Theorem ccat1st1st 11217

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 11131	. . . . 5 |- ( W e. Word V -> W e. Fin )
2		fihasheq0 11054	. . . . 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 4216	. . . . . . . 8 |- (/) e. _V
6		s1cl 11197	. . . . . . . 8 |- ( (/) e. _V -> <" (/) "> e. Word _V )
7	5, 6	ax-mp 5	. . . . . . 7 |- <" (/) "> e. Word _V
8		ccatlid 11182	. . . . . . 7 |- ( <" (/) "> e. Word _V -> ( (/) ++ <" (/) "> ) = <" (/) "> )
9	7, 8	ax-mp 5	. . . . . 6 |- ( (/) ++ <" (/) "> ) = <" (/) ">
10	9	fveq1i 5640	. . . . 5 |- ( ( (/) ++ <" (/) "> ) ` 0 ) = ( <" (/) "> ` 0 )
11		s1fv 11202	. . . . . 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 5638	. . . . . . . 8 |- ( W = (/) -> ( W ` 0 ) = ( (/) ` 0 ) )
16		0fv 5677	. . . . . . . 8 |- ( (/) ` 0 ) = (/)
17	15, 16	eqtrdi 2280	. . . . . . 7 |- ( W = (/) -> ( W ` 0 ) = (/) )
18	17	s1eqd 11196	. . . . . 6 |- ( W = (/) -> <" ( W ` 0 ) "> = <" (/) "> )
19	14, 18	oveq12d 6035	. . . . 5 |- ( W = (/) -> ( W ++ <" ( W ` 0 ) "> ) = ( (/) ++ <" (/) "> ) )
20	19	fveq1d 5641	. . . 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 2443	. . . . 5 |- ( W e. Word V -> ( (♯ ` W ) =/= 0 <-> W =/= (/) ) )
25	24	biimpa 296	. . . 4 |- ( ( W e. Word V /\ (♯ ` W ) =/= 0 ) -> W =/= (/) )
26		fstwrdne 11151	. . . 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 11132	. . . . 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 10419	. . . 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 11215	. . 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 1273	. 2 |- ( ( W e. Word V /\ (♯ ` W ) =/= 0 ) -> ( ( W ++ <" ( W ` 0 ) "> ) ` 0 ) = ( W ` 0 ) )
34		lencl 11116	. . . . 5 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
35	34	nn0zd 9599	. . . 4 |- ( W e. Word V -> (♯ ` W ) e. ZZ )
36		0z 9489	. . . 4 |- 0 e. ZZ
37		zdceq 9554	. . . 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 2413	. . 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 805	1 |- ( W e. Word V -> ( ( W ++ <" ( W ` 0 ) "> ) ` 0 ) = ( W ` 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715  DECID wdc 841   = wceq 1397   e. wcel 2202   =/= wne 2402   _Vcvv 2802   (/)c0 3494   ` cfv 5326  (class class class)co 6017   Fincfn 6908   0cc0 8031   NNcn 9142   ZZcz 9478  ..^cfzo 10376  ♯chash 11036  Word cword 11112   ++ cconcat 11166   <"cs1 11191

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-inn 9143  df-n0 9402  df-z 9479  df-uz 9755  df-fz 10243  df-fzo 10377  df-ihash 11037  df-word 11113  df-concat 11167  df-s1 11192

This theorem is referenced by: (None)