Theorem 2swrdeqwrdeq 12469

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 12387	. . 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 1008	. 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 11688	. . . . . . . . 9 |- ( I e. ( 0..^ ( # ` W ) ) -> I e. ( 0 ... ( # ` W ) ) )
4		fzosplit 11703	. . . . . . . . 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 1011	. . . . . . 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 3029	. . . . 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 3648	. . . . 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 985	. . . . . . 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 11712	. . . . . . . . 9 |- ( I e. ( 0..^ ( # ` W ) ) -> I e. NN0 )
14	13	3ad2ant3 1011	. . . . . . . 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 10709	. . . . . . 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 11711	. . . . . . . 8 |- ( I e. ( 0..^ ( # ` W ) ) -> I <_ ( # ` W ) )
19	18	3ad2ant3 1011	. . . . . . 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 4407	. . . . . . . 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 12464	. . . . . . 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 1223	. . . . 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 12371	. . . . . . . . 9 |- ( W e. Word V -> ( # ` W ) e. NN0 )
28	27	3ad2ant1 1009	. . . . . . . 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 10703	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. RR )
32	31	leidd 10021	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( # ` W ) <_ ( # ` W ) )
33	27, 32	syl 16	. . . . . . . 8 |- ( W e. Word V -> ( # ` W ) <_ ( # ` W ) )
34	33	3ad2ant1 1009	. . . . . . 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 4407	. . . . . . . 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 12464	. . . . . . 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 1223	. . . . 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 965   = wceq 1370   e. wcel 1758   A.wral 2799   u. cun 3437   <.cop 3994   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   0cc0 9397   <_ cle 9534   NN0cn0 10694   ...cfz 11558  ..^cfzo 11669   #chash 12224  Word cword 12343   substr csubstr 12347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-fzo 11670  df-hash 12225  df-word 12351  df-substr 12355

This theorem is referenced by:  2swrd1eqwrdeq  12470  2swrd2eqwrdeq  12675