Theorem reuccatpfxs1lem 41433

Description: Lemma for reuccatpfxs1 41434. Could replace reuccats1lem 13479. (Contributed by AV, 9-May-2020.)

Assertion

Ref	Expression
reuccatpfxs1lem	|- ( ( ( W e. Word V /\ U e. X ) /\ A. s e. V ( ( W ++ <" s "> ) e. X -> S = s ) /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) -> ( W = ( U prefix ( # ` W ) ) -> U = ( W ++ <" S "> ) ) )

Distinct variable groups:   S, s   x, U   V, s, x   W, s, x   X, s, x

Allowed substitution hints:   S( x)   U( s)

Proof of Theorem reuccatpfxs1lem

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . . . . 7 |- ( x = U -> ( x e. Word V <-> U e. Word V ) )
2		fveq2 6191	. . . . . . . 8 |- ( x = U -> ( # ` x ) = ( # ` U ) )
3	2	eqeq1d 2624	. . . . . . 7 |- ( x = U -> ( ( # ` x ) = ( ( # ` W ) + 1 ) <-> ( # ` U ) = ( ( # ` W ) + 1 ) ) )
4	1, 3	anbi12d 747	. . . . . 6 |- ( x = U -> ( ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) <-> ( U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) ) )
5	4	rspcv 3305	. . . . 5 |- ( U e. X -> ( A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) -> ( U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) ) )
6	5	adantl 482	. . . 4 |- ( ( W e. Word V /\ U e. X ) -> ( A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) -> ( U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) ) )
7		simpl 473	. . . . . . . . 9 |- ( ( W e. Word V /\ U e. X ) -> W e. Word V )
8	7	adantr 481	. . . . . . . 8 |- ( ( ( W e. Word V /\ U e. X ) /\ ( U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) ) -> W e. Word V )
9		simpl 473	. . . . . . . . 9 |- ( ( U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) -> U e. Word V )
10	9	adantl 482	. . . . . . . 8 |- ( ( ( W e. Word V /\ U e. X ) /\ ( U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) ) -> U e. Word V )
11		simprr 796	. . . . . . . 8 |- ( ( ( W e. Word V /\ U e. X ) /\ ( U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) ) -> ( # ` U ) = ( ( # ` W ) + 1 ) )
12		ccats1pfxeqrex 41422	. . . . . . . 8 |- ( ( W e. Word V /\ U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) -> ( W = ( U prefix ( # ` W ) ) -> E. u e. V U = ( W ++ <" u "> ) ) )
13	8, 10, 11, 12	syl3anc 1326	. . . . . . 7 |- ( ( ( W e. Word V /\ U e. X ) /\ ( U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) ) -> ( W = ( U prefix ( # ` W ) ) -> E. u e. V U = ( W ++ <" u "> ) ) )
14		s1eq 13380	. . . . . . . . . . . . . . . 16 |- ( s = u -> <" s "> = <" u "> )
15	14	oveq2d 6666	. . . . . . . . . . . . . . 15 |- ( s = u -> ( W ++ <" s "> ) = ( W ++ <" u "> ) )
16	15	eleq1d 2686	. . . . . . . . . . . . . 14 |- ( s = u -> ( ( W ++ <" s "> ) e. X <-> ( W ++ <" u "> ) e. X ) )
17		eqeq2 2633	. . . . . . . . . . . . . 14 |- ( s = u -> ( S = s <-> S = u ) )
18	16, 17	imbi12d 334	. . . . . . . . . . . . 13 |- ( s = u -> ( ( ( W ++ <" s "> ) e. X -> S = s ) <-> ( ( W ++ <" u "> ) e. X -> S = u ) ) )
19	18	rspcv 3305	. . . . . . . . . . . 12 |- ( u e. V -> ( A. s e. V ( ( W ++ <" s "> ) e. X -> S = s ) -> ( ( W ++ <" u "> ) e. X -> S = u ) ) )
20		eleq1 2689	. . . . . . . . . . . . . 14 |- ( U = ( W ++ <" u "> ) -> ( U e. X <-> ( W ++ <" u "> ) e. X ) )
21		id 22	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( W ++ <" u "> ) e. X -> S = u ) -> ( ( W ++ <" u "> ) e. X -> S = u ) )
22	21	imp 445	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( W ++ <" u "> ) e. X -> S = u ) /\ ( W ++ <" u "> ) e. X ) -> S = u )
23	22	eqcomd 2628	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( W ++ <" u "> ) e. X -> S = u ) /\ ( W ++ <" u "> ) e. X ) -> u = S )
24	23	s1eqd 13381	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( W ++ <" u "> ) e. X -> S = u ) /\ ( W ++ <" u "> ) e. X ) -> <" u "> = <" S "> )
25	24	oveq2d 6666	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( W ++ <" u "> ) e. X -> S = u ) /\ ( W ++ <" u "> ) e. X ) -> ( W ++ <" u "> ) = ( W ++ <" S "> ) )
26	25	eqeq2d 2632	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( W ++ <" u "> ) e. X -> S = u ) /\ ( W ++ <" u "> ) e. X ) -> ( U = ( W ++ <" u "> ) <-> U = ( W ++ <" S "> ) ) )
27	26	biimpd 219	. . . . . . . . . . . . . . . 16 |- ( ( ( ( W ++ <" u "> ) e. X -> S = u ) /\ ( W ++ <" u "> ) e. X ) -> ( U = ( W ++ <" u "> ) -> U = ( W ++ <" S "> ) ) )
28	27	ex 450	. . . . . . . . . . . . . . 15 |- ( ( ( W ++ <" u "> ) e. X -> S = u ) -> ( ( W ++ <" u "> ) e. X -> ( U = ( W ++ <" u "> ) -> U = ( W ++ <" S "> ) ) ) )
29	28	com13 88	. . . . . . . . . . . . . 14 |- ( U = ( W ++ <" u "> ) -> ( ( W ++ <" u "> ) e. X -> ( ( ( W ++ <" u "> ) e. X -> S = u ) -> U = ( W ++ <" S "> ) ) ) )
30	20, 29	sylbid 230	. . . . . . . . . . . . 13 |- ( U = ( W ++ <" u "> ) -> ( U e. X -> ( ( ( W ++ <" u "> ) e. X -> S = u ) -> U = ( W ++ <" S "> ) ) ) )
31	30	com3l 89	. . . . . . . . . . . 12 |- ( U e. X -> ( ( ( W ++ <" u "> ) e. X -> S = u ) -> ( U = ( W ++ <" u "> ) -> U = ( W ++ <" S "> ) ) ) )
32	19, 31	sylan9r 690	. . . . . . . . . . 11 |- ( ( U e. X /\ u e. V ) -> ( A. s e. V ( ( W ++ <" s "> ) e. X -> S = s ) -> ( U = ( W ++ <" u "> ) -> U = ( W ++ <" S "> ) ) ) )
33	32	com23 86	. . . . . . . . . 10 |- ( ( U e. X /\ u e. V ) -> ( U = ( W ++ <" u "> ) -> ( A. s e. V ( ( W ++ <" s "> ) e. X -> S = s ) -> U = ( W ++ <" S "> ) ) ) )
34	33	rexlimdva 3031	. . . . . . . . 9 |- ( U e. X -> ( E. u e. V U = ( W ++ <" u "> ) -> ( A. s e. V ( ( W ++ <" s "> ) e. X -> S = s ) -> U = ( W ++ <" S "> ) ) ) )
35	34	adantl 482	. . . . . . . 8 |- ( ( W e. Word V /\ U e. X ) -> ( E. u e. V U = ( W ++ <" u "> ) -> ( A. s e. V ( ( W ++ <" s "> ) e. X -> S = s ) -> U = ( W ++ <" S "> ) ) ) )
36	35	adantr 481	. . . . . . 7 |- ( ( ( W e. Word V /\ U e. X ) /\ ( U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) ) -> ( E. u e. V U = ( W ++ <" u "> ) -> ( A. s e. V ( ( W ++ <" s "> ) e. X -> S = s ) -> U = ( W ++ <" S "> ) ) ) )
37	13, 36	syld 47	. . . . . 6 |- ( ( ( W e. Word V /\ U e. X ) /\ ( U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) ) -> ( W = ( U prefix ( # ` W ) ) -> ( A. s e. V ( ( W ++ <" s "> ) e. X -> S = s ) -> U = ( W ++ <" S "> ) ) ) )
38	37	com23 86	. . . . 5 |- ( ( ( W e. Word V /\ U e. X ) /\ ( U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) ) -> ( A. s e. V ( ( W ++ <" s "> ) e. X -> S = s ) -> ( W = ( U prefix ( # ` W ) ) -> U = ( W ++ <" S "> ) ) ) )
39	38	ex 450	. . . 4 |- ( ( W e. Word V /\ U e. X ) -> ( ( U e. Word V /\ ( # ` U ) = ( ( # ` W ) + 1 ) ) -> ( A. s e. V ( ( W ++ <" s "> ) e. X -> S = s ) -> ( W = ( U prefix ( # ` W ) ) -> U = ( W ++ <" S "> ) ) ) ) )
40	6, 39	syld 47	. . 3 |- ( ( W e. Word V /\ U e. X ) -> ( A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) -> ( A. s e. V ( ( W ++ <" s "> ) e. X -> S = s ) -> ( W = ( U prefix ( # ` W ) ) -> U = ( W ++ <" S "> ) ) ) ) )
41	40	com23 86	. 2 |- ( ( W e. Word V /\ U e. X ) -> ( A. s e. V ( ( W ++ <" s "> ) e. X -> S = s ) -> ( A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) -> ( W = ( U prefix ( # ` W ) ) -> U = ( W ++ <" S "> ) ) ) ) )
42	41	3imp 1256	1 |- ( ( ( W e. Word V /\ U e. X ) /\ A. s e. V ( ( W ++ <" s "> ) e. X -> S = s ) /\ A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) ) -> ( W = ( U prefix ( # ` W ) ) -> U = ( W ++ <" S "> ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   ` cfv 5888  (class class class)co 6650   1c1 9937   + caddc 9939   #chash 13117  Word cword 13291   ++ 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-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:  reuccatpfxs1  41434