Theorem ccat1st1st 11167

Description: The first symbol of a word concatenated with its first symbol is the first symbol of the word. This theorem holds even if W is the empty word. (Contributed by AV, 26-Mar-2022.)

Assertion

Ref	Expression
ccat1st1st	|- ( W e. Word V -> ( ( W ++ <" ( W ` 0 ) "> ) ` 0 ) = ( W ` 0 ) )

Proof of Theorem ccat1st1st

Step	Hyp	Ref	Expression
1		wrdfin 11085	. . . . 5 |- ( W e. Word V -> W e. Fin )
2		fihasheq0 11010	. . . . 5 |- ( W e. Fin -> ( (♯ ` W ) = 0 <-> W = (/) ) )
3	1, 2	syl 14	. . . 4 |- ( W e. Word V -> ( (♯ ` W ) = 0 <-> W = (/) ) )
4	3	biimpa 296	. . 3 |- ( ( W e. Word V /\ (♯ ` W ) = 0 ) -> W = (/) )
5		0ex 4210	. . . . . . . 8 |- (/) e. _V
6		s1cl 11149	. . . . . . . 8 |- ( (/) e. _V -> <" (/) "> e. Word _V )
7	5, 6	ax-mp 5	. . . . . . 7 |- <" (/) "> e. Word _V
8		ccatlid 11136	. . . . . . 7 |- ( <" (/) "> e. Word _V -> ( (/) ++ <" (/) "> ) = <" (/) "> )
9	7, 8	ax-mp 5	. . . . . 6 |- ( (/) ++ <" (/) "> ) = <" (/) ">
10	9	fveq1i 5627	. . . . 5 |- ( ( (/) ++ <" (/) "> ) ` 0 ) = ( <" (/) "> ` 0 )
11		s1fv 11154	. . . . . 6 |- ( (/) e. _V -> ( <" (/) "> ` 0 ) = (/) )
12	5, 11	ax-mp 5	. . . . 5 |- ( <" (/) "> ` 0 ) = (/)
13	10, 12	eqtri 2250	. . . 4 |- ( ( (/) ++ <" (/) "> ) ` 0 ) = (/)
14		id 19	. . . . . 6 |- ( W = (/) -> W = (/) )
15		fveq1 5625	. . . . . . . 8 |- ( W = (/) -> ( W ` 0 ) = ( (/) ` 0 ) )
16		0fv 5664	. . . . . . . 8 |- ( (/) ` 0 ) = (/)
17	15, 16	eqtrdi 2278	. . . . . . 7 |- ( W = (/) -> ( W ` 0 ) = (/) )
18	17	s1eqd 11148	. . . . . 6 |- ( W = (/) -> <" ( W ` 0 ) "> = <" (/) "> )
19	14, 18	oveq12d 6018	. . . . 5 |- ( W = (/) -> ( W ++ <" ( W ` 0 ) "> ) = ( (/) ++ <" (/) "> ) )
20	19	fveq1d 5628	. . . 4 |- ( W = (/) -> ( ( W ++ <" ( W ` 0 ) "> ) ` 0 ) = ( ( (/) ++ <" (/) "> ) ` 0 ) )
21	13, 20, 17	3eqtr4a 2288	. . 3 |- ( W = (/) -> ( ( W ++ <" ( W ` 0 ) "> ) ` 0 ) = ( W ` 0 ) )
22	4, 21	syl 14	. 2 |- ( ( W e. Word V /\ (♯ ` W ) = 0 ) -> ( ( W ++ <" ( W ` 0 ) "> ) ` 0 ) = ( W ` 0 ) )
23		simpl 109	. . 3 |- ( ( W e. Word V /\ (♯ ` W ) =/= 0 ) -> W e. Word V )
24	3	necon3bid 2441	. . . . 5 |- ( W e. Word V -> ( (♯ ` W ) =/= 0 <-> W =/= (/) ) )
25	24	biimpa 296	. . . 4 |- ( ( W e. Word V /\ (♯ ` W ) =/= 0 ) -> W =/= (/) )
26		fstwrdne 11105	. . . 4 |- ( ( W e. Word V /\ W =/= (/) ) -> ( W ` 0 ) e. V )
27	25, 26	syldan 282	. . 3 |- ( ( W e. Word V /\ (♯ ` W ) =/= 0 ) -> ( W ` 0 ) e. V )
28		lennncl 11086	. . . . 5 |- ( ( W e. Word V /\ W =/= (/) ) -> (♯ ` W ) e. NN )
29	25, 28	syldan 282	. . . 4 |- ( ( W e. Word V /\ (♯ ` W ) =/= 0 ) -> (♯ ` W ) e. NN )
30		lbfzo0 10377	. . . 4 |- ( 0 e. ( 0..^ (♯ ` W ) ) <-> (♯ ` W ) e. NN )
31	29, 30	sylibr 134	. . 3 |- ( ( W e. Word V /\ (♯ ` W ) =/= 0 ) -> 0 e. ( 0..^ (♯ ` W ) ) )
32		ccats1val1g 11165	. . 3 |- ( ( W e. Word V /\ ( W ` 0 ) e. V /\ 0 e. ( 0..^ (♯ ` W ) ) ) -> ( ( W ++ <" ( W ` 0 ) "> ) ` 0 ) = ( W ` 0 ) )
33	23, 27, 31, 32	syl3anc 1271	. 2 |- ( ( W e. Word V /\ (♯ ` W ) =/= 0 ) -> ( ( W ++ <" ( W ` 0 ) "> ) ` 0 ) = ( W ` 0 ) )
34		lencl 11070	. . . . 5 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
35	34	nn0zd 9563	. . . 4 |- ( W e. Word V -> (♯ ` W ) e. ZZ )
36		0z 9453	. . . 4 |- 0 e. ZZ
37		zdceq 9518	. . . 4 |- ( ( (♯ ` W ) e. ZZ /\ 0 e. ZZ ) -> DECID (♯ ` W ) = 0 )
38	35, 36, 37	sylancl 413	. . 3 |- ( W e. Word V -> DECID (♯ ` W ) = 0 )
39		dcne 2411	. . 3 |- (DECID (♯ ` W ) = 0 <-> ( (♯ ` W ) = 0 \/ (♯ ` W ) =/= 0 ) )
40	38, 39	sylib 122	. 2 |- ( W e. Word V -> ( (♯ ` W ) = 0 \/ (♯ ` W ) =/= 0 ) )
41	22, 33, 40	mpjaodan 803	1 |- ( W e. Word V -> ( ( W ++ <" ( W ` 0 ) "> ) ` 0 ) = ( W ` 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   = wceq 1395   e. wcel 2200   =/= wne 2400   _Vcvv 2799   (/)c0 3491   ` cfv 5317  (class class class)co 6000   Fincfn 6885   0cc0 7995   NNcn 9106   ZZcz 9442  ..^cfzo 10334  ♯chash 10992  Word cword 11066   ++ cconcat 11120   <"cs1 11143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-1o 6560  df-er 6678  df-en 6886  df-dom 6887  df-fin 6888  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-fz 10201  df-fzo 10335  df-ihash 10993  df-word 11067  df-concat 11121  df-s1 11144

This theorem is referenced by: (None)