Theorem 2swrdeqwrdeq 13453

Description: Two words are equal if and only if they have the same prefix and the same suffix. (Contributed by Alexander van der Vekens, 23-Sep-2018.)

Assertion

Ref	Expression
2swrdeqwrdeq	|- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> ( W = S <-> ( ( # ` W ) = ( # ` S ) /\ ( ( W substr <. 0 , I >. ) = ( S substr <. 0 , I >. ) /\ ( W substr <. I , ( # ` W ) >. ) = ( S substr <. I , ( # ` W ) >. ) ) ) ) )

Proof of Theorem 2swrdeqwrdeq

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqwrd 13346	. . 3 |- ( ( W e. Word V /\ S e. Word V ) -> ( W = S <-> ( ( # ` W ) = ( # ` S ) /\ A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) ) ) )
2	1	3adant3 1081	. 2 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> ( W = S <-> ( ( # ` W ) = ( # ` S ) /\ A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) ) ) )
3		elfzofz 12485	. . . . . . . . 9 |- ( I e. ( 0..^ ( # ` W ) ) -> I e. ( 0 ... ( # ` W ) ) )
4		fzosplit 12501	. . . . . . . . 9 |- ( I e. ( 0 ... ( # ` W ) ) -> ( 0..^ ( # ` W ) ) = ( ( 0..^ I ) u. ( I..^ ( # ` W ) ) ) )
5	3, 4	syl 17	. . . . . . . 8 |- ( I e. ( 0..^ ( # ` W ) ) -> ( 0..^ ( # ` W ) ) = ( ( 0..^ I ) u. ( I..^ ( # ` W ) ) ) )
6	5	3ad2ant3 1084	. . . . . . 7 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> ( 0..^ ( # ` W ) ) = ( ( 0..^ I ) u. ( I..^ ( # ` W ) ) ) )
7	6	adantr 481	. . . . . 6 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( 0..^ ( # ` W ) ) = ( ( 0..^ I ) u. ( I..^ ( # ` W ) ) ) )
8	7	raleqdv 3144	. . . . 5 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) <-> A. i e. ( ( 0..^ I ) u. ( I..^ ( # ` W ) ) ) ( W ` i ) = ( S ` i ) ) )
9		ralunb 3794	. . . . 5 |- ( A. i e. ( ( 0..^ I ) u. ( I..^ ( # ` W ) ) ) ( W ` i ) = ( S ` i ) <-> ( A. i e. ( 0..^ I ) ( W ` i ) = ( S ` i ) /\ A. i e. ( I..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) ) )
10	8, 9	syl6bb 276	. . . 4 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) <-> ( A. i e. ( 0..^ I ) ( W ` i ) = ( S ` i ) /\ A. i e. ( I..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) ) ) )
11		3simpa 1058	. . . . . . 7 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> ( W e. Word V /\ S e. Word V ) )
12	11	adantr 481	. . . . . 6 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( W e. Word V /\ S e. Word V ) )
13		elfzonn0 12512	. . . . . . . . 9 |- ( I e. ( 0..^ ( # ` W ) ) -> I e. NN0 )
14	13	3ad2ant3 1084	. . . . . . . 8 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> I e. NN0 )
15	14	adantr 481	. . . . . . 7 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> I e. NN0 )
16		0nn0 11307	. . . . . . 7 |- 0 e. NN0
17	15, 16	jctil 560	. . . . . 6 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( 0 e. NN0 /\ I e. NN0 ) )
18		elfzo0le 12511	. . . . . . . 8 |- ( I e. ( 0..^ ( # ` W ) ) -> I <_ ( # ` W ) )
19	18	3ad2ant3 1084	. . . . . . 7 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> I <_ ( # ` W ) )
20	19	adantr 481	. . . . . 6 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> I <_ ( # ` W ) )
21		breq2 4657	. . . . . . . 8 |- ( ( # ` W ) = ( # ` S ) -> ( I <_ ( # ` W ) <-> I <_ ( # ` S ) ) )
22	21	adantl 482	. . . . . . 7 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( I <_ ( # ` W ) <-> I <_ ( # ` S ) ) )
23	20, 22	mpbid 222	. . . . . 6 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> I <_ ( # ` S ) )
24		swrdspsleq 13449	. . . . . . 7 |- ( ( ( W e. Word V /\ S e. Word V ) /\ ( 0 e. NN0 /\ I e. NN0 ) /\ ( I <_ ( # ` W ) /\ I <_ ( # ` S ) ) ) -> ( ( W substr <. 0 , I >. ) = ( S substr <. 0 , I >. ) <-> A. i e. ( 0..^ I ) ( W ` i ) = ( S ` i ) ) )
25	24	bicomd 213	. . . . . 6 |- ( ( ( W e. Word V /\ S e. Word V ) /\ ( 0 e. NN0 /\ I e. NN0 ) /\ ( I <_ ( # ` W ) /\ I <_ ( # ` S ) ) ) -> ( A. i e. ( 0..^ I ) ( W ` i ) = ( S ` i ) <-> ( W substr <. 0 , I >. ) = ( S substr <. 0 , I >. ) ) )
26	12, 17, 20, 23, 25	syl112anc 1330	. . . . 5 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( A. i e. ( 0..^ I ) ( W ` i ) = ( S ` i ) <-> ( W substr <. 0 , I >. ) = ( S substr <. 0 , I >. ) ) )
27		lencl 13324	. . . . . . . . 9 |- ( W e. Word V -> ( # ` W ) e. NN0 )
28	27	3ad2ant1 1082	. . . . . . . 8 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> ( # ` W ) e. NN0 )
29	14, 28	jca 554	. . . . . . 7 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> ( I e. NN0 /\ ( # ` W ) e. NN0 ) )
30	29	adantr 481	. . . . . 6 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( I e. NN0 /\ ( # ` W ) e. NN0 ) )
31		nn0re 11301	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. RR )
32	31	leidd 10594	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( # ` W ) <_ ( # ` W ) )
33	27, 32	syl 17	. . . . . . . 8 |- ( W e. Word V -> ( # ` W ) <_ ( # ` W ) )
34	33	3ad2ant1 1082	. . . . . . 7 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> ( # ` W ) <_ ( # ` W ) )
35	34	adantr 481	. . . . . 6 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( # ` W ) <_ ( # ` W ) )
36		breq2 4657	. . . . . . . 8 |- ( ( # ` W ) = ( # ` S ) -> ( ( # ` W ) <_ ( # ` W ) <-> ( # ` W ) <_ ( # ` S ) ) )
37	36	adantl 482	. . . . . . 7 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( ( # ` W ) <_ ( # ` W ) <-> ( # ` W ) <_ ( # ` S ) ) )
38	35, 37	mpbid 222	. . . . . 6 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( # ` W ) <_ ( # ` S ) )
39		swrdspsleq 13449	. . . . . . 7 |- ( ( ( W e. Word V /\ S e. Word V ) /\ ( I e. NN0 /\ ( # ` W ) e. NN0 ) /\ ( ( # ` W ) <_ ( # ` W ) /\ ( # ` W ) <_ ( # ` S ) ) ) -> ( ( W substr <. I , ( # ` W ) >. ) = ( S substr <. I , ( # ` W ) >. ) <-> A. i e. ( I..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) ) )
40	39	bicomd 213	. . . . . 6 |- ( ( ( W e. Word V /\ S e. Word V ) /\ ( I e. NN0 /\ ( # ` W ) e. NN0 ) /\ ( ( # ` W ) <_ ( # ` W ) /\ ( # ` W ) <_ ( # ` S ) ) ) -> ( A. i e. ( I..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) <-> ( W substr <. I , ( # ` W ) >. ) = ( S substr <. I , ( # ` W ) >. ) ) )
41	12, 30, 35, 38, 40	syl112anc 1330	. . . . 5 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( A. i e. ( I..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) <-> ( W substr <. I , ( # ` W ) >. ) = ( S substr <. I , ( # ` W ) >. ) ) )
42	26, 41	anbi12d 747	. . . 4 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( ( A. i e. ( 0..^ I ) ( W ` i ) = ( S ` i ) /\ A. i e. ( I..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) ) <-> ( ( W substr <. 0 , I >. ) = ( S substr <. 0 , I >. ) /\ ( W substr <. I , ( # ` W ) >. ) = ( S substr <. I , ( # ` W ) >. ) ) ) )
43	10, 42	bitrd 268	. . 3 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) <-> ( ( W substr <. 0 , I >. ) = ( S substr <. 0 , I >. ) /\ ( W substr <. I , ( # ` W ) >. ) = ( S substr <. I , ( # ` W ) >. ) ) ) )
44	43	pm5.32da 673	. 2 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> ( ( ( # ` W ) = ( # ` S ) /\ A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( S ` i ) ) <-> ( ( # ` W ) = ( # ` S ) /\ ( ( W substr <. 0 , I >. ) = ( S substr <. 0 , I >. ) /\ ( W substr <. I , ( # ` W ) >. ) = ( S substr <. I , ( # ` W ) >. ) ) ) ) )
45	2, 44	bitrd 268	1 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> ( W = S <-> ( ( # ` W ) = ( # ` S ) /\ ( ( W substr <. 0 , I >. ) = ( S substr <. 0 , I >. ) /\ ( W substr <. I , ( # ` W ) >. ) = ( S substr <. I , ( # ` W ) >. ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   u. cun 3572   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   0cc0 9936   <_ cle 10075   NN0cn0 11292   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   substr csubstr 13295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-substr 13303

This theorem is referenced by:  2swrd1eqwrdeq  13454  2swrd2eqwrdeq  13696