Theorem reuccatpfxs1lem 11237

Description: Lemma for reuccatpfxs1 11238. (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 2270	. . . . . . 7 |- ( x = U -> ( x e. Word V <-> U e. Word V ) )
2		fveqeq2 5608	. . . . . . 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 2880	. . . . 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 11206	. . . . . . . 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 1250	. . . . . . 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 11111	. . . . . . . . . . . . . . . 16 |- ( s = u -> <" s "> = <" u "> )
14	13	oveq2d 5983	. . . . . . . . . . . . . . 15 |- ( s = u -> ( W ++ <" s "> ) = ( W ++ <" u "> ) )
15	14	eleq1d 2276	. . . . . . . . . . . . . 14 |- ( s = u -> ( ( W ++ <" s "> ) e. X <-> ( W ++ <" u "> ) e. X ) )
16		eqeq2 2217	. . . . . . . . . . . . . 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 2880	. . . . . . . . . . . 12 |- ( u e. V -> ( A. s e. V ( ( W ++ <" s "> ) e. X -> S = s ) -> ( ( W ++ <" u "> ) e. X -> S = u ) ) )
19		eleq1 2270	. . . . . . . . . . . . . 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 2213	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( W ++ <" u "> ) e. X -> S = u ) /\ ( W ++ <" u "> ) e. X ) -> u = S )
23	22	s1eqd 11112	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( W ++ <" u "> ) e. X -> S = u ) /\ ( W ++ <" u "> ) e. X ) -> <" u "> = <" S "> )
24	23	oveq2d 5983	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( W ++ <" u "> ) e. X -> S = u ) /\ ( W ++ <" u "> ) e. X ) -> ( W ++ <" u "> ) = ( W ++ <" S "> ) )
25	24	eqeq2d 2219	. . . . . . . . . . . . . . . . 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 2625	. . . . . . . . 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 1196	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 981   = wceq 1373   e. wcel 2178   A.wral 2486   E.wrex 2487   ` cfv 5290  (class class class)co 5967   1c1 7961   + caddc 7963  ♯chash 10957  Word cword 11031   ++ cconcat 11084   <"cs1 11107   prefix cpfx 11163

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-1o 6525  df-er 6643  df-en 6851  df-dom 6852  df-fin 6853  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-inn 9072  df-n0 9331  df-z 9408  df-uz 9684  df-fz 10166  df-fzo 10300  df-ihash 10958  df-word 11032  df-lsw 11076  df-concat 11085  df-s1 11108  df-substr 11137  df-pfx 11164

This theorem is referenced by:  reuccatpfxs1  11238