Theorem ccat2s1fvwd 11223

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 11128	. . . . 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 2816	. . . . 5 |- ( ph -> X e. _V )
6	5	s1cld 11198	. . . 4 |- ( ph -> <" X "> e. Word _V )
7		ccat2s1fvwd.y	. . . . . 6 |- ( ph -> Y e. B )
8	7	elexd 2816	. . . . 5 |- ( ph -> Y e. _V )
9	8	s1cld 11198	. . . 4 |- ( ph -> <" Y "> e. Word _V )
10		ccatass 11184	. . . 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 1273	. . 3 |- ( ph -> ( ( W ++ <" X "> ) ++ <" Y "> ) = ( W ++ ( <" X "> ++ <" Y "> ) ) )
12	11	fveq1d 5641	. 2 |- ( ph -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` I ) = ( ( W ++ ( <" X "> ++ <" Y "> ) ) ` I ) )
13		ccatws1cl 11208	. . . 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 1024	. . . . 5 |- ( ( W e. Word V /\ I e. NN0 /\ I < (♯ ` W ) ) -> I e. NN0 )
18		lencl 11116	. . . . . . 7 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
19	18	3ad2ant1 1044	. . . . . 6 |- ( ( W e. Word V /\ I e. NN0 /\ I < (♯ ` W ) ) -> (♯ ` W ) e. NN0 )
20		nn0ge0 9426	. . . . . . . . 9 |- ( I e. NN0 -> 0 <_ I )
21	20	adantl 277	. . . . . . . 8 |- ( ( W e. Word V /\ I e. NN0 ) -> 0 <_ I )
22		0red 8179	. . . . . . . . 9 |- ( ( W e. Word V /\ I e. NN0 ) -> 0 e. RR )
23		nn0re 9410	. . . . . . . . . 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 9455	. . . . . . . . . 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 8267	. . . . . . . . 9 |- ( ( 0 e. RR /\ I e. RR /\ (♯ ` W ) e. RR ) -> ( ( 0 <_ I /\ I < (♯ ` W ) ) -> 0 < (♯ ` W ) ) )
28	22, 24, 26, 27	syl3anc 1273	. . . . . . . 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 1226	. . . . . 6 |- ( ( W e. Word V /\ I e. NN0 /\ I < (♯ ` W ) ) -> 0 < (♯ ` W ) )
31		elnnnn0b 9445	. . . . . 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 1025	. . . . 5 |- ( ( W e. Word V /\ I e. NN0 /\ I < (♯ ` W ) ) -> I < (♯ ` W ) )
34		elfzo0 10420	. . . . 5 |- ( I e. ( 0..^ (♯ ` W ) ) <-> ( I e. NN0 /\ (♯ ` W ) e. NN /\ I < (♯ ` W ) ) )
35	17, 32, 33, 34	syl3anbrc 1207	. . . 4 |- ( ( W e. Word V /\ I e. NN0 /\ I < (♯ ` W ) ) -> I e. ( 0..^ (♯ ` W ) ) )
36	1, 15, 16, 35	syl3anc 1273	. . 3 |- ( ph -> I e. ( 0..^ (♯ ` W ) ) )
37		ccatval1 11173	. . 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 1273	. 2 |- ( ph -> ( ( W ++ ( <" X "> ++ <" Y "> ) ) ` I ) = ( W ` I ) )
39	12, 38	eqtrd 2264	1 |- ( ph -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` I ) = ( W ` I ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   _Vcvv 2802   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   RRcr 8030   0cc0 8031   < clt 8213   <_ cle 8214   NNcn 9142   NN0cn0 9401  ..^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-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147

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-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:  ccat2s1fstg  11224  clwwlknonex2lem2  16288