Theorem pfxccatpfx2 11454

Description: A prefix of a concatenation of two words being the first word concatenated with a prefix of the second word. (Contributed by AV, 10-May-2020.)

Hypotheses

Ref	Expression
swrdccatin2.l	|- L = (♯ ` A )
pfxccatpfx2.m	|- M = (♯ ` B )

Assertion

Ref	Expression
pfxccatpfx2	|- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> ( ( A ++ B ) prefix N ) = ( A ++ ( B prefix ( N - L ) ) ) )

Proof of Theorem pfxccatpfx2

Step	Hyp	Ref	Expression
1		ccatcl 11306	. . . 4 |- ( ( A e. Word V /\ B e. Word V ) -> ( A ++ B ) e. Word V )
2	1	3adant3 1044	. . 3 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> ( A ++ B ) e. Word V )
3		swrdccatin2.l	. . . . . . 7 |- L = (♯ ` A )
4		lencl 11253	. . . . . . 7 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
5	3, 4	eqeltrid 2321	. . . . . 6 |- ( A e. Word V -> L e. NN0 )
6		elfzuz 10374	. . . . . 6 |- ( N e. ( ( L + 1 ) ... ( L + M ) ) -> N e. ( ZZ>= ` ( L + 1 ) ) )
7		peano2nn0 9553	. . . . . . 7 |- ( L e. NN0 -> ( L + 1 ) e. NN0 )
8	7	anim1i 340	. . . . . 6 |- ( ( L e. NN0 /\ N e. ( ZZ>= ` ( L + 1 ) ) ) -> ( ( L + 1 ) e. NN0 /\ N e. ( ZZ>= ` ( L + 1 ) ) ) )
9	5, 6, 8	syl2an 289	. . . . 5 |- ( ( A e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> ( ( L + 1 ) e. NN0 /\ N e. ( ZZ>= ` ( L + 1 ) ) ) )
10	9	3adant2 1043	. . . 4 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> ( ( L + 1 ) e. NN0 /\ N e. ( ZZ>= ` ( L + 1 ) ) ) )
11		eluznn0 9949	. . . 4 |- ( ( ( L + 1 ) e. NN0 /\ N e. ( ZZ>= ` ( L + 1 ) ) ) -> N e. NN0 )
12	10, 11	syl 14	. . 3 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> N e. NN0 )
13		pfxval 11391	. . 3 |- ( ( ( A ++ B ) e. Word V /\ N e. NN0 ) -> ( ( A ++ B ) prefix N ) = ( ( A ++ B ) substr <. 0 , N >. ) )
14	2, 12, 13	syl2anc 411	. 2 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> ( ( A ++ B ) prefix N ) = ( ( A ++ B ) substr <. 0 , N >. ) )
15		3simpa 1021	. . 3 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> ( A e. Word V /\ B e. Word V ) )
16	5	3ad2ant1 1045	. . . . 5 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> L e. NN0 )
17		0elfz 10474	. . . . 5 |- ( L e. NN0 -> 0 e. ( 0 ... L ) )
18	16, 17	syl 14	. . . 4 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> 0 e. ( 0 ... L ) )
19	4	nn0zd 9716	. . . . . . . . . 10 |- ( A e. Word V -> (♯ ` A ) e. ZZ )
20	3, 19	eqeltrid 2321	. . . . . . . . 9 |- ( A e. Word V -> L e. ZZ )
21	20	adantr 276	. . . . . . . 8 |- ( ( A e. Word V /\ B e. Word V ) -> L e. ZZ )
22		uzid 9886	. . . . . . . 8 |- ( L e. ZZ -> L e. ( ZZ>= ` L ) )
23		peano2uz 9933	. . . . . . . 8 |- ( L e. ( ZZ>= ` L ) -> ( L + 1 ) e. ( ZZ>= ` L ) )
24		fzss1 10418	. . . . . . . 8 |- ( ( L + 1 ) e. ( ZZ>= ` L ) -> ( ( L + 1 ) ... ( L + M ) ) C_ ( L ... ( L + M ) ) )
25	21, 22, 23, 24	4syl 18	. . . . . . 7 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( L + 1 ) ... ( L + M ) ) C_ ( L ... ( L + M ) ) )
26		pfxccatpfx2.m	. . . . . . . . . 10 |- M = (♯ ` B )
27	26	eqcomi 2238	. . . . . . . . 9 |- (♯ ` B ) = M
28	27	oveq2i 6069	. . . . . . . 8 |- ( L + (♯ ` B ) ) = ( L + M )
29	28	oveq2i 6069	. . . . . . 7 |- ( L ... ( L + (♯ ` B ) ) ) = ( L ... ( L + M ) )
30	25, 29	sseqtrrdi 3291	. . . . . 6 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( L + 1 ) ... ( L + M ) ) C_ ( L ... ( L + (♯ ` B ) ) ) )
31	30	sseld 3241	. . . . 5 |- ( ( A e. Word V /\ B e. Word V ) -> ( N e. ( ( L + 1 ) ... ( L + M ) ) -> N e. ( L ... ( L + (♯ ` B ) ) ) ) )
32	31	3impia 1227	. . . 4 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> N e. ( L ... ( L + (♯ ` B ) ) ) )
33	18, 32	jca 306	. . 3 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> ( 0 e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) )
34	3	pfxccatin12 11450	. . 3 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( 0 e. ( 0 ... L ) /\ N e. ( L ... ( L + (♯ ` B ) ) ) ) -> ( ( A ++ B ) substr <. 0 , N >. ) = ( ( A substr <. 0 , L >. ) ++ ( B prefix ( N - L ) ) ) ) )
35	15, 33, 34	sylc 62	. 2 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> ( ( A ++ B ) substr <. 0 , N >. ) = ( ( A substr <. 0 , L >. ) ++ ( B prefix ( N - L ) ) ) )
36	3	opeq2i 3892	. . . . . 6 |- <. 0 , L >. = <. 0 , (♯ ` A ) >.
37	36	oveq2i 6069	. . . . 5 |- ( A substr <. 0 , L >. ) = ( A substr <. 0 , (♯ ` A ) >. )
38		pfxval 11391	. . . . . . 7 |- ( ( A e. Word V /\ (♯ ` A ) e. NN0 ) -> ( A prefix (♯ ` A ) ) = ( A substr <. 0 , (♯ ` A ) >. ) )
39	4, 38	mpdan 421	. . . . . 6 |- ( A e. Word V -> ( A prefix (♯ ` A ) ) = ( A substr <. 0 , (♯ ` A ) >. ) )
40		pfxid 11403	. . . . . 6 |- ( A e. Word V -> ( A prefix (♯ ` A ) ) = A )
41	39, 40	eqtr3d 2269	. . . . 5 |- ( A e. Word V -> ( A substr <. 0 , (♯ ` A ) >. ) = A )
42	37, 41	eqtrid 2279	. . . 4 |- ( A e. Word V -> ( A substr <. 0 , L >. ) = A )
43	42	3ad2ant1 1045	. . 3 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> ( A substr <. 0 , L >. ) = A )
44	43	oveq1d 6073	. 2 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> ( ( A substr <. 0 , L >. ) ++ ( B prefix ( N - L ) ) ) = ( A ++ ( B prefix ( N - L ) ) ) )
45	14, 35, 44	3eqtrd 2271	1 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> ( ( A ++ B ) prefix N ) = ( A ++ ( B prefix ( N - L ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   C_ wss 3214   <.cop 3697   ` cfv 5357  (class class class)co 6058   0cc0 8143   1c1 8144   + caddc 8146   - cmin 8460   NN0cn0 9513   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361  ♯chash 11163  Word cword 11249   ++ cconcat 11303   substr csubstr 11362   prefix cpfx 11389

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-concat 11304  df-substr 11363  df-pfx 11390

This theorem is referenced by:  pfxccat3a  11455