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Theorem ccat2s1fvw 12634
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 997 . . . 4  |-  ( ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W ) )  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  W  e. Word  V )
2 simprl 754 . . . 4  |-  ( ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W ) )  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  X  e.  V )
3 simpr 459 . . . . 5  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  Y  e.  V )
43adantl 464 . . . 4  |-  ( ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W ) )  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  Y  e.  V )
5 ccatw2s1ass 12626 . . . 4  |-  ( ( W  e. Word  V  /\  X  e.  V  /\  Y  e.  V )  ->  ( ( W ++  <" X "> ) ++  <" Y "> )  =  ( W ++  ( <" X "> ++  <" Y "> ) ) )
61, 2, 4, 5syl3anc 1226 . . 3  |-  ( ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W ) )  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  (
( W ++  <" X "> ) ++  <" Y "> )  =  ( W ++  ( <" X "> ++  <" Y "> ) ) )
76fveq1d 5850 . 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 12617 . . . 4  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  ( <" X "> ++  <" Y "> )  e. Word  V )
98adantl 464 . . 3  |-  ( ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W ) )  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  ( <" X "> ++  <" Y "> )  e. Word  V )
10 simp2 995 . . . . . 6  |-  ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W
) )  ->  I  e.  NN0 )
11 lencl 12552 . . . . . . . 8  |-  ( W  e. Word  V  ->  ( # `
 W )  e. 
NN0 )
12113ad2ant1 1015 . . . . . . 7  |-  ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W
) )  ->  ( # `
 W )  e. 
NN0 )
13 nn0ge0 10817 . . . . . . . . . 10  |-  ( I  e.  NN0  ->  0  <_  I )
1413adantl 464 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  I  e.  NN0 )  -> 
0  <_  I )
15 0red 9586 . . . . . . . . . 10  |-  ( ( W  e. Word  V  /\  I  e.  NN0 )  -> 
0  e.  RR )
16 simpr 459 . . . . . . . . . . 11  |-  ( ( W  e. Word  V  /\  I  e.  NN0 )  ->  I  e.  NN0 )
1716nn0red 10849 . . . . . . . . . 10  |-  ( ( W  e. Word  V  /\  I  e.  NN0 )  ->  I  e.  RR )
1811nn0red 10849 . . . . . . . . . . 11  |-  ( W  e. Word  V  ->  ( # `
 W )  e.  RR )
1918adantr 463 . . . . . . . . . 10  |-  ( ( W  e. Word  V  /\  I  e.  NN0 )  -> 
( # `  W )  e.  RR )
20 lelttr 9664 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  I  e.  RR  /\  ( # `
 W )  e.  RR )  ->  (
( 0  <_  I  /\  I  <  ( # `  W ) )  -> 
0  <  ( # `  W
) ) )
2115, 17, 19, 20syl3anc 1226 . . . . . . . . 9  |-  ( ( W  e. Word  V  /\  I  e.  NN0 )  -> 
( ( 0  <_  I  /\  I  <  ( # `
 W ) )  ->  0  <  ( # `
 W ) ) )
2214, 21mpand 673 . . . . . . . 8  |-  ( ( W  e. Word  V  /\  I  e.  NN0 )  -> 
( I  <  ( # `
 W )  -> 
0  <  ( # `  W
) ) )
23223impia 1191 . . . . . . 7  |-  ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W
) )  ->  0  <  ( # `  W
) )
24 elnnnn0b 10836 . . . . . . 7  |-  ( (
# `  W )  e.  NN  <->  ( ( # `  W )  e.  NN0  /\  0  <  ( # `  W ) ) )
2512, 23, 24sylanbrc 662 . . . . . 6  |-  ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W
) )  ->  ( # `
 W )  e.  NN )
26 simp3 996 . . . . . 6  |-  ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W
) )  ->  I  <  ( # `  W
) )
2710, 25, 263jca 1174 . . . . 5  |-  ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W
) )  ->  (
I  e.  NN0  /\  ( # `  W )  e.  NN  /\  I  <  ( # `  W
) ) )
2827adantr 463 . . . 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 11840 . . . 4  |-  ( I  e.  ( 0..^ (
# `  W )
)  <->  ( I  e. 
NN0  /\  ( # `  W
)  e.  NN  /\  I  <  ( # `  W
) ) )
3028, 29sylibr 212 . . 3  |-  ( ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W ) )  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  I  e.  ( 0..^ ( # `  W ) ) )
31 ccatval1 12587 . . 3  |-  ( ( W  e. Word  V  /\  ( <" X "> ++  <" Y "> )  e. Word  V  /\  I  e.  ( 0..^ ( # `  W
) ) )  -> 
( ( W ++  ( <" X "> ++  <" Y "> ) ) `  I
)  =  ( W `
 I ) )
321, 9, 30, 31syl3anc 1226 . 2  |-  ( ( ( W  e. Word  V  /\  I  e.  NN0  /\  I  <  ( # `  W ) )  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  (
( W ++  ( <" X "> ++  <" Y "> ) ) `  I
)  =  ( W `
 I ) )
337, 32eqtrd 2495 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   RRcr 9480   0cc0 9481    < clt 9617    <_ cle 9618   NNcn 10531   NN0cn0 10791  ..^cfzo 11799   #chash 12390  Word cword 12521   ++ cconcat 12523   <"cs1 12524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12391  df-word 12529  df-concat 12531  df-s1 12532
This theorem is referenced by:  ccat2s1fst  12635  numclwwlkovf2ex  25291
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