Theorem ccat1st1st 11267

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 11181	. . . . 5 |- ( W e. Word V -> W e. Fin )
2		fihasheq0 11101	. . . . 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 4221	. . . . . . . 8 |- (/) e. _V
6		s1cl 11247	. . . . . . . 8 |- ( (/) e. _V -> <" (/) "> e. Word _V )
7	5, 6	ax-mp 5	. . . . . . 7 |- <" (/) "> e. Word _V
8		ccatlid 11232	. . . . . . 7 |- ( <" (/) "> e. Word _V -> ( (/) ++ <" (/) "> ) = <" (/) "> )
9	7, 8	ax-mp 5	. . . . . 6 |- ( (/) ++ <" (/) "> ) = <" (/) ">
10	9	fveq1i 5649	. . . . 5 |- ( ( (/) ++ <" (/) "> ) ` 0 ) = ( <" (/) "> ` 0 )
11		s1fv 11252	. . . . . 6 |- ( (/) e. _V -> ( <" (/) "> ` 0 ) = (/) )
12	5, 11	ax-mp 5	. . . . 5 |- ( <" (/) "> ` 0 ) = (/)
13	10, 12	eqtri 2252	. . . 4 |- ( ( (/) ++ <" (/) "> ) ` 0 ) = (/)
14		id 19	. . . . . 6 |- ( W = (/) -> W = (/) )
15		fveq1 5647	. . . . . . . 8 |- ( W = (/) -> ( W ` 0 ) = ( (/) ` 0 ) )
16		0fv 5686	. . . . . . . 8 |- ( (/) ` 0 ) = (/)
17	15, 16	eqtrdi 2280	. . . . . . 7 |- ( W = (/) -> ( W ` 0 ) = (/) )
18	17	s1eqd 11246	. . . . . 6 |- ( W = (/) -> <" ( W ` 0 ) "> = <" (/) "> )
19	14, 18	oveq12d 6046	. . . . 5 |- ( W = (/) -> ( W ++ <" ( W ` 0 ) "> ) = ( (/) ++ <" (/) "> ) )
20	19	fveq1d 5650	. . . 4 |- ( W = (/) -> ( ( W ++ <" ( W ` 0 ) "> ) ` 0 ) = ( ( (/) ++ <" (/) "> ) ` 0 ) )
21	13, 20, 17	3eqtr4a 2290	. . 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 2444	. . . . 5 |- ( W e. Word V -> ( (♯ ` W ) =/= 0 <-> W =/= (/) ) )
25	24	biimpa 296	. . . 4 |- ( ( W e. Word V /\ (♯ ` W ) =/= 0 ) -> W =/= (/) )
26		fstwrdne 11201	. . . 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 11182	. . . . 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 10465	. . . 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 11265	. . 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 11166	. . . . 5 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
35	34	nn0zd 9644	. . . 4 |- ( W e. Word V -> (♯ ` W ) e. ZZ )
36		0z 9534	. . . 4 |- 0 e. ZZ
37		zdceq 9599	. . . 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 2414	. . 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 2202   =/= wne 2403   _Vcvv 2803   (/)c0 3496   ` cfv 5333  (class class class)co 6028   Fincfn 6952   0cc0 8075   NNcn 9185   ZZcz 9523  ..^cfzo 10422  ♯chash 11083  Word cword 11162   ++ cconcat 11216   <"cs1 11241

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-inn 9186  df-n0 9445  df-z 9524  df-uz 9800  df-fz 10289  df-fzo 10423  df-ihash 11084  df-word 11163  df-concat 11217  df-s1 11242

This theorem is referenced by: (None)