Theorem eqwrds3 13704

Description: A word is equal with a length 3 string iff it has length 3 and the same symbol at each position. (Contributed by AV, 12-May-2021.)

Assertion

Ref	Expression
eqwrds3	|- ( ( 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 ) ) ) )

Proof of Theorem eqwrds3

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		s3cl 13624	. . 3 |- ( ( A e. V /\ B e. V /\ C e. V ) -> <" A B C "> e. Word V )
2		eqwrd 13346	. . 3 |- ( ( W e. Word V /\ <" A B C "> e. Word V ) -> ( W = <" A B C "> <-> ( ( # ` W ) = ( # ` <" A B C "> ) /\ A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( <" A B C "> ` i ) ) ) )
3	1, 2	sylan2 491	. 2 |- ( ( W e. Word V /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( W = <" A B C "> <-> ( ( # ` W ) = ( # ` <" A B C "> ) /\ A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( <" A B C "> ` i ) ) ) )
4		s3len 13639	. . . . 5 |- ( # ` <" A B C "> ) = 3
5	4	eqeq2i 2634	. . . 4 |- ( ( # ` W ) = ( # ` <" A B C "> ) <-> ( # ` W ) = 3 )
6	5	a1i 11	. . 3 |- ( ( W e. Word V /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( # ` W ) = ( # ` <" A B C "> ) <-> ( # ` W ) = 3 ) )
7	6	anbi1d 741	. 2 |- ( ( W e. Word V /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( ( # ` W ) = ( # ` <" A B C "> ) /\ A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( <" A B C "> ` i ) ) <-> ( ( # ` W ) = 3 /\ A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( <" A B C "> ` i ) ) ) )
8		oveq2 6658	. . . . . 6 |- ( ( # ` W ) = 3 -> ( 0..^ ( # ` W ) ) = ( 0..^ 3 ) )
9		fzo0to3tp 12554	. . . . . 6 |- ( 0..^ 3 ) = { 0 , 1 , 2 }
10	8, 9	syl6eq 2672	. . . . 5 |- ( ( # ` W ) = 3 -> ( 0..^ ( # ` W ) ) = { 0 , 1 , 2 } )
11	10	raleqdv 3144	. . . 4 |- ( ( # ` W ) = 3 -> ( A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( <" A B C "> ` i ) <-> A. i e. { 0 , 1 , 2 } ( W ` i ) = ( <" A B C "> ` i ) ) )
12		fveq2 6191	. . . . . . . 8 |- ( i = 0 -> ( W ` i ) = ( W ` 0 ) )
13		fveq2 6191	. . . . . . . 8 |- ( i = 0 -> ( <" A B C "> ` i ) = ( <" A B C "> ` 0 ) )
14	12, 13	eqeq12d 2637	. . . . . . 7 |- ( i = 0 -> ( ( W ` i ) = ( <" A B C "> ` i ) <-> ( W ` 0 ) = ( <" A B C "> ` 0 ) ) )
15		s3fv0 13636	. . . . . . . . 9 |- ( A e. V -> ( <" A B C "> ` 0 ) = A )
16	15	3ad2ant1 1082	. . . . . . . 8 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( <" A B C "> ` 0 ) = A )
17	16	eqeq2d 2632	. . . . . . 7 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( ( W ` 0 ) = ( <" A B C "> ` 0 ) <-> ( W ` 0 ) = A ) )
18	14, 17	sylan9bbr 737	. . . . . 6 |- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ i = 0 ) -> ( ( W ` i ) = ( <" A B C "> ` i ) <-> ( W ` 0 ) = A ) )
19		fveq2 6191	. . . . . . . 8 |- ( i = 1 -> ( W ` i ) = ( W ` 1 ) )
20		fveq2 6191	. . . . . . . 8 |- ( i = 1 -> ( <" A B C "> ` i ) = ( <" A B C "> ` 1 ) )
21	19, 20	eqeq12d 2637	. . . . . . 7 |- ( i = 1 -> ( ( W ` i ) = ( <" A B C "> ` i ) <-> ( W ` 1 ) = ( <" A B C "> ` 1 ) ) )
22		s3fv1 13637	. . . . . . . . 9 |- ( B e. V -> ( <" A B C "> ` 1 ) = B )
23	22	eqeq2d 2632	. . . . . . . 8 |- ( B e. V -> ( ( W ` 1 ) = ( <" A B C "> ` 1 ) <-> ( W ` 1 ) = B ) )
24	23	3ad2ant2 1083	. . . . . . 7 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( ( W ` 1 ) = ( <" A B C "> ` 1 ) <-> ( W ` 1 ) = B ) )
25	21, 24	sylan9bbr 737	. . . . . 6 |- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ i = 1 ) -> ( ( W ` i ) = ( <" A B C "> ` i ) <-> ( W ` 1 ) = B ) )
26		fveq2 6191	. . . . . . . 8 |- ( i = 2 -> ( W ` i ) = ( W ` 2 ) )
27		fveq2 6191	. . . . . . . 8 |- ( i = 2 -> ( <" A B C "> ` i ) = ( <" A B C "> ` 2 ) )
28	26, 27	eqeq12d 2637	. . . . . . 7 |- ( i = 2 -> ( ( W ` i ) = ( <" A B C "> ` i ) <-> ( W ` 2 ) = ( <" A B C "> ` 2 ) ) )
29		s3fv2 13638	. . . . . . . . 9 |- ( C e. V -> ( <" A B C "> ` 2 ) = C )
30	29	eqeq2d 2632	. . . . . . . 8 |- ( C e. V -> ( ( W ` 2 ) = ( <" A B C "> ` 2 ) <-> ( W ` 2 ) = C ) )
31	30	3ad2ant3 1084	. . . . . . 7 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( ( W ` 2 ) = ( <" A B C "> ` 2 ) <-> ( W ` 2 ) = C ) )
32	28, 31	sylan9bbr 737	. . . . . 6 |- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ i = 2 ) -> ( ( W ` i ) = ( <" A B C "> ` i ) <-> ( W ` 2 ) = C ) )
33		0zd 11389	. . . . . 6 |- ( ( A e. V /\ B e. V /\ C e. V ) -> 0 e. ZZ )
34		1zzd 11408	. . . . . 6 |- ( ( A e. V /\ B e. V /\ C e. V ) -> 1 e. ZZ )
35		2z 11409	. . . . . . 7 |- 2 e. ZZ
36	35	a1i 11	. . . . . 6 |- ( ( A e. V /\ B e. V /\ C e. V ) -> 2 e. ZZ )
37	18, 25, 32, 33, 34, 36	raltpd 4315	. . . . 5 |- ( ( A e. V /\ B e. V /\ C e. V ) -> ( A. i e. { 0 , 1 , 2 } ( W ` i ) = ( <" A B C "> ` i ) <-> ( ( W ` 0 ) = A /\ ( W ` 1 ) = B /\ ( W ` 2 ) = C ) ) )
38	37	adantl 482	. . . 4 |- ( ( W e. Word V /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( A. i e. { 0 , 1 , 2 } ( W ` i ) = ( <" A B C "> ` i ) <-> ( ( W ` 0 ) = A /\ ( W ` 1 ) = B /\ ( W ` 2 ) = C ) ) )
39	11, 38	sylan9bbr 737	. . 3 |- ( ( ( W e. Word V /\ ( A e. V /\ B e. V /\ C e. V ) ) /\ ( # ` W ) = 3 ) -> ( A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( <" A B C "> ` i ) <-> ( ( W ` 0 ) = A /\ ( W ` 1 ) = B /\ ( W ` 2 ) = C ) ) )
40	39	pm5.32da 673	. 2 |- ( ( W e. Word V /\ ( A e. V /\ B e. V /\ C e. V ) ) -> ( ( ( # ` W ) = 3 /\ A. i e. ( 0..^ ( # ` W ) ) ( W ` i ) = ( <" A B C "> ` i ) ) <-> ( ( # ` W ) = 3 /\ ( ( W ` 0 ) = A /\ ( W ` 1 ) = B /\ ( W ` 2 ) = C ) ) ) )
41	3, 7, 40	3bitrd 294	1 |- ( ( 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 ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {ctp 4181   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   2c2 11070   3c3 11071   ZZcz 11377  ..^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:  wrdl3s3  13705  s3sndisj  13706  s3iunsndisj  13707  elwwlks2ons3  26848  umgrwwlks2on  26850  elwwlks2  26861  elwspths2spth  26862