Theorem ccat1st1st 11387

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 11301	. . . . 5 |- ( W e. Word V -> W e. Fin )
2		fihasheq0 11210	. . . . 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 4255	. . . . . . . 8 |- (/) e. _V
6		s1cl 11367	. . . . . . . 8 |- ( (/) e. _V -> <" (/) "> e. Word _V )
7	5, 6	ax-mp 5	. . . . . . 7 |- <" (/) "> e. Word _V
8		ccatlid 11352	. . . . . . 7 |- ( <" (/) "> e. Word _V -> ( (/) ++ <" (/) "> ) = <" (/) "> )
9	7, 8	ax-mp 5	. . . . . 6 |- ( (/) ++ <" (/) "> ) = <" (/) ">
10	9	fveq1i 5691	. . . . 5 |- ( ( (/) ++ <" (/) "> ) ` 0 ) = ( <" (/) "> ` 0 )
11		s1fv 11372	. . . . . 6 |- ( (/) e. _V -> ( <" (/) "> ` 0 ) = (/) )
12	5, 11	ax-mp 5	. . . . 5 |- ( <" (/) "> ` 0 ) = (/)
13	10, 12	eqtri 2259	. . . 4 |- ( ( (/) ++ <" (/) "> ) ` 0 ) = (/)
14		id 19	. . . . . 6 |- ( W = (/) -> W = (/) )
15		fveq1 5689	. . . . . . . 8 |- ( W = (/) -> ( W ` 0 ) = ( (/) ` 0 ) )
16		0fv 5728	. . . . . . . 8 |- ( (/) ` 0 ) = (/)
17	15, 16	eqtrdi 2287	. . . . . . 7 |- ( W = (/) -> ( W ` 0 ) = (/) )
18	17	s1eqd 11366	. . . . . 6 |- ( W = (/) -> <" ( W ` 0 ) "> = <" (/) "> )
19	14, 18	oveq12d 6093	. . . . 5 |- ( W = (/) -> ( W ++ <" ( W ` 0 ) "> ) = ( (/) ++ <" (/) "> ) )
20	19	fveq1d 5692	. . . 4 |- ( W = (/) -> ( ( W ++ <" ( W ` 0 ) "> ) ` 0 ) = ( ( (/) ++ <" (/) "> ) ` 0 ) )
21	13, 20, 17	3eqtr4a 2297	. . 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 2461	. . . . 5 |- ( W e. Word V -> ( (♯ ` W ) =/= 0 <-> W =/= (/) ) )
25	24	biimpa 296	. . . 4 |- ( ( W e. Word V /\ (♯ ` W ) =/= 0 ) -> W =/= (/) )
26		fstwrdne 11321	. . . 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 11302	. . . . 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 10570	. . . 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 11385	. . 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 1278	. 2 |- ( ( W e. Word V /\ (♯ ` W ) =/= 0 ) -> ( ( W ++ <" ( W ` 0 ) "> ) ` 0 ) = ( W ` 0 ) )
34		lencl 11286	. . . . 5 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
35	34	nn0zd 9745	. . . 4 |- ( W e. Word V -> (♯ ` W ) e. ZZ )
36		0z 9634	. . . 4 |- 0 e. ZZ
37		zdceq 9699	. . . 4 |- ( ( (♯ ` W ) e. ZZ /\ 0 e. ZZ ) -> DECID (♯ ` W ) = 0 )
38	35, 36, 37	sylancl 417	. . 3 |- ( W e. Word V -> DECID (♯ ` W ) = 0 )
39		dcne 2431	. . 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 810	1 |- ( W e. Word V -> ( ( W ++ <" ( W ` 0 ) "> ) ` 0 ) = ( W ` 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   =/= wne 2420   _Vcvv 2821   (/)c0 3520   ` cfv 5372  (class class class)co 6075   Fincfn 7012   0cc0 8169   NNcn 9283   ZZcz 9623  ..^cfzo 10527  ♯chash 11192  Word cword 11282   ++ cconcat 11336   <"cs1 11361

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-ihash 11193  df-word 11283  df-concat 11337  df-s1 11362

This theorem is referenced by: (None)