Theorem ccat2s1fvwd 11437

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 11342	. . . . 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 2835	. . . . 5 |- ( ph -> X e. _V )
6	5	s1cld 11412	. . . 4 |- ( ph -> <" X "> e. Word _V )
7		ccat2s1fvwd.y	. . . . . 6 |- ( ph -> Y e. B )
8	7	elexd 2835	. . . . 5 |- ( ph -> Y e. _V )
9	8	s1cld 11412	. . . 4 |- ( ph -> <" Y "> e. Word _V )
10		ccatass 11398	. . . 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 1278	. . 3 |- ( ph -> ( ( W ++ <" X "> ) ++ <" Y "> ) = ( W ++ ( <" X "> ++ <" Y "> ) ) )
12	11	fveq1d 5700	. 2 |- ( ph -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` I ) = ( ( W ++ ( <" X "> ++ <" Y "> ) ) ` I ) )
13		ccatws1cl 11422	. . . 4 |- ( ( <" X "> e. Word _V /\ Y e. _V ) -> ( <" X "> ++ <" Y "> ) e. Word _V )
14	6, 8, 13	syl2anc 415	. . 3 |- ( ph -> ( <" X "> ++ <" Y "> ) e. Word _V )
15		ccat2s1fvwd.i	. . . 4 |- ( ph -> I e. NN0 )
16		ccat2s1fvwd.1	. . . 4 |- ( ph -> I < (♯ ` W ) )
17		simp2 1029	. . . . 5 |- ( ( W e. Word V /\ I e. NN0 /\ I < (♯ ` W ) ) -> I e. NN0 )
18		lencl 11330	. . . . . . 7 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
19	18	3ad2ant1 1049	. . . . . 6 |- ( ( W e. Word V /\ I e. NN0 /\ I < (♯ ` W ) ) -> (♯ ` W ) e. NN0 )
20		nn0ge0 9599	. . . . . . . . 9 |- ( I e. NN0 -> 0 <_ I )
21	20	adantl 277	. . . . . . . 8 |- ( ( W e. Word V /\ I e. NN0 ) -> 0 <_ I )
22		0red 8333	. . . . . . . . 9 |- ( ( W e. Word V /\ I e. NN0 ) -> 0 e. RR )
23		nn0re 9583	. . . . . . . . . 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 9632	. . . . . . . . . 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 8420	. . . . . . . . 9 |- ( ( 0 e. RR /\ I e. RR /\ (♯ ` W ) e. RR ) -> ( ( 0 <_ I /\ I < (♯ ` W ) ) -> 0 < (♯ ` W ) ) )
28	22, 24, 26, 27	syl3anc 1278	. . . . . . . 8 |- ( ( W e. Word V /\ I e. NN0 ) -> ( ( 0 <_ I /\ I < (♯ ` W ) ) -> 0 < (♯ ` W ) ) )
29	21, 28	mpand 433	. . . . . . 7 |- ( ( W e. Word V /\ I e. NN0 ) -> ( I < (♯ ` W ) -> 0 < (♯ ` W ) ) )
30	29	3impia 1231	. . . . . 6 |- ( ( W e. Word V /\ I e. NN0 /\ I < (♯ ` W ) ) -> 0 < (♯ ` W ) )
31		elnnnn0b 9618	. . . . . 6 |- ( (♯ ` W ) e. NN <-> ( (♯ ` W ) e. NN0 /\ 0 < (♯ ` W ) ) )
32	19, 30, 31	sylanbrc 421	. . . . 5 |- ( ( W e. Word V /\ I e. NN0 /\ I < (♯ ` W ) ) -> (♯ ` W ) e. NN )
33		simp3 1030	. . . . 5 |- ( ( W e. Word V /\ I e. NN0 /\ I < (♯ ` W ) ) -> I < (♯ ` W ) )
34		elfzo0 10610	. . . . 5 |- ( I e. ( 0..^ (♯ ` W ) ) <-> ( I e. NN0 /\ (♯ ` W ) e. NN /\ I < (♯ ` W ) ) )
35	17, 32, 33, 34	syl3anbrc 1212	. . . 4 |- ( ( W e. Word V /\ I e. NN0 /\ I < (♯ ` W ) ) -> I e. ( 0..^ (♯ ` W ) ) )
36	1, 15, 16, 35	syl3anc 1278	. . 3 |- ( ph -> I e. ( 0..^ (♯ ` W ) ) )
37		ccatval1 11387	. . 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 1278	. 2 |- ( ph -> ( ( W ++ ( <" X "> ++ <" Y "> ) ) ` I ) = ( W ` I ) )
39	12, 38	eqtrd 2271	1 |- ( ph -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` I ) = ( W ` I ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   RRcr 8184   0cc0 8185   < clt 8366   <_ cle 8367   NNcn 9313   NN0cn0 9574  ..^cfzo 10566  ♯chash 11236  Word cword 11326   ++ cconcat 11380   <"cs1 11405

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-er 6811  df-en 7027  df-dom 7028  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429  df-fzo 10567  df-ihash 11237  df-word 11327  df-concat 11381  df-s1 11406

This theorem is used by:  ccat2s1fstg  11438  clwwlknonex2lem2  16884