Theorem pfxccatpfx2 41428

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
pfxccatin12.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 13359	. . . 4 |- ( ( A e. Word V /\ B e. Word V ) -> ( A ++ B ) e. Word V )
2	1	3adant3 1081	. . 3 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> ( A ++ B ) e. Word V )
3		pfxccatin12.l	. . . . . . 7 |- L = ( # ` A )
4		lencl 13324	. . . . . . 7 |- ( A e. Word V -> ( # ` A ) e. NN0 )
5	3, 4	syl5eqel 2705	. . . . . 6 |- ( A e. Word V -> L e. NN0 )
6		elfzuz 12338	. . . . . 6 |- ( N e. ( ( L + 1 ) ... ( L + M ) ) -> N e. ( ZZ>= ` ( L + 1 ) ) )
7		peano2nn0 11333	. . . . . . 7 |- ( L e. NN0 -> ( L + 1 ) e. NN0 )
8	7	anim1i 592	. . . . . 6 |- ( ( L e. NN0 /\ N e. ( ZZ>= ` ( L + 1 ) ) ) -> ( ( L + 1 ) e. NN0 /\ N e. ( ZZ>= ` ( L + 1 ) ) ) )
9	5, 6, 8	syl2an 494	. . . . 5 |- ( ( A e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> ( ( L + 1 ) e. NN0 /\ N e. ( ZZ>= ` ( L + 1 ) ) ) )
10	9	3adant2 1080	. . . 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 11757	. . . 4 |- ( ( ( L + 1 ) e. NN0 /\ N e. ( ZZ>= ` ( L + 1 ) ) ) -> N e. NN0 )
12	10, 11	syl 17	. . 3 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> N e. NN0 )
13		pfxval 41383	. . 3 |- ( ( ( A ++ B ) e. Word V /\ N e. NN0 ) -> ( ( A ++ B ) prefix N ) = ( ( A ++ B ) substr <. 0 , N >. ) )
14	2, 12, 13	syl2anc 693	. 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 1058	. . 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 1082	. . . . 5 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> L e. NN0 )
17		0elfz 12436	. . . . 5 |- ( L e. NN0 -> 0 e. ( 0 ... L ) )
18	16, 17	syl 17	. . . 4 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> 0 e. ( 0 ... L ) )
19	4	nn0zd 11480	. . . . . . . . . . 11 |- ( A e. Word V -> ( # ` A ) e. ZZ )
20	3, 19	syl5eqel 2705	. . . . . . . . . 10 |- ( A e. Word V -> L e. ZZ )
21	20	adantr 481	. . . . . . . . 9 |- ( ( A e. Word V /\ B e. Word V ) -> L e. ZZ )
22		uzid 11702	. . . . . . . . 9 |- ( L e. ZZ -> L e. ( ZZ>= ` L ) )
23	21, 22	syl 17	. . . . . . . 8 |- ( ( A e. Word V /\ B e. Word V ) -> L e. ( ZZ>= ` L ) )
24		peano2uz 11741	. . . . . . . 8 |- ( L e. ( ZZ>= ` L ) -> ( L + 1 ) e. ( ZZ>= ` L ) )
25		fzss1 12380	. . . . . . . 8 |- ( ( L + 1 ) e. ( ZZ>= ` L ) -> ( ( L + 1 ) ... ( L + M ) ) C_ ( L ... ( L + M ) ) )
26	23, 24, 25	3syl 18	. . . . . . 7 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( L + 1 ) ... ( L + M ) ) C_ ( L ... ( L + M ) ) )
27		pfxccatpfx2.m	. . . . . . . . . 10 |- M = ( # ` B )
28	27	eqcomi 2631	. . . . . . . . 9 |- ( # ` B ) = M
29	28	oveq2i 6661	. . . . . . . 8 |- ( L + ( # ` B ) ) = ( L + M )
30	29	oveq2i 6661	. . . . . . 7 |- ( L ... ( L + ( # ` B ) ) ) = ( L ... ( L + M ) )
31	26, 30	syl6sseqr 3652	. . . . . 6 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( L + 1 ) ... ( L + M ) ) C_ ( L ... ( L + ( # ` B ) ) ) )
32	31	sseld 3602	. . . . 5 |- ( ( A e. Word V /\ B e. Word V ) -> ( N e. ( ( L + 1 ) ... ( L + M ) ) -> N e. ( L ... ( L + ( # ` B ) ) ) ) )
33	32	3impia 1261	. . . 4 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> N e. ( L ... ( L + ( # ` B ) ) ) )
34	18, 33	jca 554	. . 3 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> ( 0 e. ( 0 ... L ) /\ N e. ( L ... ( L + ( # ` B ) ) ) ) )
35	3	pfxccatin12 41425	. . 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 ) ) ) ) )
36	15, 34, 35	sylc 65	. 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 ) ) ) )
37	3	opeq2i 4406	. . . . . 6 |- <. 0 , L >. = <. 0 , ( # ` A ) >.
38	37	oveq2i 6661	. . . . 5 |- ( A substr <. 0 , L >. ) = ( A substr <. 0 , ( # ` A ) >. )
39		swrdid 13428	. . . . 5 |- ( A e. Word V -> ( A substr <. 0 , ( # ` A ) >. ) = A )
40	38, 39	syl5eq 2668	. . . 4 |- ( A e. Word V -> ( A substr <. 0 , L >. ) = A )
41	40	3ad2ant1 1082	. . 3 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> ( A substr <. 0 , L >. ) = A )
42	41	oveq1d 6665	. 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 ) ) ) )
43	14, 36, 42	3eqtrd 2660	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 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   <.cop 4183   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   #chash 13117  Word cword 13291   ++ cconcat 13293   substr csubstr 13295   prefix cpfx 41381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-substr 13303  df-pfx 41382

This theorem is referenced by:  pfxccat3a  41429