ccatswrd - Metamath Proof Explorer

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Theorem ccatswrd 13456

Description: Joining two adjacent subwords makes a longer subword. (Contributed by Stefan O'Rear, 20-Aug-2015.)

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

Proof of Theorem ccatswrd Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		swrdcl 13419	. . . . . 6 Word substr Word
2	1	adantr 481	. . . . 5 Word $/\$ $/\$ $/\$ substr Word
3		swrdcl 13419	. . . . . 6 Word substr Word
4	3	adantr 481	. . . . 5 Word $/\$ $/\$ $/\$ substr Word
5		ccatcl 13359	. . . . 5 substr Word $/\$ substr Word substr ++ substr Word
6	2, 4, 5	syl2anc 693	. . . 4 Word $/\$ $/\$ $/\$ substr ++ substr Word
7		wrdf 13310	. . . 4 substr ++ substr Word substr ++ substr ..^ substr ++ substr
8		ffn 6045	. . . 4 substr ++ substr ..^ substr ++ substr substr ++ substr ..^ substr ++ substr
9	6, 7, 8	3syl 18	. . 3 Word $/\$ $/\$ $/\$ substr ++ substr ..^ substr ++ substr
10		ccatlen 13360	. . . . . . 7 substr Word $/\$ substr Word substr ++ substr substr substr
11	2, 4, 10	syl2anc 693	. . . . . 6 Word $/\$ $/\$ $/\$ substr ++ substr substr substr
12		simpl 473	. . . . . . . . 9 Word $/\$ $/\$ $/\$ Word
13		simpr1 1067	. . . . . . . . 9 Word $/\$ $/\$ $/\$
14		simpr2 1068	. . . . . . . . . 10 Word $/\$ $/\$ $/\$
15		simpr3 1069	. . . . . . . . . 10 Word $/\$ $/\$ $/\$
16		fzass4 12379	. . . . . . . . . . . 12 $/\$ $/\$
17	16	biimpri 218	. . . . . . . . . . 11 $/\$ $/\$
18	17	simpld 475	. . . . . . . . . 10 $/\$
19	14, 15, 18	syl2anc 693	. . . . . . . . 9 Word $/\$ $/\$ $/\$
20		swrdlen 13423	. . . . . . . . 9 Word $/\$ $/\$ substr
21	12, 13, 19, 20	syl3anc 1326	. . . . . . . 8 Word $/\$ $/\$ $/\$ substr
22		swrdlen 13423	. . . . . . . . 9 Word $/\$ $/\$ substr
23	12, 14, 15, 22	syl3anc 1326	. . . . . . . 8 Word $/\$ $/\$ $/\$ substr
24	21, 23	oveq12d 6668	. . . . . . 7 Word $/\$ $/\$ $/\$ substr substr
25		elfzelz 12342	. . . . . . . . . 10
26	14, 25	syl 17	. . . . . . . . 9 Word $/\$ $/\$ $/\$
27	26	zcnd 11483	. . . . . . . 8 Word $/\$ $/\$ $/\$
28		elfzelz 12342	. . . . . . . . . 10
29	13, 28	syl 17	. . . . . . . . 9 Word $/\$ $/\$ $/\$
30	29	zcnd 11483	. . . . . . . 8 Word $/\$ $/\$ $/\$
31		elfzelz 12342	. . . . . . . . . 10
32	15, 31	syl 17	. . . . . . . . 9 Word $/\$ $/\$ $/\$
33	32	zcnd 11483	. . . . . . . 8 Word $/\$ $/\$ $/\$
34	27, 30, 33	npncan3d 10428	. . . . . . 7 Word $/\$ $/\$ $/\$
35	24, 34	eqtrd 2656	. . . . . 6 Word $/\$ $/\$ $/\$ substr substr
36	11, 35	eqtrd 2656	. . . . 5 Word $/\$ $/\$ $/\$ substr ++ substr
37	36	oveq2d 6666	. . . 4 Word $/\$ $/\$ $/\$ ..^ substr ++ substr ..^
38	37	fneq2d 5982	. . 3 Word $/\$ $/\$ $/\$ substr ++ substr ..^ substr ++ substr substr ++ substr ..^
39	9, 38	mpbid 222	. 2 Word $/\$ $/\$ $/\$ substr ++ substr ..^
40		swrdcl 13419	. . . . 5 Word substr Word
41	40	adantr 481	. . . 4 Word $/\$ $/\$ $/\$ substr Word
42		wrdf 13310	. . . 4 substr Word substr ..^ substr
43		ffn 6045	. . . 4 substr ..^ substr substr ..^ substr
44	41, 42, 43	3syl 18	. . 3 Word $/\$ $/\$ $/\$ substr ..^ substr
45		fzass4 12379	. . . . . . . . 9 $/\$ $/\$
46	45	biimpri 218	. . . . . . . 8 $/\$ $/\$
47	46	simpld 475	. . . . . . 7 $/\$
48	13, 14, 47	syl2anc 693	. . . . . 6 Word $/\$ $/\$ $/\$
49		swrdlen 13423	. . . . . 6 Word $/\$ $/\$ substr
50	12, 48, 15, 49	syl3anc 1326	. . . . 5 Word $/\$ $/\$ $/\$ substr
51	50	oveq2d 6666	. . . 4 Word $/\$ $/\$ $/\$ ..^ substr ..^
52	51	fneq2d 5982	. . 3 Word $/\$ $/\$ $/\$ substr ..^ substr substr ..^
53	44, 52	mpbid 222	. 2 Word $/\$ $/\$ $/\$ substr ..^
54		simpr 477	. . . . 5 Word $/\$ $/\$ $/\$ $/\$ ..^ ..^
55	26, 29	zsubcld 11487	. . . . . 6 Word $/\$ $/\$ $/\$
56	55	adantr 481	. . . . 5 Word $/\$ $/\$ $/\$ $/\$ ..^
57		fzospliti 12500	. . . . 5 ..^ $/\$ ..^ $\/$ ..^
58	54, 56, 57	syl2anc 693	. . . 4 Word $/\$ $/\$ $/\$ $/\$ ..^ ..^ $\/$ ..^
59	2	adantr 481	. . . . . . 7 Word $/\$ $/\$ $/\$ $/\$ ..^ substr Word
60	4	adantr 481	. . . . . . 7 Word $/\$ $/\$ $/\$ $/\$ ..^ substr Word
61	21	oveq2d 6666	. . . . . . . . 9 Word $/\$ $/\$ $/\$ ..^ substr ..^
62	61	eleq2d 2687	. . . . . . . 8 Word $/\$ $/\$ $/\$ ..^ substr ..^
63	62	biimpar 502	. . . . . . 7 Word $/\$ $/\$ $/\$ $/\$ ..^ ..^ substr
64		ccatval1 13361	. . . . . . 7 substr Word $/\$ substr Word $/\$ ..^ substr substr ++ substr substr
65	59, 60, 63, 64	syl3anc 1326	. . . . . 6 Word $/\$ $/\$ $/\$ $/\$ ..^ substr ++ substr substr
66		simpll 790	. . . . . . 7 Word $/\$ $/\$ $/\$ $/\$ ..^ Word
67		simplr1 1103	. . . . . . 7 Word $/\$ $/\$ $/\$ $/\$ ..^
68	19	adantr 481	. . . . . . 7 Word $/\$ $/\$ $/\$ $/\$ ..^
69		simpr 477	. . . . . . 7 Word $/\$ $/\$ $/\$ $/\$ ..^ ..^
70		swrdfv 13424	. . . . . . 7 Word $/\$ $/\$ $/\$ ..^ substr
71	66, 67, 68, 69, 70	syl31anc 1329	. . . . . 6 Word $/\$ $/\$ $/\$ $/\$ ..^ substr
72	65, 71	eqtrd 2656	. . . . 5 Word $/\$ $/\$ $/\$ $/\$ ..^ substr ++ substr
73	2	adantr 481	. . . . . . 7 Word $/\$ $/\$ $/\$ $/\$ ..^ substr Word
74	4	adantr 481	. . . . . . 7 Word $/\$ $/\$ $/\$ $/\$ ..^ substr Word
75	21, 35	oveq12d 6668	. . . . . . . . 9 Word $/\$ $/\$ $/\$ substr ..^ substr substr ..^
76	75	eleq2d 2687	. . . . . . . 8 Word $/\$ $/\$ $/\$ substr ..^ substr substr ..^
77	76	biimpar 502	. . . . . . 7 Word $/\$ $/\$ $/\$ $/\$ ..^ substr ..^ substr substr
78		ccatval2 13362	. . . . . . 7 substr Word $/\$ substr Word $/\$ substr ..^ substr substr substr ++ substr substr substr
79	73, 74, 77, 78	syl3anc 1326	. . . . . 6 Word $/\$ $/\$ $/\$ $/\$ ..^ substr ++ substr substr substr
80		simpll 790	. . . . . . 7 Word $/\$ $/\$ $/\$ $/\$ ..^ Word
81		simplr2 1104	. . . . . . 7 Word $/\$ $/\$ $/\$ $/\$ ..^
82		simplr3 1105	. . . . . . 7 Word $/\$ $/\$ $/\$ $/\$ ..^
83	21	oveq2d 6666	. . . . . . . . 9 Word $/\$ $/\$ $/\$ substr
84	83	adantr 481	. . . . . . . 8 Word $/\$ $/\$ $/\$ $/\$ ..^ substr
85	34	oveq2d 6666	. . . . . . . . . . 11 Word $/\$ $/\$ $/\$ ..^ ..^
86	85	eleq2d 2687	. . . . . . . . . 10 Word $/\$ $/\$ $/\$ ..^ ..^
87	86	biimpar 502	. . . . . . . . 9 Word $/\$ $/\$ $/\$ $/\$ ..^ ..^
88	32, 26	zsubcld 11487	. . . . . . . . . 10 Word $/\$ $/\$ $/\$
89	88	adantr 481	. . . . . . . . 9 Word $/\$ $/\$ $/\$ $/\$ ..^
90		fzosubel3 12528	. . . . . . . . 9 ..^ $/\$ ..^
91	87, 89, 90	syl2anc 693	. . . . . . . 8 Word $/\$ $/\$ $/\$ $/\$ ..^ ..^
92	84, 91	eqeltrd 2701	. . . . . . 7 Word $/\$ $/\$ $/\$ $/\$ ..^ substr ..^
93		swrdfv 13424	. . . . . . 7 Word $/\$ $/\$ $/\$ substr ..^ substr substr substr
94	80, 81, 82, 92, 93	syl31anc 1329	. . . . . 6 Word $/\$ $/\$ $/\$ $/\$ ..^ substr substr substr
95	83	oveq1d 6665	. . . . . . . . 9 Word $/\$ $/\$ $/\$ substr
96	95	adantr 481	. . . . . . . 8 Word $/\$ $/\$ $/\$ $/\$ ..^ substr
97		elfzoelz 12470	. . . . . . . . . . 11 ..^
98	97	zcnd 11483	. . . . . . . . . 10 ..^
99	98	adantl 482	. . . . . . . . 9 Word $/\$ $/\$ $/\$ $/\$ ..^
100	27, 30	subcld 10392	. . . . . . . . . 10 Word $/\$ $/\$ $/\$
101	100	adantr 481	. . . . . . . . 9 Word $/\$ $/\$ $/\$ $/\$ ..^
102	27	adantr 481	. . . . . . . . 9 Word $/\$ $/\$ $/\$ $/\$ ..^
103	99, 101, 102	subadd23d 10414	. . . . . . . 8 Word $/\$ $/\$ $/\$ $/\$ ..^
104	27, 30	nncand 10397	. . . . . . . . . 10 Word $/\$ $/\$ $/\$
105	104	oveq2d 6666	. . . . . . . . 9 Word $/\$ $/\$ $/\$
106	105	adantr 481	. . . . . . . 8 Word $/\$ $/\$ $/\$ $/\$ ..^
107	96, 103, 106	3eqtrd 2660	. . . . . . 7 Word $/\$ $/\$ $/\$ $/\$ ..^ substr
108	107	fveq2d 6195	. . . . . 6 Word $/\$ $/\$ $/\$ $/\$ ..^ substr
109	79, 94, 108	3eqtrd 2660	. . . . 5 Word $/\$ $/\$ $/\$ $/\$ ..^ substr ++ substr
110	72, 109	jaodan 826	. . . 4 Word $/\$ $/\$ $/\$ $/\$ ..^ $\/$ ..^ substr ++ substr
111	58, 110	syldan 487	. . 3 Word $/\$ $/\$ $/\$ $/\$ ..^ substr ++ substr
112		simpll 790	. . . 4 Word $/\$ $/\$ $/\$ $/\$ ..^ Word
113	48	adantr 481	. . . 4 Word $/\$ $/\$ $/\$ $/\$ ..^
114		simplr3 1105	. . . 4 Word $/\$ $/\$ $/\$ $/\$ ..^
115		swrdfv 13424	. . . 4 Word $/\$ $/\$ $/\$ ..^ substr
116	112, 113, 114, 54, 115	syl31anc 1329	. . 3 Word $/\$ $/\$ $/\$ $/\$ ..^ substr
117	111, 116	eqtr4d 2659	. 2 Word $/\$ $/\$ $/\$ $/\$ ..^ substr ++ substr substr
118	39, 53, 117	eqfnfvd 6314	1 Word $/\$ $/\$ $/\$ substr ++ substr substr

Colors of variables: wff setvar class

Syntax hints:

wi 4 $\/$ wo 383 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

cop 4183

wfn 5883

wf 5884

cfv 5888 (class class class)co 6650

cc 9934

cc0 9936

caddc 9939

cmin 10266

cz 11377

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: wrdcctswrd 13465 swrdccatwrd 13468 wrdeqs1cat 13474 splid 13504 splval2 13508 swrds2 13685 efgredleme 18156 efgredlemc 18158 efgcpbllemb 18168 frgpuplem 18185 wrdsplex 30618

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