Theorem pfxeq 11150

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 1026	. . . . 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 1024	. . . . 5 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> M e. NN0 )
3		pfxclg 11133	. . . . 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 1027	. . . . 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 1025	. . . . 5 |- ( ( ( M e. NN0 /\ N e. NN0 ) /\ ( W e. Word V /\ U e. Word V ) /\ ( M <_ (♯ ` W ) /\ N <_ (♯ ` U ) ) ) -> N e. NN0 )
7		pfxclg 11133	. . . . 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 11036	. . . 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 11000	. . . . . . . . 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 1187	. . . . . . 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 10236	. . . . . . 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 11139	. . . . . 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 11000	. . . . . . . . 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 1187	. . . . . . 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 10236	. . . . . . 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 11139	. . . . . 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 2220	. . . 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 5962	. . . . . 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 2708	. . . . 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 11138	. . . . . . . 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 1250	. . . . . . 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 5954	. . . . . . . . . . 11 |- ( M = N -> ( 0..^ M ) = ( 0..^ N ) )
42	41	eleq2d 2275	. . . . . . . . . 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 11138	. . . . . . . 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 1250	. . . . . . 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 2220	. . . . . 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 2502	. . . . 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 1210	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 981   = wceq 1373   e. wcel 2176   A.wral 2484   class class class wbr 4045   ` cfv 5272  (class class class)co 5946   0cc0 7927   <_ cle 8110   NN0cn0 9297   ...cfz 10132  ..^cfzo 10266  ♯chash 10922  Word cword 10996   prefix cpfx 11128

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-addass 8029  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-0id 8035  ax-rnegex 8036  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043

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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-frec 6479  df-1o 6504  df-er 6622  df-en 6830  df-dom 6831  df-fin 6832  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-inn 9039  df-n0 9298  df-z 9375  df-uz 9651  df-fz 10133  df-fzo 10267  df-ihash 10923  df-word 10997  df-substr 11102  df-pfx 11129

This theorem is referenced by:  pfxsuffeqwrdeq  11152