Theorem ccat2s1fvwd 11393

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 11298	. . . . 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 11368	. . . 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 11368	. . . 4 |- ( ph -> <" Y "> e. Word _V )
10		ccatass 11354	. . . 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 5692	. 2 |- ( ph -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` I ) = ( ( W ++ ( <" X "> ++ <" Y "> ) ) ` I ) )
13		ccatws1cl 11378	. . . 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 11286	. . . . . . 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 9567	. . . . . . . . 9 |- ( I e. NN0 -> 0 <_ I )
21	20	adantl 277	. . . . . . . 8 |- ( ( W e. Word V /\ I e. NN0 ) -> 0 <_ I )
22		0red 8317	. . . . . . . . 9 |- ( ( W e. Word V /\ I e. NN0 ) -> 0 e. RR )
23		nn0re 9551	. . . . . . . . . 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 9600	. . . . . . . . . 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 8404	. . . . . . . . 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 9586	. . . . . 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 10571	. . . . 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 11343	. . 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   RRcr 8168   0cc0 8169   < clt 8350   <_ cle 8351   NNcn 9283   NN0cn0 9542  ..^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-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

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-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:  ccat2s1fstg  11394  clwwlknonex2lem2  16593