Theorem ccatw2s1p1 13413

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.)

Assertion

Ref	Expression
ccatw2s1p1	|- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` N ) = X )

Proof of Theorem ccatw2s1p1

Step	Hyp	Ref	Expression
1		ccatws1cl 13396	. . . 4 |- ( ( W e. Word V /\ X e. V ) -> ( W ++ <" X "> ) e. Word V )
2	1	ad2ant2r 783	. . 3 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( W ++ <" X "> ) e. Word V )
3		simpr 477	. . . 4 |- ( ( X e. V /\ Y e. V ) -> Y e. V )
4	3	adantl 482	. . 3 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> Y e. V )
5		lencl 13324	. . . . . . 7 |- ( W e. Word V -> ( # ` W ) e. NN0 )
6		fzonn0p1 12544	. . . . . . 7 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. ( 0..^ ( ( # ` W ) + 1 ) ) )
7	5, 6	syl 17	. . . . . 6 |- ( W e. Word V -> ( # ` W ) e. ( 0..^ ( ( # ` W ) + 1 ) ) )
8	7	adantr 481	. . . . 5 |- ( ( W e. Word V /\ ( # ` W ) = N ) -> ( # ` W ) e. ( 0..^ ( ( # ` W ) + 1 ) ) )
9	8	adantr 481	. . . 4 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( # ` W ) e. ( 0..^ ( ( # ` W ) + 1 ) ) )
10		simpr 477	. . . . . 6 |- ( ( W e. Word V /\ ( # ` W ) = N ) -> ( # ` W ) = N )
11	10	eqcomd 2628	. . . . 5 |- ( ( W e. Word V /\ ( # ` W ) = N ) -> N = ( # ` W ) )
12	11	adantr 481	. . . 4 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> N = ( # ` W ) )
13		ccatws1len 13398	. . . . . 6 |- ( ( W e. Word V /\ X e. V ) -> ( # ` ( W ++ <" X "> ) ) = ( ( # ` W ) + 1 ) )
14	13	ad2ant2r 783	. . . . 5 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( # ` ( W ++ <" X "> ) ) = ( ( # ` W ) + 1 ) )
15	14	oveq2d 6666	. . . 4 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( 0..^ ( # ` ( W ++ <" X "> ) ) ) = ( 0..^ ( ( # ` W ) + 1 ) ) )
16	9, 12, 15	3eltr4d 2716	. . 3 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> N e. ( 0..^ ( # ` ( W ++ <" X "> ) ) ) )
17		ccats1val1 13403	. . 3 |- ( ( ( W ++ <" X "> ) e. Word V /\ Y e. V /\ N e. ( 0..^ ( # ` ( W ++ <" X "> ) ) ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` N ) = ( ( W ++ <" X "> ) ` N ) )
18	2, 4, 16, 17	syl3anc 1326	. 2 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` N ) = ( ( W ++ <" X "> ) ` N ) )
19		simpl 473	. . . 4 |- ( ( W e. Word V /\ ( # ` W ) = N ) -> W e. Word V )
20	19	adantr 481	. . 3 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> W e. Word V )
21		simpl 473	. . . 4 |- ( ( X e. V /\ Y e. V ) -> X e. V )
22	21	adantl 482	. . 3 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> X e. V )
23		eqcom 2629	. . . . . 6 |- ( ( # ` W ) = N <-> N = ( # ` W ) )
24	23	biimpi 206	. . . . 5 |- ( ( # ` W ) = N -> N = ( # ` W ) )
25	24	adantl 482	. . . 4 |- ( ( W e. Word V /\ ( # ` W ) = N ) -> N = ( # ` W ) )
26	25	adantr 481	. . 3 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> N = ( # ` W ) )
27		ccats1val2 13404	. . 3 |- ( ( W e. Word V /\ X e. V /\ N = ( # ` W ) ) -> ( ( W ++ <" X "> ) ` N ) = X )
28	20, 22, 26, 27	syl3anc 1326	. 2 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( ( W ++ <" X "> ) ` N ) = X )
29	18, 28	eqtrd 2656	1 |- ( ( ( W e. Word V /\ ( # ` W ) = N ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` N ) = X )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   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:  numclwwlkovf2exlem2  27212  numclwlk1lem2foalem  27222