Theorem pfxccatpfx2 11284

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 11141	. . . 4 |- ( ( A e. Word V /\ B e. Word V ) -> ( A ++ B ) e. Word V )
2	1	3adant3 1041	. . 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 11088	. . . . . . 7 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
5	3, 4	eqeltrid 2316	. . . . . 6 |- ( A e. Word V -> L e. NN0 )
6		elfzuz 10229	. . . . . 6 |- ( N e. ( ( L + 1 ) ... ( L + M ) ) -> N e. ( ZZ>= ` ( L + 1 ) ) )
7		peano2nn0 9420	. . . . . . 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 1040	. . . 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 9806	. . . 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 11221	. . 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 1018	. . 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 1042	. . . . 5 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> L e. NN0 )
17		0elfz 10326	. . . . 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 9578	. . . . . . . . . 10 |- ( A e. Word V -> (♯ ` A ) e. ZZ )
20	3, 19	eqeltrid 2316	. . . . . . . . 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 9748	. . . . . . . 8 |- ( L e. ZZ -> L e. ( ZZ>= ` L ) )
23		peano2uz 9790	. . . . . . . 8 |- ( L e. ( ZZ>= ` L ) -> ( L + 1 ) e. ( ZZ>= ` L ) )
24		fzss1 10271	. . . . . . . 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 2233	. . . . . . . . 9 |- (♯ ` B ) = M
28	27	oveq2i 6018	. . . . . . . 8 |- ( L + (♯ ` B ) ) = ( L + M )
29	28	oveq2i 6018	. . . . . . 7 |- ( L ... ( L + (♯ ` B ) ) ) = ( L ... ( L + M ) )
30	25, 29	sseqtrrdi 3273	. . . . . 6 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( L + 1 ) ... ( L + M ) ) C_ ( L ... ( L + (♯ ` B ) ) ) )
31	30	sseld 3223	. . . . 5 |- ( ( A e. Word V /\ B e. Word V ) -> ( N e. ( ( L + 1 ) ... ( L + M ) ) -> N e. ( L ... ( L + (♯ ` B ) ) ) ) )
32	31	3impia 1224	. . . 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 11280	. . 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 3861	. . . . . 6 |- <. 0 , L >. = <. 0 , (♯ ` A ) >.
37	36	oveq2i 6018	. . . . 5 |- ( A substr <. 0 , L >. ) = ( A substr <. 0 , (♯ ` A ) >. )
38		pfxval 11221	. . . . . . 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 11233	. . . . . 6 |- ( A e. Word V -> ( A prefix (♯ ` A ) ) = A )
41	39, 40	eqtr3d 2264	. . . . 5 |- ( A e. Word V -> ( A substr <. 0 , (♯ ` A ) >. ) = A )
42	37, 41	eqtrid 2274	. . . 4 |- ( A e. Word V -> ( A substr <. 0 , L >. ) = A )
43	42	3ad2ant1 1042	. . 3 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> ( A substr <. 0 , L >. ) = A )
44	43	oveq1d 6022	. 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 2266	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 1002   = wceq 1395   e. wcel 2200   C_ wss 3197   <.cop 3669   ` cfv 5318  (class class class)co 6007   0cc0 8010   1c1 8011   + caddc 8013   - cmin 8328   NN0cn0 9380   ZZcz 9457   ZZ>=cuz 9733   ...cfz 10216  ♯chash 11009  Word cword 11084   ++ cconcat 11138   substr csubstr 11192   prefix cpfx 11219

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126

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 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-er 6688  df-en 6896  df-dom 6897  df-fin 6898  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-inn 9122  df-n0 9381  df-z 9458  df-uz 9734  df-fz 10217  df-fzo 10351  df-ihash 11010  df-word 11085  df-concat 11139  df-substr 11193  df-pfx 11220

This theorem is referenced by:  pfxccat3a  11285