Theorem reuccatpfxs1lem 11496

Description: Lemma for reuccatpfxs1 11497. (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 5699	. . . . . . 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 11465	. . . . . . . 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 11365	. . . . . . . . . . . . . . . 16 |- ( s = u -> <" s "> = <" u "> )
14	13	oveq2d 6091	. . . . . . . . . . . . . . 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 11366	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( W ++ <" u "> ) e. X -> S = u ) /\ ( W ++ <" u "> ) e. X ) -> <" u "> = <" S "> )
24	23	oveq2d 6091	. . . . . . . . . . . . . . . . . 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 "> ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   ` cfv 5372  (class class class)co 6075   1c1 8170   + caddc 8172  ♯chash 11192  Word cword 11282   ++ cconcat 11336   <"cs1 11361   prefix cpfx 11422

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-ihash 11193  df-word 11283  df-lsw 11328  df-concat 11337  df-s1 11362  df-substr 11396  df-pfx 11423

This theorem is referenced by:  reuccatpfxs1  11497