Theorem reuccatpfxs1 41434

Description: There is a unique word having the length of a given word increased by 1 with the given word as prefix if there is a unique symbol which extends the given word. Could replace reuccats1 13480. (Contributed by AV, 10-May-2020.)

Assertion

Ref	Expression
reuccatpfxs1	|- ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) -> ( E! v e. V ( W ++ <" v "> ) e. X -> E! w e. X W = ( w prefix ( # ` W ) ) ) )

Distinct variable groups:   v, V, w, x   v, W, w, x   v, X, w, x

Proof of Theorem reuccatpfxs1

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		s1eq 13380	. . . . 5 |- ( v = u -> <" v "> = <" u "> )
2	1	oveq2d 6666	. . . 4 |- ( v = u -> ( W ++ <" v "> ) = ( W ++ <" u "> ) )
3	2	eleq1d 2686	. . 3 |- ( v = u -> ( ( W ++ <" v "> ) e. X <-> ( W ++ <" u "> ) e. X ) )
4	3	reu8 3402	. 2 |- ( E! v e. V ( W ++ <" v "> ) e. X <-> E. v e. V ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) )
5		simprl 794	. . . . 5 |- ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) -> ( W ++ <" v "> ) e. X )
6		simpl 473	. . . . . . . . . 10 |- ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) -> W e. Word V )
7	6	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) -> W e. Word V )
8	7	anim1i 592	. . . . . . . 8 |- ( ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ w e. X ) -> ( W e. Word V /\ w e. X ) )
9		simplrr 801	. . . . . . . 8 |- ( ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ w e. X ) -> A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) )
10		simp-4r 807	. . . . . . . 8 |- ( ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ w e. X ) -> A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) )
11		reuccatpfxs1lem 41433	. . . . . . . 8 |- ( ( ( W e. Word V /\ w e. X ) /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) -> ( W = ( w prefix ( # ` W ) ) -> w = ( W ++ <" v "> ) ) )
12	8, 9, 10, 11	syl3anc 1326	. . . . . . 7 |- ( ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ w e. X ) -> ( W = ( w prefix ( # ` W ) ) -> w = ( W ++ <" v "> ) ) )
13	6	anim1i 592	. . . . . . . . . . . . . . . 16 |- ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) -> ( W e. Word V /\ v e. V ) )
14	13	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) -> ( W e. Word V /\ v e. V ) )
15	14	ad2antrr 762	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ w e. X ) /\ w = ( W ++ <" v "> ) ) -> ( W e. Word V /\ v e. V ) )
16		lswccats1 13411	. . . . . . . . . . . . . 14 |- ( ( W e. Word V /\ v e. V ) -> ( lastS ` ( W ++ <" v "> ) ) = v )
17	15, 16	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ w e. X ) /\ w = ( W ++ <" v "> ) ) -> ( lastS ` ( W ++ <" v "> ) ) = v )
18	17	eqcomd 2628	. . . . . . . . . . . 12 |- ( ( ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ w e. X ) /\ w = ( W ++ <" v "> ) ) -> v = ( lastS ` ( W ++ <" v "> ) ) )
19	18	s1eqd 13381	. . . . . . . . . . 11 |- ( ( ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ w e. X ) /\ w = ( W ++ <" v "> ) ) -> <" v "> = <" ( lastS ` ( W ++ <" v "> ) ) "> )
20	19	oveq2d 6666	. . . . . . . . . 10 |- ( ( ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ w e. X ) /\ w = ( W ++ <" v "> ) ) -> ( W ++ <" v "> ) = ( W ++ <" ( lastS ` ( W ++ <" v "> ) ) "> ) )
21		id 22	. . . . . . . . . . . 12 |- ( w = ( W ++ <" v "> ) -> w = ( W ++ <" v "> ) )
22		fveq2 6191	. . . . . . . . . . . . . 14 |- ( w = ( W ++ <" v "> ) -> ( lastS ` w ) = ( lastS ` ( W ++ <" v "> ) ) )
23	22	s1eqd 13381	. . . . . . . . . . . . 13 |- ( w = ( W ++ <" v "> ) -> <" ( lastS ` w ) "> = <" ( lastS ` ( W ++ <" v "> ) ) "> )
24	23	oveq2d 6666	. . . . . . . . . . . 12 |- ( w = ( W ++ <" v "> ) -> ( W ++ <" ( lastS ` w ) "> ) = ( W ++ <" ( lastS ` ( W ++ <" v "> ) ) "> ) )
25	21, 24	eqeq12d 2637	. . . . . . . . . . 11 |- ( w = ( W ++ <" v "> ) -> ( w = ( W ++ <" ( lastS ` w ) "> ) <-> ( W ++ <" v "> ) = ( W ++ <" ( lastS ` ( W ++ <" v "> ) ) "> ) ) )
26	25	adantl 482	. . . . . . . . . 10 |- ( ( ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ w e. X ) /\ w = ( W ++ <" v "> ) ) -> ( w = ( W ++ <" ( lastS ` w ) "> ) <-> ( W ++ <" v "> ) = ( W ++ <" ( lastS ` ( W ++ <" v "> ) ) "> ) ) )
27	20, 26	mpbird 247	. . . . . . . . 9 |- ( ( ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ w e. X ) /\ w = ( W ++ <" v "> ) ) -> w = ( W ++ <" ( lastS ` w ) "> ) )
28		eleq1 2689	. . . . . . . . . . . . . . . . . . 19 |- ( x = w -> ( x e. Word V <-> w e. Word V ) )
29		fveq2 6191	. . . . . . . . . . . . . . . . . . . 20 |- ( x = w -> ( # ` x ) = ( # ` w ) )
30	29	eqeq1d 2624	. . . . . . . . . . . . . . . . . . 19 |- ( x = w -> ( ( # ` x ) = ( ( # ` W ) + 1 ) <-> ( # ` w ) = ( ( # ` W ) + 1 ) ) )
31	28, 30	anbi12d 747	. . . . . . . . . . . . . . . . . 18 |- ( x = w -> ( ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) <-> ( w e. Word V /\ ( # ` w ) = ( ( # ` W ) + 1 ) ) ) )
32	31	rspcva 3307	. . . . . . . . . . . . . . . . 17 |- ( ( w e. X /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) -> ( w e. Word V /\ ( # ` w ) = ( ( # ` W ) + 1 ) ) )
33		3anass 1042	. . . . . . . . . . . . . . . . . 18 |- ( ( W e. Word V /\ w e. Word V /\ ( # ` w ) = ( ( # ` W ) + 1 ) ) <-> ( W e. Word V /\ ( w e. Word V /\ ( # ` w ) = ( ( # ` W ) + 1 ) ) ) )
34	33	simplbi2com 657	. . . . . . . . . . . . . . . . 17 |- ( ( w e. Word V /\ ( # ` w ) = ( ( # ` W ) + 1 ) ) -> ( W e. Word V -> ( W e. Word V /\ w e. Word V /\ ( # ` w ) = ( ( # ` W ) + 1 ) ) ) )
35	32, 34	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( w e. X /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) -> ( W e. Word V -> ( W e. Word V /\ w e. Word V /\ ( # ` w ) = ( ( # ` W ) + 1 ) ) ) )
36	35	ex 450	. . . . . . . . . . . . . . 15 |- ( w e. X -> ( A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) -> ( W e. Word V -> ( W e. Word V /\ w e. Word V /\ ( # ` w ) = ( ( # ` W ) + 1 ) ) ) ) )
37	36	com13 88	. . . . . . . . . . . . . 14 |- ( W e. Word V -> ( A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) -> ( w e. X -> ( W e. Word V /\ w e. Word V /\ ( # ` w ) = ( ( # ` W ) + 1 ) ) ) ) )
38	37	imp 445	. . . . . . . . . . . . 13 |- ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) -> ( w e. X -> ( W e. Word V /\ w e. Word V /\ ( # ` w ) = ( ( # ` W ) + 1 ) ) ) )
39	38	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) -> ( w e. X -> ( W e. Word V /\ w e. Word V /\ ( # ` w ) = ( ( # ` W ) + 1 ) ) ) )
40	39	imp 445	. . . . . . . . . . 11 |- ( ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ w e. X ) -> ( W e. Word V /\ w e. Word V /\ ( # ` w ) = ( ( # ` W ) + 1 ) ) )
41	40	adantr 481	. . . . . . . . . 10 |- ( ( ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ w e. X ) /\ w = ( W ++ <" v "> ) ) -> ( W e. Word V /\ w e. Word V /\ ( # ` w ) = ( ( # ` W ) + 1 ) ) )
42		ccats1pfxeqbi 41431	. . . . . . . . . 10 |- ( ( W e. Word V /\ w e. Word V /\ ( # ` w ) = ( ( # ` W ) + 1 ) ) -> ( W = ( w prefix ( # ` W ) ) <-> w = ( W ++ <" ( lastS ` w ) "> ) ) )
43	41, 42	syl 17	. . . . . . . . 9 |- ( ( ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ w e. X ) /\ w = ( W ++ <" v "> ) ) -> ( W = ( w prefix ( # ` W ) ) <-> w = ( W ++ <" ( lastS ` w ) "> ) ) )
44	27, 43	mpbird 247	. . . . . . . 8 |- ( ( ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ w e. X ) /\ w = ( W ++ <" v "> ) ) -> W = ( w prefix ( # ` W ) ) )
45	44	ex 450	. . . . . . 7 |- ( ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ w e. X ) -> ( w = ( W ++ <" v "> ) -> W = ( w prefix ( # ` W ) ) ) )
46	12, 45	impbid 202	. . . . . 6 |- ( ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ w e. X ) -> ( W = ( w prefix ( # ` W ) ) <-> w = ( W ++ <" v "> ) ) )
47	46	ralrimiva 2966	. . . . 5 |- ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) -> A. w e. X ( W = ( w prefix ( # ` W ) ) <-> w = ( W ++ <" v "> ) ) )
48		reu6i 3397	. . . . 5 |- ( ( ( W ++ <" v "> ) e. X /\ A. w e. X ( W = ( w prefix ( # ` W ) ) <-> w = ( W ++ <" v "> ) ) ) -> E! w e. X W = ( w prefix ( # ` W ) ) )
49	5, 47, 48	syl2anc 693	. . . 4 |- ( ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) -> E! w e. X W = ( w prefix ( # ` W ) ) )
50	49	ex 450	. . 3 |- ( ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) /\ v e. V ) -> ( ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) -> E! w e. X W = ( w prefix ( # ` W ) ) ) )
51	50	rexlimdva 3031	. 2 |- ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) -> ( E. v e. V ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) -> E! w e. X W = ( w prefix ( # ` W ) ) ) )
52	4, 51	syl5bi 232	1 |- ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) -> ( E! v e. V ( W ++ <" v "> ) e. X -> E! w e. X W = ( w prefix ( # ` W ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   E!wreu 2914   ` cfv 5888  (class class class)co 6650   1c1 9937   + caddc 9939   #chash 13117  Word cword 13291   lastS clsw 13292   ++ cconcat 13293   <"cs1 13294   prefix cpfx 41381

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-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-pfx 41382

This theorem is referenced by: (None)