Theorem reuccatpfxs1lem 11518

Description: Lemma for reuccatpfxs1 11519. (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 2301	. . . . . . 7 |- ( x = U -> ( x e. Word V <-> U e. Word V ) )
2		fveqeq2 5704	. . . . . . 7 |- ( x = U -> ( (♯ ` x ) = ( (♯ ` W ) + 1 ) <-> (♯ ` U ) = ( (♯ ` W ) + 1 ) ) )
3	1, 2	anbi12d 477	. . . . . 6 |- ( x = U -> ( ( x e. Word V /\ (♯ ` x ) = ( (♯ ` W ) + 1 ) ) <-> ( U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) ) )
4	3	rspcv 2925	. . . . 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 537	. . . . . . . 8 |- ( ( ( W e. Word V /\ U e. X ) /\ ( U e. Word V /\ (♯ ` U ) = ( (♯ ` W ) + 1 ) ) ) -> (♯ ` U ) = ( (♯ ` W ) + 1 ) )
11		ccats1pfxeqrex 11487	. . . . . . . 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 1278	. . . . . . 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 11387	. . . . . . . . . . . . . . . 16 |- ( s = u -> <" s "> = <" u "> )
14	13	oveq2d 6101	. . . . . . . . . . . . . . 15 |- ( s = u -> ( W ++ <" s "> ) = ( W ++ <" u "> ) )
15	14	eleq1d 2307	. . . . . . . . . . . . . 14 |- ( s = u -> ( ( W ++ <" s "> ) e. X <-> ( W ++ <" u "> ) e. X ) )
16		eqeq2 2248	. . . . . . . . . . . . . 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 2925	. . . . . . . . . . . 12 |- ( u e. V -> ( A. s e. V ( ( W ++ <" s "> ) e. X -> S = s ) -> ( ( W ++ <" u "> ) e. X -> S = u ) ) )
19		eleq1 2301	. . . . . . . . . . . . . 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 2244	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( W ++ <" u "> ) e. X -> S = u ) /\ ( W ++ <" u "> ) e. X ) -> u = S )
23	22	s1eqd 11388	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( W ++ <" u "> ) e. X -> S = u ) /\ ( W ++ <" u "> ) e. X ) -> <" u "> = <" S "> )
24	23	oveq2d 6101	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( W ++ <" u "> ) e. X -> S = u ) /\ ( W ++ <" u "> ) e. X ) -> ( W ++ <" u "> ) = ( W ++ <" S "> ) )
25	24	eqeq2d 2250	. . . . . . . . . . . . . . . . 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 414	. . . . . . . . . . 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 2668	. . . . . . . . 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 1224	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 "> ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   ` cfv 5377  (class class class)co 6085   1c1 8180   + caddc 8182  ♯chash 11214  Word cword 11304   ++ cconcat 11358   <"cs1 11383   prefix cpfx 11444

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412  df-fzo 10550  df-ihash 11215  df-word 11305  df-lsw 11350  df-concat 11359  df-s1 11384  df-substr 11418  df-pfx 11445

This theorem is used by:  reuccatpfxs1  11519