Theorem ccatw2s1p1g 11209

Description: Extract the symbol of the first singleton word of a word concatenated with this singleton word and another singleton word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Proof shortened by AV, 1-May-2020.) (Revised by AV, 1-May-2020.) (Revised by AV, 29-Jan-2024.)

Assertion

Ref	Expression
ccatw2s1p1g	|- ( ( ( W e. Word V /\ (♯ ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` N ) = X )

Proof of Theorem ccatw2s1p1g

Step	Hyp	Ref	Expression
1		ccatws1cl 11196	. . . 4 |- ( ( W e. Word V /\ X e. V ) -> ( W ++ <" X "> ) e. Word V )
2	1	ad2ant2r 509	. . 3 |- ( ( ( W e. Word V /\ (♯ ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( W ++ <" X "> ) e. Word V )
3		simprr 531	. . 3 |- ( ( ( W e. Word V /\ (♯ ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> Y e. V )
4		lencl 11104	. . . . . . 7 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
5		fzonn0p1 10444	. . . . . . 7 |- ( (♯ ` W ) e. NN0 -> (♯ ` W ) e. ( 0..^ ( (♯ ` W ) + 1 ) ) )
6	4, 5	syl 14	. . . . . 6 |- ( W e. Word V -> (♯ ` W ) e. ( 0..^ ( (♯ ` W ) + 1 ) ) )
7	6	adantr 276	. . . . 5 |- ( ( W e. Word V /\ (♯ ` W ) = N ) -> (♯ ` W ) e. ( 0..^ ( (♯ ` W ) + 1 ) ) )
8	7	adantr 276	. . . 4 |- ( ( ( W e. Word V /\ (♯ ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> (♯ ` W ) e. ( 0..^ ( (♯ ` W ) + 1 ) ) )
9		simpr 110	. . . . . 6 |- ( ( W e. Word V /\ (♯ ` W ) = N ) -> (♯ ` W ) = N )
10	9	eqcomd 2235	. . . . 5 |- ( ( W e. Word V /\ (♯ ` W ) = N ) -> N = (♯ ` W ) )
11	10	adantr 276	. . . 4 |- ( ( ( W e. Word V /\ (♯ ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> N = (♯ ` W ) )
12		ccatws1leng 11198	. . . . . 6 |- ( ( W e. Word V /\ X e. V ) -> (♯ ` ( W ++ <" X "> ) ) = ( (♯ ` W ) + 1 ) )
13	12	ad2ant2r 509	. . . . 5 |- ( ( ( W e. Word V /\ (♯ ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> (♯ ` ( W ++ <" X "> ) ) = ( (♯ ` W ) + 1 ) )
14	13	oveq2d 6027	. . . 4 |- ( ( ( W e. Word V /\ (♯ ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( 0..^ (♯ ` ( W ++ <" X "> ) ) ) = ( 0..^ ( (♯ ` W ) + 1 ) ) )
15	8, 11, 14	3eltr4d 2313	. . 3 |- ( ( ( W e. Word V /\ (♯ ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> N e. ( 0..^ (♯ ` ( W ++ <" X "> ) ) ) )
16		ccats1val1g 11203	. . 3 |- ( ( ( W ++ <" X "> ) e. Word V /\ Y e. V /\ N e. ( 0..^ (♯ ` ( W ++ <" X "> ) ) ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` N ) = ( ( W ++ <" X "> ) ` N ) )
17	2, 3, 15, 16	syl3anc 1271	. 2 |- ( ( ( W e. Word V /\ (♯ ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` N ) = ( ( W ++ <" X "> ) ` N ) )
18		simpll 527	. . 3 |- ( ( ( W e. Word V /\ (♯ ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> W e. Word V )
19		simprl 529	. . 3 |- ( ( ( W e. Word V /\ (♯ ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> X e. V )
20		ccats1val2 11204	. . 3 |- ( ( W e. Word V /\ X e. V /\ N = (♯ ` W ) ) -> ( ( W ++ <" X "> ) ` N ) = X )
21	18, 19, 11, 20	syl3anc 1271	. 2 |- ( ( ( W e. Word V /\ (♯ ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( ( W ++ <" X "> ) ` N ) = X )
22	17, 21	eqtrd 2262	1 |- ( ( ( W e. Word V /\ (♯ ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` N ) = X )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   ` cfv 5322  (class class class)co 6011   0cc0 8020   1c1 8021   + caddc 8023   NN0cn0 9390  ..^cfzo 10365  ♯chash 11025  Word cword 11100   ++ cconcat 11154   <"cs1 11179

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4200  ax-sep 4203  ax-nul 4211  ax-pow 4260  ax-pr 4295  ax-un 4526  ax-setind 4631  ax-iinf 4682  ax-cnex 8111  ax-resscn 8112  ax-1cn 8113  ax-1re 8114  ax-icn 8115  ax-addcl 8116  ax-addrcl 8117  ax-mulcl 8118  ax-addcom 8120  ax-addass 8122  ax-distr 8124  ax-i2m1 8125  ax-0lt1 8126  ax-0id 8128  ax-rnegex 8129  ax-cnre 8131  ax-pre-ltirr 8132  ax-pre-ltwlin 8133  ax-pre-lttrn 8134  ax-pre-apti 8135  ax-pre-ltadd 8136

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-int 3925  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-tr 4184  df-id 4386  df-iord 4459  df-on 4461  df-ilim 4462  df-suc 4464  df-iom 4685  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-riota 5964  df-ov 6014  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297  df-recs 6464  df-frec 6550  df-1o 6575  df-er 6695  df-en 6903  df-dom 6904  df-fin 6905  df-pnf 8204  df-mnf 8205  df-xr 8206  df-ltxr 8207  df-le 8208  df-sub 8340  df-neg 8341  df-inn 9132  df-n0 9391  df-z 9468  df-uz 9744  df-fz 10232  df-fzo 10366  df-ihash 11026  df-word 11101  df-concat 11155  df-s1 11180

This theorem is referenced by:  clwwlknonex2lem2  16223