Theorem ccat2s1fvwd 11211

Description: Extract a symbol of a word from the concatenation of the word with two single symbols. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 13-Jan-2020.) (Proof shortened by AV, 1-May-2020.) (Revised by AV, 28-Jan-2024.)

Hypotheses

Ref	Expression
ccat2s1fvwd.w	|- ( ph -> W e. Word V )
ccat2s1fvwd.i	|- ( ph -> I e. NN0 )
ccat2s1fvwd.1	|- ( ph -> I < (♯ ` W ) )
ccat2s1fvwd.x	|- ( ph -> X e. A )
ccat2s1fvwd.y	|- ( ph -> Y e. B )

Assertion

Ref	Expression
ccat2s1fvwd	|- ( ph -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` I ) = ( W ` I ) )

Proof of Theorem ccat2s1fvwd

Step	Hyp	Ref	Expression
1		ccat2s1fvwd.w	. . . . 5 |- ( ph -> W e. Word V )
2		wrdv 11116	. . . . 5 |- ( W e. Word V -> W e. Word _V )
3	1, 2	syl 14	. . . 4 |- ( ph -> W e. Word _V )
4		ccat2s1fvwd.x	. . . . . 6 |- ( ph -> X e. A )
5	4	elexd 2814	. . . . 5 |- ( ph -> X e. _V )
6	5	s1cld 11186	. . . 4 |- ( ph -> <" X "> e. Word _V )
7		ccat2s1fvwd.y	. . . . . 6 |- ( ph -> Y e. B )
8	7	elexd 2814	. . . . 5 |- ( ph -> Y e. _V )
9	8	s1cld 11186	. . . 4 |- ( ph -> <" Y "> e. Word _V )
10		ccatass 11172	. . . 4 |- ( ( W e. Word _V /\ <" X "> e. Word _V /\ <" Y "> e. Word _V ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) = ( W ++ ( <" X "> ++ <" Y "> ) ) )
11	3, 6, 9, 10	syl3anc 1271	. . 3 |- ( ph -> ( ( W ++ <" X "> ) ++ <" Y "> ) = ( W ++ ( <" X "> ++ <" Y "> ) ) )
12	11	fveq1d 5635	. 2 |- ( ph -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` I ) = ( ( W ++ ( <" X "> ++ <" Y "> ) ) ` I ) )
13		ccatws1cl 11196	. . . 4 |- ( ( <" X "> e. Word _V /\ Y e. _V ) -> ( <" X "> ++ <" Y "> ) e. Word _V )
14	6, 8, 13	syl2anc 411	. . 3 |- ( ph -> ( <" X "> ++ <" Y "> ) e. Word _V )
15		ccat2s1fvwd.i	. . . 4 |- ( ph -> I e. NN0 )
16		ccat2s1fvwd.1	. . . 4 |- ( ph -> I < (♯ ` W ) )
17		simp2 1022	. . . . 5 |- ( ( W e. Word V /\ I e. NN0 /\ I < (♯ ` W ) ) -> I e. NN0 )
18		lencl 11104	. . . . . . 7 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
19	18	3ad2ant1 1042	. . . . . 6 |- ( ( W e. Word V /\ I e. NN0 /\ I < (♯ ` W ) ) -> (♯ ` W ) e. NN0 )
20		nn0ge0 9415	. . . . . . . . 9 |- ( I e. NN0 -> 0 <_ I )
21	20	adantl 277	. . . . . . . 8 |- ( ( W e. Word V /\ I e. NN0 ) -> 0 <_ I )
22		0red 8168	. . . . . . . . 9 |- ( ( W e. Word V /\ I e. NN0 ) -> 0 e. RR )
23		nn0re 9399	. . . . . . . . . 10 |- ( I e. NN0 -> I e. RR )
24	23	adantl 277	. . . . . . . . 9 |- ( ( W e. Word V /\ I e. NN0 ) -> I e. RR )
25	18	nn0red 9444	. . . . . . . . . 10 |- ( W e. Word V -> (♯ ` W ) e. RR )
26	25	adantr 276	. . . . . . . . 9 |- ( ( W e. Word V /\ I e. NN0 ) -> (♯ ` W ) e. RR )
27		lelttr 8256	. . . . . . . . 9 |- ( ( 0 e. RR /\ I e. RR /\ (♯ ` W ) e. RR ) -> ( ( 0 <_ I /\ I < (♯ ` W ) ) -> 0 < (♯ ` W ) ) )
28	22, 24, 26, 27	syl3anc 1271	. . . . . . . 8 |- ( ( W e. Word V /\ I e. NN0 ) -> ( ( 0 <_ I /\ I < (♯ ` W ) ) -> 0 < (♯ ` W ) ) )
29	21, 28	mpand 429	. . . . . . 7 |- ( ( W e. Word V /\ I e. NN0 ) -> ( I < (♯ ` W ) -> 0 < (♯ ` W ) ) )
30	29	3impia 1224	. . . . . 6 |- ( ( W e. Word V /\ I e. NN0 /\ I < (♯ ` W ) ) -> 0 < (♯ ` W ) )
31		elnnnn0b 9434	. . . . . 6 |- ( (♯ ` W ) e. NN <-> ( (♯ ` W ) e. NN0 /\ 0 < (♯ ` W ) ) )
32	19, 30, 31	sylanbrc 417	. . . . 5 |- ( ( W e. Word V /\ I e. NN0 /\ I < (♯ ` W ) ) -> (♯ ` W ) e. NN )
33		simp3 1023	. . . . 5 |- ( ( W e. Word V /\ I e. NN0 /\ I < (♯ ` W ) ) -> I < (♯ ` W ) )
34		elfzo0 10409	. . . . 5 |- ( I e. ( 0..^ (♯ ` W ) ) <-> ( I e. NN0 /\ (♯ ` W ) e. NN /\ I < (♯ ` W ) ) )
35	17, 32, 33, 34	syl3anbrc 1205	. . . 4 |- ( ( W e. Word V /\ I e. NN0 /\ I < (♯ ` W ) ) -> I e. ( 0..^ (♯ ` W ) ) )
36	1, 15, 16, 35	syl3anc 1271	. . 3 |- ( ph -> I e. ( 0..^ (♯ ` W ) ) )
37		ccatval1 11161	. . 3 |- ( ( W e. Word V /\ ( <" X "> ++ <" Y "> ) e. Word _V /\ I e. ( 0..^ (♯ ` W ) ) ) -> ( ( W ++ ( <" X "> ++ <" Y "> ) ) ` I ) = ( W ` I ) )
38	1, 14, 36, 37	syl3anc 1271	. 2 |- ( ph -> ( ( W ++ ( <" X "> ++ <" Y "> ) ) ` I ) = ( W ` I ) )
39	12, 38	eqtrd 2262	1 |- ( ph -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` I ) = ( W ` I ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   _Vcvv 2800   class class class wbr 4084   ` cfv 5322  (class class class)co 6011   RRcr 8019   0cc0 8020   < clt 8202   <_ cle 8203   NNcn 9131   NN0cn0 9390  ..^cfzo 10365  ♯chash 11025  Word cword 11100   ++ cconcat 11154   <"cs1 11179

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 4200  ax-sep 4203  ax-nul 4211  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-iinf 4682  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-addcom 8120  ax-addass 8122  ax-distr 8124  ax-i2m1 8125  ax-0lt1 8126  ax-0id 8128  ax-rnegex 8129  ax-cnre 8131  ax-pre-ltirr 8132  ax-pre-ltwlin 8133  ax-pre-lttrn 8134  ax-pre-apti 8135  ax-pre-ltadd 8136

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-tr 4184  df-id 4386  df-iord 4459  df-on 4461  df-ilim 4462  df-suc 4464  df-iom 4685  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-recs 6464  df-frec 6550  df-1o 6575  df-er 6695  df-en 6903  df-dom 6904  df-fin 6905  df-pnf 8204  df-mnf 8205  df-xr 8206  df-ltxr 8207  df-le 8208  df-sub 8340  df-neg 8341  df-inn 9132  df-n0 9391  df-z 9468  df-uz 9744  df-fz 10232  df-fzo 10366  df-ihash 11026  df-word 11101  df-concat 11155  df-s1 11180

This theorem is referenced by:  ccat2s1fstg  11212  clwwlknonex2lem2  16223