Theorem 2swrdeqwrdeq 12658

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 12562	. . 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 1016	. 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 11823	. . . . . . . . 9 |- ( I e. ( 0..^ ( # ` W ) ) -> I e. ( 0 ... ( # ` W ) ) )
4		fzosplit 11838	. . . . . . . . 9 |- ( I e. ( 0 ... ( # ` W ) ) -> ( 0..^ ( # ` W ) ) = ( ( 0..^ I ) u. ( I..^ ( # ` W ) ) ) )
5	3, 4	syl 16	. . . . . . . 8 |- ( I e. ( 0..^ ( # ` W ) ) -> ( 0..^ ( # ` W ) ) = ( ( 0..^ I ) u. ( I..^ ( # ` W ) ) ) )
6	5	3ad2ant3 1019	. . . . . . 7 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> ( 0..^ ( # ` W ) ) = ( ( 0..^ I ) u. ( I..^ ( # ` W ) ) ) )
7	6	adantr 465	. . . . . 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 3069	. . . . 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 3690	. . . . 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 261	. . . 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 993	. . . . . . 7 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> ( W e. Word V /\ S e. Word V ) )
12	11	adantr 465	. . . . . 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 11847	. . . . . . . . 9 |- ( I e. ( 0..^ ( # ` W ) ) -> I e. NN0 )
14	13	3ad2ant3 1019	. . . . . . . 8 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> I e. NN0 )
15	14	adantr 465	. . . . . . 7 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> I e. NN0 )
16		0nn0 10822	. . . . . . 7 |- 0 e. NN0
17	15, 16	jctil 537	. . . . . 6 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( 0 e. NN0 /\ I e. NN0 ) )
18		elfzo0le 11846	. . . . . . . 8 |- ( I e. ( 0..^ ( # ` W ) ) -> I <_ ( # ` W ) )
19	18	3ad2ant3 1019	. . . . . . 7 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> I <_ ( # ` W ) )
20	19	adantr 465	. . . . . 6 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> I <_ ( # ` W ) )
21		breq2 4457	. . . . . . . 8 |- ( ( # ` W ) = ( # ` S ) -> ( I <_ ( # ` W ) <-> I <_ ( # ` S ) ) )
22	21	adantl 466	. . . . . . 7 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( I <_ ( # ` W ) <-> I <_ ( # ` S ) ) )
23	20, 22	mpbid 210	. . . . . 6 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> I <_ ( # ` S ) )
24		swrdspsleq 12653	. . . . . . 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 201	. . . . . 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 1232	. . . . 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 12543	. . . . . . . . 9 |- ( W e. Word V -> ( # ` W ) e. NN0 )
28	27	3ad2ant1 1017	. . . . . . . 8 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> ( # ` W ) e. NN0 )
29	14, 28	jca 532	. . . . . . 7 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> ( I e. NN0 /\ ( # ` W ) e. NN0 ) )
30	29	adantr 465	. . . . . 6 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( I e. NN0 /\ ( # ` W ) e. NN0 ) )
31		nn0re 10816	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. RR )
32	31	leidd 10131	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( # ` W ) <_ ( # ` W ) )
33	27, 32	syl 16	. . . . . . . 8 |- ( W e. Word V -> ( # ` W ) <_ ( # ` W ) )
34	33	3ad2ant1 1017	. . . . . . 7 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> ( # ` W ) <_ ( # ` W ) )
35	34	adantr 465	. . . . . 6 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( # ` W ) <_ ( # ` W ) )
36		breq2 4457	. . . . . . . 8 |- ( ( # ` W ) = ( # ` S ) -> ( ( # ` W ) <_ ( # ` W ) <-> ( # ` W ) <_ ( # ` S ) ) )
37	36	adantl 466	. . . . . . 7 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( ( # ` W ) <_ ( # ` W ) <-> ( # ` W ) <_ ( # ` S ) ) )
38	35, 37	mpbid 210	. . . . . 6 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( # ` W ) <_ ( # ` S ) )
39		swrdspsleq 12653	. . . . . . 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 201	. . . . . 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 1232	. . . . 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 710	. . . 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 253	. . 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 641	. 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 253	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 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   A.wral 2817   u. cun 3479   <.cop 4039   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   0cc0 9504   <_ cle 9641   NN0cn0 10807   ...cfz 11684  ..^cfzo 11804   #chash 12385  Word cword 12515   substr csubstr 12519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-hash 12386  df-word 12523  df-substr 12527

This theorem is referenced by:  2swrd1eqwrdeq  12659  2swrd2eqwrdeq  12871