Theorem pfxccatpfx2 11228

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 11087	. . . 4 |- ( ( A e. Word V /\ B e. Word V ) -> ( A ++ B ) e. Word V )
2	1	3adant3 1020	. . 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 11035	. . . . . . 7 |- ( A e. Word V -> (♯ ` A ) e. NN0 )
5	3, 4	eqeltrid 2294	. . . . . 6 |- ( A e. Word V -> L e. NN0 )
6		elfzuz 10178	. . . . . 6 |- ( N e. ( ( L + 1 ) ... ( L + M ) ) -> N e. ( ZZ>= ` ( L + 1 ) ) )
7		peano2nn0 9370	. . . . . . 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 1019	. . . 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 9755	. . . 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 11165	. . 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 997	. . 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 1021	. . . . 5 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> L e. NN0 )
17		0elfz 10275	. . . . 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 9528	. . . . . . . . . 10 |- ( A e. Word V -> (♯ ` A ) e. ZZ )
20	3, 19	eqeltrid 2294	. . . . . . . . 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 9697	. . . . . . . 8 |- ( L e. ZZ -> L e. ( ZZ>= ` L ) )
23		peano2uz 9739	. . . . . . . 8 |- ( L e. ( ZZ>= ` L ) -> ( L + 1 ) e. ( ZZ>= ` L ) )
24		fzss1 10220	. . . . . . . 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 2211	. . . . . . . . 9 |- (♯ ` B ) = M
28	27	oveq2i 5978	. . . . . . . 8 |- ( L + (♯ ` B ) ) = ( L + M )
29	28	oveq2i 5978	. . . . . . 7 |- ( L ... ( L + (♯ ` B ) ) ) = ( L ... ( L + M ) )
30	25, 29	sseqtrrdi 3250	. . . . . 6 |- ( ( A e. Word V /\ B e. Word V ) -> ( ( L + 1 ) ... ( L + M ) ) C_ ( L ... ( L + (♯ ` B ) ) ) )
31	30	sseld 3200	. . . . 5 |- ( ( A e. Word V /\ B e. Word V ) -> ( N e. ( ( L + 1 ) ... ( L + M ) ) -> N e. ( L ... ( L + (♯ ` B ) ) ) ) )
32	31	3impia 1203	. . . 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 11224	. . 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 3837	. . . . . 6 |- <. 0 , L >. = <. 0 , (♯ ` A ) >.
37	36	oveq2i 5978	. . . . 5 |- ( A substr <. 0 , L >. ) = ( A substr <. 0 , (♯ ` A ) >. )
38		pfxval 11165	. . . . . . 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 11177	. . . . . 6 |- ( A e. Word V -> ( A prefix (♯ ` A ) ) = A )
41	39, 40	eqtr3d 2242	. . . . 5 |- ( A e. Word V -> ( A substr <. 0 , (♯ ` A ) >. ) = A )
42	37, 41	eqtrid 2252	. . . 4 |- ( A e. Word V -> ( A substr <. 0 , L >. ) = A )
43	42	3ad2ant1 1021	. . 3 |- ( ( A e. Word V /\ B e. Word V /\ N e. ( ( L + 1 ) ... ( L + M ) ) ) -> ( A substr <. 0 , L >. ) = A )
44	43	oveq1d 5982	. 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 2244	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 981   = wceq 1373   e. wcel 2178   C_ wss 3174   <.cop 3646   ` cfv 5290  (class class class)co 5967   0cc0 7960   1c1 7961   + caddc 7963   - cmin 8278   NN0cn0 9330   ZZcz 9407   ZZ>=cuz 9683   ...cfz 10165  ♯chash 10957  Word cword 11031   ++ cconcat 11084   substr csubstr 11136   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-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076

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-inn 9072  df-n0 9331  df-z 9408  df-uz 9684  df-fz 10166  df-fzo 10300  df-ihash 10958  df-word 11032  df-concat 11085  df-substr 11137  df-pfx 11164

This theorem is referenced by:  pfxccat3a  11229