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Theorem ccat2s1fvw 13415
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.)
Assertion
Ref Expression
ccat2s1fvw |- ( ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` I ) = ( W ` I ) )
Proof of Theorem ccat2s1fvw
StepHypRef Expression
1 simpl1 1064 . . . 4 |- ( ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> W e. Word V )
2 simprl 794 . . . 4 |- ( ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> X e. V )
3 simpr 477 . . . . 5 |- ( ( X e. V /\ Y e. V ) -> Y e. V )
43adantl 482 . . . 4 |- ( ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> Y e. V )
5 ccatw2s1ass 13407 . . . 4 |- ( ( W e. Word V /\ X e. V /\ Y e. V ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) = ( W ++ ( <" X "> ++ <" Y "> ) ) )
61, 2, 4, 5syl3anc 1326 . . 3 |- ( ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) = ( W ++ ( <" X "> ++ <" Y "> ) ) )
76fveq1d 6193 . 2 |- ( ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` I ) = ( ( W ++ ( <" X "> ++ <" Y "> ) ) ` I ) )
8 ccat2s1cl 13397 . . . 4 |- ( ( X e. V /\ Y e. V ) -> ( <" X "> ++ <" Y "> ) e. Word V )
98adantl 482 . . 3 |- ( ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( <" X "> ++ <" Y "> ) e. Word V )
10 simp2 1062 . . . . . 6 |- ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) -> I e. NN0 )
11 lencl 13324 . . . . . . . 8 |- ( W e. Word V -> ( # ` W ) e. NN0 )
12113ad2ant1 1082 . . . . . . 7 |- ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) -> ( # ` W ) e. NN0 )
13 nn0ge0 11318 . . . . . . . . . 10 |- ( I e. NN0 -> 0 <_ I )
1413adantl 482 . . . . . . . . 9 |- ( ( W e. Word V /\ I e. NN0 ) -> 0 <_ I )
15 0red 10041 . . . . . . . . . 10 |- ( ( W e. Word V /\ I e. NN0 ) -> 0 e. RR )
16 simpr 477 . . . . . . . . . . 11 |- ( ( W e. Word V /\ I e. NN0 ) -> I e. NN0 )
1716nn0red 11352 . . . . . . . . . 10 |- ( ( W e. Word V /\ I e. NN0 ) -> I e. RR )
1811nn0red 11352 . . . . . . . . . . 11 |- ( W e. Word V -> ( # ` W ) e. RR )
1918adantr 481 . . . . . . . . . 10 |- ( ( W e. Word V /\ I e. NN0 ) -> ( # ` W ) e. RR )
20 lelttr 10128 . . . . . . . . . 10 |- ( ( 0 e. RR /\ I e. RR /\ ( # ` W ) e. RR ) -> ( ( 0 <_ I /\ I < ( # ` W ) ) -> 0 < ( # ` W ) ) )
2115, 17, 19, 20syl3anc 1326 . . . . . . . . 9 |- ( ( W e. Word V /\ I e. NN0 ) -> ( ( 0 <_ I /\ I < ( # ` W ) ) -> 0 < ( # ` W ) ) )
2214, 21mpand 711 . . . . . . . 8 |- ( ( W e. Word V /\ I e. NN0 ) -> ( I < ( # ` W ) -> 0 < ( # ` W ) ) )
23223impia 1261 . . . . . . 7 |- ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) -> 0 < ( # ` W ) )
24 elnnnn0b 11337 . . . . . . 7 |- ( ( # ` W ) e. NN <-> ( ( # ` W ) e. NN0 /\ 0 < ( # ` W ) ) )
2512, 23, 24sylanbrc 698 . . . . . 6 |- ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) -> ( # ` W ) e. NN )
26 simp3 1063 . . . . . 6 |- ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) -> I < ( # ` W ) )
2710, 25, 263jca 1242 . . . . 5 |- ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) -> ( I e. NN0 /\ ( # ` W ) e. NN /\ I < ( # ` W ) ) )
2827adantr 481 . . . 4 |- ( ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( I e. NN0 /\ ( # ` W ) e. NN /\ I < ( # ` W ) ) )
29 elfzo0 12508 . . . 4 |- ( I e. ( 0..^ ( # ` W ) ) <-> ( I e. NN0 /\ ( # ` W ) e. NN /\ I < ( # ` W ) ) )
3028, 29sylibr 224 . . 3 |- ( ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> I e. ( 0..^ ( # ` W ) ) )
31 ccatval1 13361 . . 3 |- ( ( W e. Word V /\ ( <" X "> ++ <" Y "> ) e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> ( ( W ++ ( <" X "> ++ <" Y "> ) ) ` I ) = ( W ` I ) )
321, 9, 30, 31syl3anc 1326 . 2 |- ( ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( W ++ ( <" X "> ++ <" Y "> ) ) ` I ) = ( W ` I ) )
337, 32eqtrd 2656 1 |- ( ( ( W e. Word V /\ I e. NN0 /\ I < ( # ` W ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` I ) = ( W ` I ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   < clt 10074   <_ cle 10075   NNcn 11020   NN0cn0 11292  ..^cfzo 12465   #chash 13117  Word cword 13291   ++ cconcat 13293   <"cs1 13294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302
This theorem is referenced by:  ccat2s1fst  13416  numclwwlkovf2exlem2  27212
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