Theorem 2swrdeqwrdeq 12809

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 12708	. . 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 1028	. 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 11935	. . . . . . . . 9 |- ( I e. ( 0..^ ( # ` W ) ) -> I e. ( 0 ... ( # ` W ) ) )
4		fzosplit 11951	. . . . . . . . 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 1031	. . . . . . 7 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> ( 0..^ ( # ` W ) ) = ( ( 0..^ I ) u. ( I..^ ( # ` W ) ) ) )
7	6	adantr 467	. . . . . 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 2993	. . . . 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 3615	. . . . 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 265	. . . 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 1005	. . . . . . 7 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> ( W e. Word V /\ S e. Word V ) )
12	11	adantr 467	. . . . . 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 11960	. . . . . . . . 9 |- ( I e. ( 0..^ ( # ` W ) ) -> I e. NN0 )
14	13	3ad2ant3 1031	. . . . . . . 8 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> I e. NN0 )
15	14	adantr 467	. . . . . . 7 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> I e. NN0 )
16		0nn0 10884	. . . . . . 7 |- 0 e. NN0
17	15, 16	jctil 540	. . . . . 6 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( 0 e. NN0 /\ I e. NN0 ) )
18		elfzo0le 11959	. . . . . . . 8 |- ( I e. ( 0..^ ( # ` W ) ) -> I <_ ( # ` W ) )
19	18	3ad2ant3 1031	. . . . . . 7 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> I <_ ( # ` W ) )
20	19	adantr 467	. . . . . 6 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> I <_ ( # ` W ) )
21		breq2 4406	. . . . . . . 8 |- ( ( # ` W ) = ( # ` S ) -> ( I <_ ( # ` W ) <-> I <_ ( # ` S ) ) )
22	21	adantl 468	. . . . . . 7 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( I <_ ( # ` W ) <-> I <_ ( # ` S ) ) )
23	20, 22	mpbid 214	. . . . . 6 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> I <_ ( # ` S ) )
24		swrdspsleq 12805	. . . . . . 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 205	. . . . . 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 1272	. . . . 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 12687	. . . . . . . . 9 |- ( W e. Word V -> ( # ` W ) e. NN0 )
28	27	3ad2ant1 1029	. . . . . . . 8 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> ( # ` W ) e. NN0 )
29	14, 28	jca 535	. . . . . . 7 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> ( I e. NN0 /\ ( # ` W ) e. NN0 ) )
30	29	adantr 467	. . . . . 6 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( I e. NN0 /\ ( # ` W ) e. NN0 ) )
31		nn0re 10878	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. RR )
32	31	leidd 10180	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( # ` W ) <_ ( # ` W ) )
33	27, 32	syl 17	. . . . . . . 8 |- ( W e. Word V -> ( # ` W ) <_ ( # ` W ) )
34	33	3ad2ant1 1029	. . . . . . 7 |- ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) -> ( # ` W ) <_ ( # ` W ) )
35	34	adantr 467	. . . . . 6 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( # ` W ) <_ ( # ` W ) )
36		breq2 4406	. . . . . . . 8 |- ( ( # ` W ) = ( # ` S ) -> ( ( # ` W ) <_ ( # ` W ) <-> ( # ` W ) <_ ( # ` S ) ) )
37	36	adantl 468	. . . . . . 7 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( ( # ` W ) <_ ( # ` W ) <-> ( # ` W ) <_ ( # ` S ) ) )
38	35, 37	mpbid 214	. . . . . 6 |- ( ( ( W e. Word V /\ S e. Word V /\ I e. ( 0..^ ( # ` W ) ) ) /\ ( # ` W ) = ( # ` S ) ) -> ( # ` W ) <_ ( # ` S ) )
39		swrdspsleq 12805	. . . . . . 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 205	. . . . . 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 1272	. . . . 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 717	. . . 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 257	. . 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 647	. 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 257	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 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   A.wral 2737   u. cun 3402   <.cop 3974   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   0cc0 9539   <_ cle 9676   NN0cn0 10869   ...cfz 11784  ..^cfzo 11915   #chash 12515  Word cword 12656   substr csubstr 12660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916  df-hash 12516  df-word 12664  df-substr 12668

This theorem is referenced by:  2swrd1eqwrdeq  12810  2swrd2eqwrdeq  13028