Theorem pfxeq 11276

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 1049	. . . . 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 1047	. . . . 5 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> M e. NN0 )
3		pfxclg 11258	. . . . 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 1050	. . . . 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 1048	. . . . 5 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> N e. NN0 )
7		pfxclg 11258	. . . . 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 11153	. . . 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 11116	. . . . . . . . 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 1210	. . . . . . 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 10346	. . . . . . 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 11265	. . . . . 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 11116	. . . . . . . . 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 1210	. . . . . . 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 10346	. . . . . . 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 11265	. . . . . 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 2246	. . . 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 6033	. . . . . 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 2736	. . . . 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 11264	. . . . . . . 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 1273	. . . . . . 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 6025	. . . . . . . . . . 11 |- ( M = N -> ( 0..^ M ) = ( 0..^ N ) )
42	41	eleq2d 2301	. . . . . . . . . 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 11264	. . . . . . . 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 1273	. . . . . . 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 2246	. . . . . 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 2528	. . . . 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 1233	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 1004   = wceq 1397   e. wcel 2202   A.wral 2510   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   0cc0 8031   <_ cle 8214   NN0cn0 9401   ...cfz 10242  ..^cfzo 10376  ♯chash 11036  Word cword 11112   prefix cpfx 11252

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-n0 9402  df-z 9479  df-uz 9755  df-fz 10243  df-fzo 10377  df-ihash 11037  df-word 11113  df-substr 11226  df-pfx 11253

This theorem is referenced by:  pfxsuffeqwrdeq  11278