Theorem wrdl3s3 13705

Description: A word of length 3 is a length 3 string. (Contributed by AV, 18-May-2021.)

Assertion

Ref	Expression
wrdl3s3	|- ( ( W e. Word V /\ ( # ` W ) = 3 ) <-> E. a e. V E. b e. V E. c e. V W = <" a b c "> )

Distinct variable groups:   V, a, b, c   W, a, b, c

Proof of Theorem wrdl3s3

Step	Hyp	Ref	Expression
1		c0ex 10034	. . . . . . . 8 |- 0 e. _V
2	1	tpid1 4303	. . . . . . 7 |- 0 e. { 0 , 1 , 2 }
3		fzo0to3tp 12554	. . . . . . 7 |- ( 0..^ 3 ) = { 0 , 1 , 2 }
4	2, 3	eleqtrri 2700	. . . . . 6 |- 0 e. ( 0..^ 3 )
5		oveq2 6658	. . . . . 6 |- ( ( # ` W ) = 3 -> ( 0..^ ( # ` W ) ) = ( 0..^ 3 ) )
6	4, 5	syl5eleqr 2708	. . . . 5 |- ( ( # ` W ) = 3 -> 0 e. ( 0..^ ( # ` W ) ) )
7		wrdsymbcl 13318	. . . . 5 |- ( ( W e. Word V /\ 0 e. ( 0..^ ( # ` W ) ) ) -> ( W ` 0 ) e. V )
8	6, 7	sylan2 491	. . . 4 |- ( ( W e. Word V /\ ( # ` W ) = 3 ) -> ( W ` 0 ) e. V )
9		1ex 10035	. . . . . . . 8 |- 1 e. _V
10	9	tpid2 4304	. . . . . . 7 |- 1 e. { 0 , 1 , 2 }
11	10, 3	eleqtrri 2700	. . . . . 6 |- 1 e. ( 0..^ 3 )
12	11, 5	syl5eleqr 2708	. . . . 5 |- ( ( # ` W ) = 3 -> 1 e. ( 0..^ ( # ` W ) ) )
13		wrdsymbcl 13318	. . . . 5 |- ( ( W e. Word V /\ 1 e. ( 0..^ ( # ` W ) ) ) -> ( W ` 1 ) e. V )
14	12, 13	sylan2 491	. . . 4 |- ( ( W e. Word V /\ ( # ` W ) = 3 ) -> ( W ` 1 ) e. V )
15		2ex 11092	. . . . . . . 8 |- 2 e. _V
16	15	tpid3 4307	. . . . . . 7 |- 2 e. { 0 , 1 , 2 }
17	16, 3	eleqtrri 2700	. . . . . 6 |- 2 e. ( 0..^ 3 )
18	17, 5	syl5eleqr 2708	. . . . 5 |- ( ( # ` W ) = 3 -> 2 e. ( 0..^ ( # ` W ) ) )
19		wrdsymbcl 13318	. . . . 5 |- ( ( W e. Word V /\ 2 e. ( 0..^ ( # ` W ) ) ) -> ( W ` 2 ) e. V )
20	18, 19	sylan2 491	. . . 4 |- ( ( W e. Word V /\ ( # ` W ) = 3 ) -> ( W ` 2 ) e. V )
21		simpr 477	. . . . 5 |- ( ( W e. Word V /\ ( # ` W ) = 3 ) -> ( # ` W ) = 3 )
22		eqid 2622	. . . . . 6 |- ( W ` 0 ) = ( W ` 0 )
23		eqid 2622	. . . . . 6 |- ( W ` 1 ) = ( W ` 1 )
24		eqid 2622	. . . . . 6 |- ( W ` 2 ) = ( W ` 2 )
25	22, 23, 24	3pm3.2i 1239	. . . . 5 |- ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = ( W ` 2 ) )
26	21, 25	jctir 561	. . . 4 |- ( ( W e. Word V /\ ( # ` W ) = 3 ) -> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = ( W ` 2 ) ) ) )
27		eqeq2 2633	. . . . . . 7 |- ( a = ( W ` 0 ) -> ( ( W ` 0 ) = a <-> ( W ` 0 ) = ( W ` 0 ) ) )
28	27	3anbi1d 1403	. . . . . 6 |- ( a = ( W ` 0 ) -> ( ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) <-> ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) )
29	28	anbi2d 740	. . . . 5 |- ( a = ( W ` 0 ) -> ( ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) <-> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) ) )
30		eqeq2 2633	. . . . . . 7 |- ( b = ( W ` 1 ) -> ( ( W ` 1 ) = b <-> ( W ` 1 ) = ( W ` 1 ) ) )
31	30	3anbi2d 1404	. . . . . 6 |- ( b = ( W ` 1 ) -> ( ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) <-> ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = c ) ) )
32	31	anbi2d 740	. . . . 5 |- ( b = ( W ` 1 ) -> ( ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) <-> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = c ) ) ) )
33		eqeq2 2633	. . . . . . 7 |- ( c = ( W ` 2 ) -> ( ( W ` 2 ) = c <-> ( W ` 2 ) = ( W ` 2 ) ) )
34	33	3anbi3d 1405	. . . . . 6 |- ( c = ( W ` 2 ) -> ( ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = c ) <-> ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = ( W ` 2 ) ) ) )
35	34	anbi2d 740	. . . . 5 |- ( c = ( W ` 2 ) -> ( ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = c ) ) <-> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = ( W ` 2 ) ) ) ) )
36	29, 32, 35	rspc3ev 3326	. . . 4 |- ( ( ( ( W ` 0 ) e. V /\ ( W ` 1 ) e. V /\ ( W ` 2 ) e. V ) /\ ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = ( W ` 0 ) /\ ( W ` 1 ) = ( W ` 1 ) /\ ( W ` 2 ) = ( W ` 2 ) ) ) ) -> E. a e. V E. b e. V E. c e. V ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) )
37	8, 14, 20, 26, 36	syl31anc 1329	. . 3 |- ( ( W e. Word V /\ ( # ` W ) = 3 ) -> E. a e. V E. b e. V E. c e. V ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) )
38		df-3an 1039	. . . . . . . . 9 |- ( ( a e. V /\ b e. V /\ c e. V ) <-> ( ( a e. V /\ b e. V ) /\ c e. V ) )
39		eqwrds3 13704	. . . . . . . . . 10 |- ( ( W e. Word V /\ ( a e. V /\ b e. V /\ c e. V ) ) -> ( W = <" a b c "> <-> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) ) )
40	39	ex 450	. . . . . . . . 9 |- ( W e. Word V -> ( ( a e. V /\ b e. V /\ c e. V ) -> ( W = <" a b c "> <-> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) ) ) )
41	38, 40	syl5bir 233	. . . . . . . 8 |- ( W e. Word V -> ( ( ( a e. V /\ b e. V ) /\ c e. V ) -> ( W = <" a b c "> <-> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) ) ) )
42	41	expd 452	. . . . . . 7 |- ( W e. Word V -> ( ( a e. V /\ b e. V ) -> ( c e. V -> ( W = <" a b c "> <-> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) ) ) ) )
43	42	adantr 481	. . . . . 6 |- ( ( W e. Word V /\ ( # ` W ) = 3 ) -> ( ( a e. V /\ b e. V ) -> ( c e. V -> ( W = <" a b c "> <-> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) ) ) ) )
44	43	imp31 448	. . . . 5 |- ( ( ( ( W e. Word V /\ ( # ` W ) = 3 ) /\ ( a e. V /\ b e. V ) ) /\ c e. V ) -> ( W = <" a b c "> <-> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) ) )
45	44	rexbidva 3049	. . . 4 |- ( ( ( W e. Word V /\ ( # ` W ) = 3 ) /\ ( a e. V /\ b e. V ) ) -> ( E. c e. V W = <" a b c "> <-> E. c e. V ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) ) )
46	45	2rexbidva 3056	. . 3 |- ( ( W e. Word V /\ ( # ` W ) = 3 ) -> ( E. a e. V E. b e. V E. c e. V W = <" a b c "> <-> E. a e. V E. b e. V E. c e. V ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = a /\ ( W ` 1 ) = b /\ ( W ` 2 ) = c ) ) ) )
47	37, 46	mpbird 247	. 2 |- ( ( W e. Word V /\ ( # ` W ) = 3 ) -> E. a e. V E. b e. V E. c e. V W = <" a b c "> )
48		s3cl 13624	. . . . . . . 8 |- ( ( a e. V /\ b e. V /\ c e. V ) -> <" a b c "> e. Word V )
49	48	ad4ant123 1294	. . . . . . 7 |- ( ( ( ( a e. V /\ b e. V ) /\ c e. V ) /\ W = <" a b c "> ) -> <" a b c "> e. Word V )
50		s3len 13639	. . . . . . 7 |- ( # ` <" a b c "> ) = 3
51	49, 50	jctir 561	. . . . . 6 |- ( ( ( ( a e. V /\ b e. V ) /\ c e. V ) /\ W = <" a b c "> ) -> ( <" a b c "> e. Word V /\ ( # ` <" a b c "> ) = 3 ) )
52		eleq1 2689	. . . . . . . 8 |- ( W = <" a b c "> -> ( W e. Word V <-> <" a b c "> e. Word V ) )
53		fveq2 6191	. . . . . . . . 9 |- ( W = <" a b c "> -> ( # ` W ) = ( # ` <" a b c "> ) )
54	53	eqeq1d 2624	. . . . . . . 8 |- ( W = <" a b c "> -> ( ( # ` W ) = 3 <-> ( # ` <" a b c "> ) = 3 ) )
55	52, 54	anbi12d 747	. . . . . . 7 |- ( W = <" a b c "> -> ( ( W e. Word V /\ ( # ` W ) = 3 ) <-> ( <" a b c "> e. Word V /\ ( # ` <" a b c "> ) = 3 ) ) )
56	55	adantl 482	. . . . . 6 |- ( ( ( ( a e. V /\ b e. V ) /\ c e. V ) /\ W = <" a b c "> ) -> ( ( W e. Word V /\ ( # ` W ) = 3 ) <-> ( <" a b c "> e. Word V /\ ( # ` <" a b c "> ) = 3 ) ) )
57	51, 56	mpbird 247	. . . . 5 |- ( ( ( ( a e. V /\ b e. V ) /\ c e. V ) /\ W = <" a b c "> ) -> ( W e. Word V /\ ( # ` W ) = 3 ) )
58	57	ex 450	. . . 4 |- ( ( ( a e. V /\ b e. V ) /\ c e. V ) -> ( W = <" a b c "> -> ( W e. Word V /\ ( # ` W ) = 3 ) ) )
59	58	rexlimdva 3031	. . 3 |- ( ( a e. V /\ b e. V ) -> ( E. c e. V W = <" a b c "> -> ( W e. Word V /\ ( # ` W ) = 3 ) ) )
60	59	rexlimivv 3036	. 2 |- ( E. a e. V E. b e. V E. c e. V W = <" a b c "> -> ( W e. Word V /\ ( # ` W ) = 3 ) )
61	47, 60	impbii 199	1 |- ( ( W e. Word V /\ ( # ` W ) = 3 ) <-> E. a e. V E. b e. V E. c e. V W = <" a b c "> )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   {ctp 4181   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   2c2 11070   3c3 11071  ..^cfzo 12465   #chash 13117  Word cword 13291   <"cs3 13587

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-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-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594

This theorem is referenced by:  elwwlks2s3  26859