Theorem ccat2s1fvwd 11360

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 11265	. . . . 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 2829	. . . . 5 |- ( ph -> X e. _V )
6	5	s1cld 11335	. . . 4 |- ( ph -> <" X "> e. Word _V )
7		ccat2s1fvwd.y	. . . . . 6 |- ( ph -> Y e. B )
8	7	elexd 2829	. . . . 5 |- ( ph -> Y e. _V )
9	8	s1cld 11335	. . . 4 |- ( ph -> <" Y "> e. Word _V )
10		ccatass 11321	. . . 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 1274	. . 3 |- ( ph -> ( ( W ++ <" X "> ) ++ <" Y "> ) = ( W ++ ( <" X "> ++ <" Y "> ) ) )
12	11	fveq1d 5677	. 2 |- ( ph -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` I ) = ( ( W ++ ( <" X "> ++ <" Y "> ) ) ` I ) )
13		ccatws1cl 11345	. . . 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 1025	. . . . 5 |- ( ( W e. Word V /\ I e. NN0 /\ I < (♯ ` W ) ) -> I e. NN0 )
18		lencl 11253	. . . . . . 7 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
19	18	3ad2ant1 1045	. . . . . 6 |- ( ( W e. Word V /\ I e. NN0 /\ I < (♯ ` W ) ) -> (♯ ` W ) e. NN0 )
20		nn0ge0 9538	. . . . . . . . 9 |- ( I e. NN0 -> 0 <_ I )
21	20	adantl 277	. . . . . . . 8 |- ( ( W e. Word V /\ I e. NN0 ) -> 0 <_ I )
22		0red 8291	. . . . . . . . 9 |- ( ( W e. Word V /\ I e. NN0 ) -> 0 e. RR )
23		nn0re 9522	. . . . . . . . . 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 9571	. . . . . . . . . 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 8378	. . . . . . . . 9 |- ( ( 0 e. RR /\ I e. RR /\ (♯ ` W ) e. RR ) -> ( ( 0 <_ I /\ I < (♯ ` W ) ) -> 0 < (♯ ` W ) ) )
28	22, 24, 26, 27	syl3anc 1274	. . . . . . . 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 1227	. . . . . 6 |- ( ( W e. Word V /\ I e. NN0 /\ I < (♯ ` W ) ) -> 0 < (♯ ` W ) )
31		elnnnn0b 9557	. . . . . 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 1026	. . . . 5 |- ( ( W e. Word V /\ I e. NN0 /\ I < (♯ ` W ) ) -> I < (♯ ` W ) )
34		elfzo0 10542	. . . . 5 |- ( I e. ( 0..^ (♯ ` W ) ) <-> ( I e. NN0 /\ (♯ ` W ) e. NN /\ I < (♯ ` W ) ) )
35	17, 32, 33, 34	syl3anbrc 1208	. . . 4 |- ( ( W e. Word V /\ I e. NN0 /\ I < (♯ ` W ) ) -> I e. ( 0..^ (♯ ` W ) ) )
36	1, 15, 16, 35	syl3anc 1274	. . 3 |- ( ph -> I e. ( 0..^ (♯ ` W ) ) )
37		ccatval1 11310	. . 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 1274	. 2 |- ( ph -> ( ( W ++ ( <" X "> ++ <" Y "> ) ) ` I ) = ( W ` I ) )
39	12, 38	eqtrd 2267	1 |- ( ph -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` I ) = ( W ` I ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   _Vcvv 2815   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   RRcr 8142   0cc0 8143   < clt 8324   <_ cle 8325   NNcn 9254   NN0cn0 9513  ..^cfzo 10498  ♯chash 11163  Word cword 11249   ++ cconcat 11303   <"cs1 11328

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-concat 11304  df-s1 11329

This theorem is referenced by:  ccat2s1fstg  11361  clwwlknonex2lem2  16559