Theorem pfxeq 11267

Description: The prefixes of two words are equal iff they have the same length and the same symbols at each position. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 4-May-2020.)

Assertion

Ref	Expression
pfxeq	|- ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> ( ( W prefix M ) = ( U prefix N ) <-> ( M = N /\ A. i e. ( 0..^ M ) ( W ` i ) = ( U ` i ) ) ) )

Distinct variable groups:   i, M   i, N   U, i   i, V   i, W

Proof of Theorem pfxeq

Step	Hyp	Ref	Expression
1		simp2l 1047	. . . . 5 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> W e. Word V )
2		simp1l 1045	. . . . 5 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> M e. NN0 )
3		pfxclg 11249	. . . . 5 |- ( ( W e. Word V /\ M e. NN0 ) -> ( W prefix M ) e. Word V )
4	1, 2, 3	syl2anc 411	. . . 4 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> ( W prefix M ) e. Word V )
5		simp2r 1048	. . . . 5 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> U e. Word V )
6		simp1r 1046	. . . . 5 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> N e. NN0 )
7		pfxclg 11249	. . . . 5 |- ( ( U e. Word V /\ N e. NN0 ) -> ( U prefix N ) e. Word V )
8	5, 6, 7	syl2anc 411	. . . 4 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> ( U prefix N ) e. Word V )
9		eqwrd 11144	. . . 4 |- ( ( ( W prefix M ) e. Word V /\ ( U prefix N ) e. Word V ) -> ( ( W prefix M ) = ( U prefix N ) <-> ( (♯ ` ( W prefix M ) ) = (♯ ` ( U prefix N ) ) /\ A. i e. ( 0..^ (♯ ` ( W prefix M ) ) ) ( ( W prefix M ) ` i ) = ( ( U prefix N ) ` i ) ) ) )
10	4, 8, 9	syl2anc 411	. . 3 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> ( ( W prefix M ) = ( U prefix N ) <-> ( (♯ ` ( W prefix M ) ) = (♯ ` ( U prefix N ) ) /\ A. i e. ( 0..^ (♯ ` ( W prefix M ) ) ) ( ( W prefix M ) ` i ) = ( ( U prefix N ) ` i ) ) ) )
11		simpl 109	. . . . . . . 8 |- ( ( M e. NN0 /\ N e. NN0 ) -> M e. NN0 )
12		lencl 11107	. . . . . . . . 9 |- ( W e. Word V -> (♯ ` W ) e. NN0 )
13	12	adantr 276	. . . . . . . 8 |- ( ( W e. Word V /\ U e. Word V ) -> (♯ ` W ) e. NN0 )
14		simpl 109	. . . . . . . 8 |- ( ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) -> M <_ (♯ ` W ) )
15	11, 13, 14	3anim123i 1208	. . . . . . 7 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> ( M e. NN0 /\ (♯ ` W ) e. NN0 /\ M <_ (♯ ` W ) ) )
16		elfz2nn0 10337	. . . . . . 7 |- ( M e. ( 0 ... (♯ ` W ) ) <-> ( M e. NN0 /\ (♯ ` W ) e. NN0 /\ M <_ (♯ ` W ) ) )
17	15, 16	sylibr 134	. . . . . 6 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> M e. ( 0 ... (♯ ` W ) ) )
18		pfxlen 11256	. . . . . 6 |- ( ( W e. Word V /\ M e. ( 0 ... (♯ ` W ) ) ) -> (♯ ` ( W prefix M ) ) = M )
19	1, 17, 18	syl2anc 411	. . . . 5 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> (♯ ` ( W prefix M ) ) = M )
20		simpr 110	. . . . . . . 8 |- ( ( M e. NN0 /\ N e. NN0 ) -> N e. NN0 )
21		lencl 11107	. . . . . . . . 9 |- ( U e. Word V -> (♯ ` U ) e. NN0 )
22	21	adantl 277	. . . . . . . 8 |- ( ( W e. Word V /\ U e. Word V ) -> (♯ ` U ) e. NN0 )
23		simpr 110	. . . . . . . 8 |- ( ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) -> N <_ (♯ ` U ) )
24	20, 22, 23	3anim123i 1208	. . . . . . 7 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> ( N e. NN0 /\ (♯ ` U ) e. NN0 /\ N <_ (♯ ` U ) ) )
25		elfz2nn0 10337	. . . . . . 7 |- ( N e. ( 0 ... (♯ ` U ) ) <-> ( N e. NN0 /\ (♯ ` U ) e. NN0 /\ N <_ (♯ ` U ) ) )
26	24, 25	sylibr 134	. . . . . 6 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> N e. ( 0 ... (♯ ` U ) ) )
27		pfxlen 11256	. . . . . 6 |- ( ( U e. Word V /\ N e. ( 0 ... (♯ ` U ) ) ) -> (♯ ` ( U prefix N ) ) = N )
28	5, 26, 27	syl2anc 411	. . . . 5 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> (♯ ` ( U prefix N ) ) = N )
29	19, 28	eqeq12d 2244	. . . 4 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> ( (♯ ` ( W prefix M ) ) = (♯ ` ( U prefix N ) ) <-> M = N ) )
30	29	anbi1d 465	. . 3 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> ( ( (♯ ` ( W prefix M ) ) = (♯ ` ( U prefix N ) ) /\ A. i e. ( 0..^ (♯ ` ( W prefix M ) ) ) ( ( W prefix M ) ` i ) = ( ( U prefix N ) ` i ) ) <-> ( M = N /\ A. i e. ( 0..^ (♯ ` ( W prefix M ) ) ) ( ( W prefix M ) ` i ) = ( ( U prefix N ) ` i ) ) ) )
31	19	adantr 276	. . . . . . 7 |- ( ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) /\ M = N ) -> (♯ ` ( W prefix M ) ) = M )
32	31	oveq2d 6029	. . . . . 6 |- ( ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) /\ M = N ) -> ( 0..^ (♯ ` ( W prefix M ) ) ) = ( 0..^ M ) )
33	32	raleqdv 2734	. . . . 5 |- ( ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) /\ M = N ) -> ( A. i e. ( 0..^ (♯ ` ( W prefix M ) ) ) ( ( W prefix M ) ` i ) = ( ( U prefix N ) ` i ) <-> A. i e. ( 0..^ M ) ( ( W prefix M ) ` i ) = ( ( U prefix N ) ` i ) ) )
34	1	ad2antrr 488	. . . . . . . 8 |- ( ( ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) /\ M = N ) /\ i e. ( 0..^ M ) ) -> W e. Word V )
35	17	ad2antrr 488	. . . . . . . 8 |- ( ( ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) /\ M = N ) /\ i e. ( 0..^ M ) ) -> M e. ( 0 ... (♯ ` W ) ) )
36		simpr 110	. . . . . . . 8 |- ( ( ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) /\ M = N ) /\ i e. ( 0..^ M ) ) -> i e. ( 0..^ M ) )
37		pfxfv 11255	. . . . . . . 8 |- ( ( W e. Word V /\ M e. ( 0 ... (♯ ` W ) ) /\ i e. ( 0..^ M ) ) -> ( ( W prefix M ) ` i ) = ( W ` i ) )
38	34, 35, 36, 37	syl3anc 1271	. . . . . . 7 |- ( ( ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) /\ M = N ) /\ i e. ( 0..^ M ) ) -> ( ( W prefix M ) ` i ) = ( W ` i ) )
39	5	ad2antrr 488	. . . . . . . 8 |- ( ( ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) /\ M = N ) /\ i e. ( 0..^ M ) ) -> U e. Word V )
40	26	ad2antrr 488	. . . . . . . 8 |- ( ( ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) /\ M = N ) /\ i e. ( 0..^ M ) ) -> N e. ( 0 ... (♯ ` U ) ) )
41		oveq2 6021	. . . . . . . . . . 11 |- ( M = N -> ( 0..^ M ) = ( 0..^ N ) )
42	41	eleq2d 2299	. . . . . . . . . 10 |- ( M = N -> ( i e. ( 0..^ M ) <-> i e. ( 0..^ N ) ) )
43	42	adantl 277	. . . . . . . . 9 |- ( ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) /\ M = N ) -> ( i e. ( 0..^ M ) <-> i e. ( 0..^ N ) ) )
44	43	biimpa 296	. . . . . . . 8 |- ( ( ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) /\ M = N ) /\ i e. ( 0..^ M ) ) -> i e. ( 0..^ N ) )
45		pfxfv 11255	. . . . . . . 8 |- ( ( U e. Word V /\ N e. ( 0 ... (♯ ` U ) ) /\ i e. ( 0..^ N ) ) -> ( ( U prefix N ) ` i ) = ( U ` i ) )
46	39, 40, 44, 45	syl3anc 1271	. . . . . . 7 |- ( ( ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) /\ M = N ) /\ i e. ( 0..^ M ) ) -> ( ( U prefix N ) ` i ) = ( U ` i ) )
47	38, 46	eqeq12d 2244	. . . . . 6 |- ( ( ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) /\ M = N ) /\ i e. ( 0..^ M ) ) -> ( ( ( W prefix M ) ` i ) = ( ( U prefix N ) ` i ) <-> ( W ` i ) = ( U ` i ) ) )
48	47	ralbidva 2526	. . . . 5 |- ( ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) /\ M = N ) -> ( A. i e. ( 0..^ M ) ( ( W prefix M ) ` i ) = ( ( U prefix N ) ` i ) <-> A. i e. ( 0..^ M ) ( W ` i ) = ( U ` i ) ) )
49	33, 48	bitrd 188	. . . 4 |- ( ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) /\ M = N ) -> ( A. i e. ( 0..^ (♯ ` ( W prefix M ) ) ) ( ( W prefix M ) ` i ) = ( ( U prefix N ) ` i ) <-> A. i e. ( 0..^ M ) ( W ` i ) = ( U ` i ) ) )
50	49	pm5.32da 452	. . 3 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> ( ( M = N /\ A. i e. ( 0..^ (♯ ` ( W prefix M ) ) ) ( ( W prefix M ) ` i ) = ( ( U prefix N ) ` i ) ) <-> ( M = N /\ A. i e. ( 0..^ M ) ( W ` i ) = ( U ` i ) ) ) )
51	10, 30, 50	3bitrd 214	. 2 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> ( ( W prefix M ) = ( U prefix N ) <-> ( M = N /\ A. i e. ( 0..^ M ) ( W ` i ) = ( U ` i ) ) ) )
52	51	3com12 1231	1 |- ( ( ( W e. Word V /\ U e. Word V ) /\ ( M e. NN0 /\ N e. NN0 ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> ( ( W prefix M ) = ( U prefix N ) <-> ( M = N /\ A. i e. ( 0..^ M ) ( W ` i ) = ( U ` i ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   0cc0 8022   <_ cle 8205   NN0cn0 9392   ...cfz 10233  ..^cfzo 10367  ♯chash 11027  Word cword 11103   prefix cpfx 11243

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 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-1o 6577  df-er 6697  df-en 6905  df-dom 6906  df-fin 6907  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-n0 9393  df-z 9470  df-uz 9746  df-fz 10234  df-fzo 10368  df-ihash 11028  df-word 11104  df-substr 11217  df-pfx 11244

This theorem is referenced by:  pfxsuffeqwrdeq  11269