123		simpl 473	. . . . . . . . . . . . . . . . 17 $/\$ Word $/\$ Word
124	122, 123, 46	syl2anc 693	. . . . . . . . . . . . . . . 16 $/\$ Word $/\$ Word
125	121, 124	jca 554	. . . . . . . . . . . . . . 15 $/\$ Word $/\$ Word Word $/\$
126	125	ex 450	. . . . . . . . . . . . . 14 Word $/\$ Word Word $/\$
127	126	adantl 482	. . . . . . . . . . . . 13 $/\$ Word $/\$ Word Word $/\$
128	127	impcom 446	. . . . . . . . . . . 12 Word $/\$ Word $/\$ $/\$ Word $/\$
129	128, 48	syl 17	. . . . . . . . . . 11 Word $/\$ Word $/\$ $/\$ substr
130	120, 129	oveq12d 6668	. . . . . . . . . 10 Word $/\$ Word $/\$ $/\$ substr substr
131	120, 130	oveq12d 6668	. . . . . . . . 9 Word $/\$ Word $/\$ $/\$ substr ..^ substr substr ..^
132	131	ad2antrl 764	. . . . . . . 8 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ substr ..^ substr substr ..^
133	119, 132	eleqtrrd 2704	. . . . . . 7 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ substr ..^ substr substr
134		df-3an 1039	. . . . . . 7 substr Word $/\$ substr Word $/\$ substr ..^ substr substr substr Word $/\$ substr Word $/\$ substr ..^ substr substr
135	101, 133, 134	sylanbrc 698	. . . . . 6 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ substr Word $/\$ substr Word $/\$ substr ..^ substr substr
136		ccatval2 13362	. . . . . 6 substr Word $/\$ substr Word $/\$ substr ..^ substr substr substr ++ substr substr substr
137	135, 136	syl 17	. . . . 5 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ substr ++ substr substr substr
138	100, 137	eqtr4d 2659	. . . 4 ..^ $/\$ Word $/\$ Word $/\$ $/\$ $/\$ ..^ ++ substr substr ++ substr
139	95, 138	pm2.61ian 831	. . 3 Word $/\$ Word $/\$ $/\$ $/\$ ..^ ++ substr substr ++ substr
140	20, 72, 139	eqfnfvd 6314	. 2 Word $/\$ Word $/\$ $/\$ ++ substr substr ++ substr
141	140	ex 450	1 Word $/\$ Word $/\$ ++ substr substr ++ substr

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

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

This theorem is referenced by: swrdccat3 13492 swrdccatin12d 13501

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