Theorem reuccatpfxs1lem 11273

Description: Lemma for reuccatpfxs1 11274. (Contributed by Alexander van der Vekens, 5-Oct-2018.) (Revised 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 2292	. . . . . . 7 |- ( x = U -> ( x e. Word V <-> U e. Word V ) )
2		fveqeq2 5635	. . . . . . 7 |- ( x = U -> ( (♯ ` x ) = ( (♯ ` W ) + 1 ) <-> (♯ ` U ) = ( (♯ ` W ) + 1 ) ) )
3	1, 2	anbi12d 473	. . . . . 6 |- ( x = U -> ( ( x e. Word V /\ (♯ ` x ) = ( (♯ ` W ) + 1 ) ) <-> ( U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) ) )
4	3	rspcv 2903	. . . . 5 |- ( U e. X -> ( A. x e. X ( x e. Word V /\ (♯ ` x ) = ( (♯ ` W ) + 1 ) ) -> ( U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) ) )
5	4	adantl 277	. . . 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 ) ) ) )
6		simpl 109	. . . . . . . . 9 |- ( ( W e. Word V /\ U e. X ) -> W e. Word V )
7	6	adantr 276	. . . . . . . 8 |- ( ( ( W e. Word V /\ U e. X ) /\ ( U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) ) -> W e. Word V )
8		simpl 109	. . . . . . . . 9 |- ( ( U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> U e. Word V )
9	8	adantl 277	. . . . . . . 8 |- ( ( ( W e. Word V /\ U e. X ) /\ ( U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) ) -> U e. Word V )
10		simprr 531	. . . . . . . 8 |- ( ( ( W e. Word V /\ U e. X ) /\ ( U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) ) -> (♯ ` U ) = ( (♯ ` W ) + 1 ) )
11		ccats1pfxeqrex 11242	. . . . . . . 8 |- ( ( W e. Word V /\ U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) -> ( W = ( U prefix (♯ ` W ) ) -> E. u e. V U = ( W ++ <" u "> ) ) )
12	7, 9, 10, 11	syl3anc 1271	. . . . . . 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 "> ) ) )
13		s1eq 11147	. . . . . . . . . . . . . . . 16 |- ( s = u -> <" s "> = <" u "> )
14	13	oveq2d 6016	. . . . . . . . . . . . . . 15 |- ( s = u -> ( W ++ <" s "> ) = ( W ++ <" u "> ) )
15	14	eleq1d 2298	. . . . . . . . . . . . . 14 |- ( s = u -> ( ( W ++ <" s "> ) e. X <-> ( W ++ <" u "> ) e. X ) )
16		eqeq2 2239	. . . . . . . . . . . . . 14 |- ( s = u -> ( S = s <-> S = u ) )
17	15, 16	imbi12d 234	. . . . . . . . . . . . 13 |- ( s = u -> ( ( ( W ++ <" s "> ) e. X -> S = s ) <-> ( ( W ++ <" u "> ) e. X -> S = u ) ) )
18	17	rspcv 2903	. . . . . . . . . . . 12 |- ( u e. V -> ( A. s e. V ( ( W ++ <" s "> ) e. X -> S = s ) -> ( ( W ++ <" u "> ) e. X -> S = u ) ) )
19		eleq1 2292	. . . . . . . . . . . . . 14 |- ( U = ( W ++ <" u "> ) -> ( U e. X <-> ( W ++ <" u "> ) e. X ) )
20		id 19	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( W ++ <" u "> ) e. X -> S = u ) -> ( ( W ++ <" u "> ) e. X -> S = u ) )
21	20	imp 124	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( W ++ <" u "> ) e. X -> S = u ) /\ ( W ++ <" u "> ) e. X ) -> S = u )
22	21	eqcomd 2235	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( W ++ <" u "> ) e. X -> S = u ) /\ ( W ++ <" u "> ) e. X ) -> u = S )
23	22	s1eqd 11148	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( W ++ <" u "> ) e. X -> S = u ) /\ ( W ++ <" u "> ) e. X ) -> <" u "> = <" S "> )
24	23	oveq2d 6016	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( W ++ <" u "> ) e. X -> S = u ) /\ ( W ++ <" u "> ) e. X ) -> ( W ++ <" u "> ) = ( W ++ <" S "> ) )
25	24	eqeq2d 2241	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( W ++ <" u "> ) e. X -> S = u ) /\ ( W ++ <" u "> ) e. X ) -> ( U = ( W ++ <" u "> ) <-> U = ( W ++ <" S "> ) ) )
26	25	biimpd 144	. . . . . . . . . . . . . . . 16 |- ( ( ( ( W ++ <" u "> ) e. X -> S = u ) /\ ( W ++ <" u "> ) e. X ) -> ( U = ( W ++ <" u "> ) -> U = ( W ++ <" S "> ) ) )
27	26	ex 115	. . . . . . . . . . . . . . 15 |- ( ( ( W ++ <" u "> ) e. X -> S = u ) -> ( ( W ++ <" u "> ) e. X -> ( U = ( W ++ <" u "> ) -> U = ( W ++ <" S "> ) ) ) )
28	27	com13 80	. . . . . . . . . . . . . 14 |- ( U = ( W ++ <" u "> ) -> ( ( W ++ <" u "> ) e. X -> ( ( ( W ++ <" u "> ) e. X -> S = u ) -> U = ( W ++ <" S "> ) ) ) )
29	19, 28	sylbid 150	. . . . . . . . . . . . 13 |- ( U = ( W ++ <" u "> ) -> ( U e. X -> ( ( ( W ++ <" u "> ) e. X -> S = u ) -> U = ( W ++ <" S "> ) ) ) )
30	29	com3l 81	. . . . . . . . . . . 12 |- ( U e. X -> ( ( ( W ++ <" u "> ) e. X -> S = u ) -> ( U = ( W ++ <" u "> ) -> U = ( W ++ <" S "> ) ) ) )
31	18, 30	sylan9r 410	. . . . . . . . . . 11 |- ( ( U e. X /\ u e. V ) -> ( A. s e. V ( ( W ++ <" s "> ) e. X -> S = s ) -> ( U = ( W ++ <" u "> ) -> U = ( W ++ <" S "> ) ) ) )
32	31	com23 78	. . . . . . . . . 10 |- ( ( U e. X /\ u e. V ) -> ( U = ( W ++ <" u "> ) -> ( A. s e. V ( ( W ++ <" s "> ) e. X -> S = s ) -> U = ( W ++ <" S "> ) ) ) )
33	32	rexlimdva 2648	. . . . . . . . 9 |- ( U e. X -> ( E. u e. V U = ( W ++ <" u "> ) -> ( A. s e. V ( ( W ++ <" s "> ) e. X -> S = s ) -> U = ( W ++ <" S "> ) ) ) )
34	33	adantl 277	. . . . . . . 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 "> ) ) ) )
35	34	adantr 276	. . . . . . 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 "> ) ) ) )
36	12, 35	syld 45	. . . . . 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 "> ) ) ) )
37	36	com23 78	. . . . 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 "> ) ) ) )
38	37	ex 115	. . . 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 "> ) ) ) ) )
39	5, 38	syld 45	. . 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 "> ) ) ) ) )
40	39	com23 78	. 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 "> ) ) ) ) )
41	40	3imp 1217	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 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   ` cfv 5317  (class class class)co 6000   1c1 7996   + caddc 7998  ♯chash 10992  Word cword 11066   ++ cconcat 11120   <"cs1 11143   prefix cpfx 11199

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 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-1o 6560  df-er 6678  df-en 6886  df-dom 6887  df-fin 6888  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-fz 10201  df-fzo 10335  df-ihash 10993  df-word 11067  df-lsw 11112  df-concat 11121  df-s1 11144  df-substr 11173  df-pfx 11200

This theorem is referenced by:  reuccatpfxs1  11274