Theorem ccat1st1st 11354

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 11268	. . . . 5 |- ( W e. Word V -> W e. Fin )
2		fihasheq0 11181	. . . . 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 4242	. . . . . . . 8 |- (/) e. _V
6		s1cl 11334	. . . . . . . 8 |- ( (/) e. _V -> <" (/) "> e. Word _V )
7	5, 6	ax-mp 5	. . . . . . 7 |- <" (/) "> e. Word _V
8		ccatlid 11319	. . . . . . 7 |- ( <" (/) "> e. Word _V -> ( (/) ++ <" (/) "> ) = <" (/) "> )
9	7, 8	ax-mp 5	. . . . . 6 |- ( (/) ++ <" (/) "> ) = <" (/) ">
10	9	fveq1i 5676	. . . . 5 |- ( ( (/) ++ <" (/) "> ) ` 0 ) = ( <" (/) "> ` 0 )
11		s1fv 11339	. . . . . 6 |- ( (/) e. _V -> ( <" (/) "> ` 0 ) = (/) )
12	5, 11	ax-mp 5	. . . . 5 |- ( <" (/) "> ` 0 ) = (/)
13	10, 12	eqtri 2255	. . . 4 |- ( ( (/) ++ <" (/) "> ) ` 0 ) = (/)
14		id 19	. . . . . 6 |- ( W = (/) -> W = (/) )
15		fveq1 5674	. . . . . . . 8 |- ( W = (/) -> ( W ` 0 ) = ( (/) ` 0 ) )
16		0fv 5713	. . . . . . . 8 |- ( (/) ` 0 ) = (/)
17	15, 16	eqtrdi 2283	. . . . . . 7 |- ( W = (/) -> ( W ` 0 ) = (/) )
18	17	s1eqd 11333	. . . . . 6 |- ( W = (/) -> <" ( W ` 0 ) "> = <" (/) "> )
19	14, 18	oveq12d 6076	. . . . 5 |- ( W = (/) -> ( W ++ <" ( W ` 0 ) "> ) = ( (/) ++ <" (/) "> ) )
20	19	fveq1d 5677	. . . 4 |- ( W = (/) -> ( ( W ++ <" ( W ` 0 ) "> ) ` 0 ) = ( ( (/) ++ <" (/) "> ) ` 0 ) )
21	13, 20, 17	3eqtr4a 2293	. . 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 2455	. . . . 5 |- ( W e. Word V -> ( (♯ ` W ) =/= 0 <-> W =/= (/) ) )
25	24	biimpa 296	. . . 4 |- ( ( W e. Word V /\ (♯ ` W ) =/= 0 ) -> W =/= (/) )
26		fstwrdne 11288	. . . 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 11269	. . . . 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 10541	. . . 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 11352	. . 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 1274	. 2 |- ( ( W e. Word V /\ (♯ ` W ) =/= 0 ) -> ( ( W ++ <" ( W ` 0 ) "> ) ` 0 ) = ( W ` 0 ) )
34		lencl 11253	. . . . 5 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
35	34	nn0zd 9716	. . . 4 |- ( W e. Word V -> (♯ ` W ) e. ZZ )
36		0z 9605	. . . 4 |- 0 e. ZZ
37		zdceq 9670	. . . 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 2425	. . 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 806	1 |- ( W e. Word V -> ( ( W ++ <" ( W ` 0 ) "> ) ` 0 ) = ( W ` 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   = wceq 1398   e. wcel 2205   =/= wne 2414   _Vcvv 2815   (/)c0 3512   ` cfv 5357  (class class class)co 6058   Fincfn 6988   0cc0 8143   NNcn 9254   ZZcz 9594  ..^cfzo 10498  ♯chash 11163  Word cword 11249   ++ cconcat 11303   <"cs1 11328

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-concat 11304  df-s1 11329

This theorem is referenced by: (None)