ccatpfx - Mathbox for Alexander van der Vekens

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Theorem ccatpfx 41409

Description: Joining a prefix with an adjacent subword makes a longer prefix. (Contributed by AV, 7-May-2020.)

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

Proof of Theorem ccatpfx Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pfxcl 41386	. . . . . 6 Word prefix Word
2	1	3ad2ant1 1082	. . . . 5 Word $/\$ $/\$ prefix Word
3		swrdcl 13419	. . . . . 6 Word substr Word
4	3	3ad2ant1 1082	. . . . 5 Word $/\$ $/\$ substr Word
5		ccatcl 13359	. . . . 5 prefix Word $/\$ substr Word prefix ++ substr Word
6	2, 4, 5	syl2anc 693	. . . 4 Word $/\$ $/\$ prefix ++ substr Word
7		wrdf 13310	. . . 4 prefix ++ substr Word prefix ++ substr ..^ prefix ++ substr
8		ffn 6045	. . . 4 prefix ++ substr ..^ prefix ++ substr prefix ++ substr ..^ prefix ++ substr
9	6, 7, 8	3syl 18	. . 3 Word $/\$ $/\$ prefix ++ substr ..^ prefix ++ substr
10		ccatlen 13360	. . . . . . 7 prefix Word $/\$ substr Word prefix ++ substr prefix substr
11	2, 4, 10	syl2anc 693	. . . . . 6 Word $/\$ $/\$ prefix ++ substr prefix substr
12		simp1 1061	. . . . . . . . 9 Word $/\$ $/\$ Word
13		fzass4 12379	. . . . . . . . . . . 12 $/\$ $/\$
14	13	biimpri 218	. . . . . . . . . . 11 $/\$ $/\$
15	14	simpld 475	. . . . . . . . . 10 $/\$
16	15	3adant1 1079	. . . . . . . . 9 Word $/\$ $/\$
17		pfxlen 41391	. . . . . . . . 9 Word $/\$ prefix
18	12, 16, 17	syl2anc 693	. . . . . . . 8 Word $/\$ $/\$ prefix
19		swrdlen 13423	. . . . . . . 8 Word $/\$ $/\$ substr
20	18, 19	oveq12d 6668	. . . . . . 7 Word $/\$ $/\$ prefix substr
21		elfzelz 12342	. . . . . . . . . . 11
22	21	ad2antrl 764	. . . . . . . . . 10 Word $/\$ $/\$
23	22	zcnd 11483	. . . . . . . . 9 Word $/\$ $/\$
24	23	3impb 1260	. . . . . . . 8 Word $/\$ $/\$
25		elfzelz 12342	. . . . . . . . . . 11
26	25	ad2antll 765	. . . . . . . . . 10 Word $/\$ $/\$
27	26	zcnd 11483	. . . . . . . . 9 Word $/\$ $/\$
28	27	3impb 1260	. . . . . . . 8 Word $/\$ $/\$
29	24, 28	pncan3d 10395	. . . . . . 7 Word $/\$ $/\$
30	20, 29	eqtrd 2656	. . . . . 6 Word $/\$ $/\$ prefix substr
31	11, 30	eqtrd 2656	. . . . 5 Word $/\$ $/\$ prefix ++ substr
32	31	oveq2d 6666	. . . 4 Word $/\$ $/\$ ..^ prefix ++ substr ..^
33	32	fneq2d 5982	. . 3 Word $/\$ $/\$ prefix ++ substr ..^ prefix ++ substr prefix ++ substr ..^
34	9, 33	mpbid 222	. 2 Word $/\$ $/\$ prefix ++ substr ..^
35		pfxfn 41390	. . 3 Word $/\$ prefix ..^
36	35	3adant2 1080	. 2 Word $/\$ $/\$ prefix ..^
37		simpr 477	. . . . 5 Word $/\$ $/\$ $/\$ ..^ ..^
38	21	3ad2ant2 1083	. . . . . 6 Word $/\$ $/\$
39	38	adantr 481	. . . . 5 Word $/\$ $/\$ $/\$ ..^
40		fzospliti 12500	. . . . 5 ..^ $/\$ ..^ $\/$ ..^
41	37, 39, 40	syl2anc 693	. . . 4 Word $/\$ $/\$ $/\$ ..^ ..^ $\/$ ..^
42	2	adantr 481	. . . . . . 7 Word $/\$ $/\$ $/\$ ..^ prefix Word
43	4	adantr 481	. . . . . . 7 Word $/\$ $/\$ $/\$ ..^ substr Word
44	18	oveq2d 6666	. . . . . . . . 9 Word $/\$ $/\$ ..^ prefix ..^
45	44	eleq2d 2687	. . . . . . . 8 Word $/\$ $/\$ ..^ prefix ..^
46	45	biimpar 502	. . . . . . 7 Word $/\$ $/\$ $/\$ ..^ ..^ prefix
47		ccatval1 13361	. . . . . . 7 prefix Word $/\$ substr Word $/\$ ..^ prefix prefix ++ substr prefix
48	42, 43, 46, 47	syl3anc 1326	. . . . . 6 Word $/\$ $/\$ $/\$ ..^ prefix ++ substr prefix
49	12	adantr 481	. . . . . . 7 Word $/\$ $/\$ $/\$ ..^ Word
50	16	adantr 481	. . . . . . 7 Word $/\$ $/\$ $/\$ ..^
51		simpr 477	. . . . . . 7 Word $/\$ $/\$ $/\$ ..^ ..^
52		pfxfv 41399	. . . . . . 7 Word $/\$ $/\$ ..^ prefix
53	49, 50, 51, 52	syl3anc 1326	. . . . . 6 Word $/\$ $/\$ $/\$ ..^ prefix
54	48, 53	eqtrd 2656	. . . . 5 Word $/\$ $/\$ $/\$ ..^ prefix ++ substr
55	2	adantr 481	. . . . . . 7 Word $/\$ $/\$ $/\$ ..^ prefix Word
56	4	adantr 481	. . . . . . 7 Word $/\$ $/\$ $/\$ ..^ substr Word
57	18, 30	oveq12d 6668	. . . . . . . . 9 Word $/\$ $/\$ prefix ..^ prefix substr ..^
58	57	eleq2d 2687	. . . . . . . 8 Word $/\$ $/\$ prefix ..^ prefix substr ..^
59	58	biimpar 502	. . . . . . 7 Word $/\$ $/\$ $/\$ ..^ prefix ..^ prefix substr
60		ccatval2 13362	. . . . . . 7 prefix Word $/\$ substr Word $/\$ prefix ..^ prefix substr prefix ++ substr substr prefix
61	55, 56, 59, 60	syl3anc 1326	. . . . . 6 Word $/\$ $/\$ $/\$ ..^ prefix ++ substr substr prefix
62	18	oveq2d 6666	. . . . . . . . 9 Word $/\$ $/\$ prefix
63	62	adantr 481	. . . . . . . 8 Word $/\$ $/\$ $/\$ ..^ prefix
64	38	anim1i 592	. . . . . . . . . . 11 Word $/\$ $/\$ $/\$ ..^ $/\$ ..^
65	64	ancomd 467	. . . . . . . . . 10 Word $/\$ $/\$ $/\$ ..^ ..^ $/\$
66		fzosubel 12526	. . . . . . . . . 10 ..^ $/\$ ..^
67	65, 66	syl 17	. . . . . . . . 9 Word $/\$ $/\$ $/\$ ..^ ..^
68	21	zcnd 11483	. . . . . . . . . . . . . . 15
69	68	subidd 10380	. . . . . . . . . . . . . 14
70	69	eqcomd 2628	. . . . . . . . . . . . 13
71	70	3ad2ant2 1083	. . . . . . . . . . . 12 Word $/\$ $/\$
72	71	oveq1d 6665	. . . . . . . . . . 11 Word $/\$ $/\$ ..^ ..^
73	72	eleq2d 2687	. . . . . . . . . 10 Word $/\$ $/\$ ..^ ..^
74	73	adantr 481	. . . . . . . . 9 Word $/\$ $/\$ $/\$ ..^ ..^ ..^
75	67, 74	mpbird 247	. . . . . . . 8 Word $/\$ $/\$ $/\$ ..^ ..^
76	63, 75	eqeltrd 2701	. . . . . . 7 Word $/\$ $/\$ $/\$ ..^ prefix ..^
77		swrdfv 13424	. . . . . . 7 Word $/\$ $/\$ $/\$ prefix ..^ substr prefix prefix
78	76, 77	syldan 487	. . . . . 6 Word $/\$ $/\$ $/\$ ..^ substr prefix prefix
79	63	oveq1d 6665	. . . . . . . 8 Word $/\$ $/\$ $/\$ ..^ prefix
80		elfzoelz 12470	. . . . . . . . . . 11 ..^
81	80	zcnd 11483	. . . . . . . . . 10 ..^
82	81	adantl 482	. . . . . . . . 9 Word $/\$ $/\$ $/\$ ..^
83	24	adantr 481	. . . . . . . . 9 Word $/\$ $/\$ $/\$ ..^
84	82, 83	npcand 10396	. . . . . . . 8 Word $/\$ $/\$ $/\$ ..^
85	79, 84	eqtrd 2656	. . . . . . 7 Word $/\$ $/\$ $/\$ ..^ prefix
86	85	fveq2d 6195	. . . . . 6 Word $/\$ $/\$ $/\$ ..^ prefix
87	61, 78, 86	3eqtrd 2660	. . . . 5 Word $/\$ $/\$ $/\$ ..^ prefix ++ substr
88	54, 87	jaodan 826	. . . 4 Word $/\$ $/\$ $/\$ ..^ $\/$ ..^ prefix ++ substr
89	41, 88	syldan 487	. . 3 Word $/\$ $/\$ $/\$ ..^ prefix ++ substr
90	12	adantr 481	. . . 4 Word $/\$ $/\$ $/\$ ..^ Word
91		simpl3 1066	. . . 4 Word $/\$ $/\$ $/\$ ..^
92		pfxfv 41399	. . . 4 Word $/\$ $/\$ ..^ prefix
93	90, 91, 37, 92	syl3anc 1326	. . 3 Word $/\$ $/\$ $/\$ ..^ prefix
94	89, 93	eqtr4d 2659	. 2 Word $/\$ $/\$ $/\$ ..^ prefix ++ substr prefix
95	34, 36, 94	eqfnfvd 6314	1 Word $/\$ $/\$ prefix ++ substr prefix

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $\/$ 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 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: pfxcctswrd 41417

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