Theorem pfxeq 11446

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 1054	. . . . 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 1052	. . . . 5 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> M e. NN0 )
3		pfxclg 11428	. . . . 5 |- ( ( W e. Word V /\ M e. NN0 ) -> ( W prefix M ) e. Word V )
4	1, 2, 3	syl2anc 415	. . . 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 1055	. . . . 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 1053	. . . . 5 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> N e. NN0 )
7		pfxclg 11428	. . . . 5 |- ( ( U e. Word V /\ N e. NN0 ) -> ( U prefix N ) e. Word V )
8	5, 6, 7	syl2anc 415	. . . 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 11323	. . . 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 415	. . 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 11286	. . . . . . . . 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 1215	. . . . . . 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 10497	. . . . . . 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 11435	. . . . . 6 |- ( ( W e. Word V /\ M e. ( 0 ... (♯ ` W ) ) ) -> (♯ ` ( W prefix M ) ) = M )
19	1, 17, 18	syl2anc 415	. . . . 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 11286	. . . . . . . . 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 1215	. . . . . . 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 10497	. . . . . . 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 11435	. . . . . 6 |- ( ( U e. Word V /\ N e. ( 0 ... (♯ ` U ) ) ) -> (♯ ` ( U prefix N ) ) = N )
28	5, 26, 27	syl2anc 415	. . . . 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 2253	. . . 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 469	. . 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 6091	. . . . . 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 2755	. . . . 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 492	. . . . . . . 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 492	. . . . . . . 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 11434	. . . . . . . 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 1278	. . . . . . 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 492	. . . . . . . 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 492	. . . . . . . 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 6083	. . . . . . . . . . 11 |- ( M = N -> ( 0..^ M ) = ( 0..^ N ) )
42	41	eleq2d 2308	. . . . . . . . . 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 11434	. . . . . . . 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 1278	. . . . . . 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 2253	. . . . . 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 2546	. . . . 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 456	. . 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 1238	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 1009   = wceq 1402   e. wcel 2209   A.wral 2528   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   0cc0 8169   <_ cle 8351   NN0cn0 9542   ...cfz 10390  ..^cfzo 10527  ♯chash 11192  Word cword 11282   prefix cpfx 11422

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-ihash 11193  df-word 11283  df-substr 11396  df-pfx 11423

This theorem is referenced by:  pfxsuffeqwrdeq  11448