Theorem eqs1 13392

Description: A word of length 1 is a singleton word. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Proof shortened by AV, 1-May-2020.)

Assertion

Ref	Expression
eqs1	|- ( ( W e. Word A /\ ( # ` W ) = 1 ) -> W = <" ( W ` 0 ) "> )

Proof of Theorem eqs1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . 3 |- ( ( W e. Word A /\ ( # ` W ) = 1 ) -> ( # ` W ) = 1 )
2		s1len 13385	. . 3 |- ( # ` <" ( W ` 0 ) "> ) = 1
3	1, 2	syl6eqr 2674	. 2 |- ( ( W e. Word A /\ ( # ` W ) = 1 ) -> ( # ` W ) = ( # ` <" ( W ` 0 ) "> ) )
4		fvex 6201	. . . . 5 |- ( W ` 0 ) e. _V
5		s1fv 13390	. . . . . 6 |- ( ( W ` 0 ) e. _V -> ( <" ( W ` 0 ) "> ` 0 ) = ( W ` 0 ) )
6	5	eqcomd 2628	. . . . 5 |- ( ( W ` 0 ) e. _V -> ( W ` 0 ) = ( <" ( W ` 0 ) "> ` 0 ) )
7	4, 6	mp1i 13	. . . 4 |- ( ( W e. Word A /\ ( # ` W ) = 1 ) -> ( W ` 0 ) = ( <" ( W ` 0 ) "> ` 0 ) )
8		c0ex 10034	. . . . 5 |- 0 e. _V
9		fveq2 6191	. . . . . 6 |- ( x = 0 -> ( W ` x ) = ( W ` 0 ) )
10		fveq2 6191	. . . . . 6 |- ( x = 0 -> ( <" ( W ` 0 ) "> ` x ) = ( <" ( W ` 0 ) "> ` 0 ) )
11	9, 10	eqeq12d 2637	. . . . 5 |- ( x = 0 -> ( ( W ` x ) = ( <" ( W ` 0 ) "> ` x ) <-> ( W ` 0 ) = ( <" ( W ` 0 ) "> ` 0 ) ) )
12	8, 11	ralsn 4222	. . . 4 |- ( A. x e. { 0 } ( W ` x ) = ( <" ( W ` 0 ) "> ` x ) <-> ( W ` 0 ) = ( <" ( W ` 0 ) "> ` 0 ) )
13	7, 12	sylibr 224	. . 3 |- ( ( W e. Word A /\ ( # ` W ) = 1 ) -> A. x e. { 0 } ( W ` x ) = ( <" ( W ` 0 ) "> ` x ) )
14		oveq2 6658	. . . . . 6 |- ( ( # ` W ) = 1 -> ( 0..^ ( # ` W ) ) = ( 0..^ 1 ) )
15	14	adantl 482	. . . . 5 |- ( ( W e. Word A /\ ( # ` W ) = 1 ) -> ( 0..^ ( # ` W ) ) = ( 0..^ 1 ) )
16		fzo01 12550	. . . . 5 |- ( 0..^ 1 ) = { 0 }
17	15, 16	syl6eq 2672	. . . 4 |- ( ( W e. Word A /\ ( # ` W ) = 1 ) -> ( 0..^ ( # ` W ) ) = { 0 } )
18	17	raleqdv 3144	. . 3 |- ( ( W e. Word A /\ ( # ` W ) = 1 ) -> ( A. x e. ( 0..^ ( # ` W ) ) ( W ` x ) = ( <" ( W ` 0 ) "> ` x ) <-> A. x e. { 0 } ( W ` x ) = ( <" ( W ` 0 ) "> ` x ) ) )
19	13, 18	mpbird 247	. 2 |- ( ( W e. Word A /\ ( # ` W ) = 1 ) -> A. x e. ( 0..^ ( # ` W ) ) ( W ` x ) = ( <" ( W ` 0 ) "> ` x ) )
20		1nn 11031	. . . . 5 |- 1 e. NN
21		fstwrdne0 13345	. . . . 5 |- ( ( 1 e. NN /\ ( W e. Word A /\ ( # ` W ) = 1 ) ) -> ( W ` 0 ) e. A )
22	20, 21	mpan 706	. . . 4 |- ( ( W e. Word A /\ ( # ` W ) = 1 ) -> ( W ` 0 ) e. A )
23	22	s1cld 13383	. . 3 |- ( ( W e. Word A /\ ( # ` W ) = 1 ) -> <" ( W ` 0 ) "> e. Word A )
24		eqwrd 13346	. . 3 |- ( ( W e. Word A /\ <" ( W ` 0 ) "> e. Word A ) -> ( W = <" ( W ` 0 ) "> <-> ( ( # ` W ) = ( # ` <" ( W ` 0 ) "> ) /\ A. x e. ( 0..^ ( # ` W ) ) ( W ` x ) = ( <" ( W ` 0 ) "> ` x ) ) ) )
25	23, 24	syldan 487	. 2 |- ( ( W e. Word A /\ ( # ` W ) = 1 ) -> ( W = <" ( W ` 0 ) "> <-> ( ( # ` W ) = ( # ` <" ( W ` 0 ) "> ) /\ A. x e. ( 0..^ ( # ` W ) ) ( W ` x ) = ( <" ( W ` 0 ) "> ` x ) ) ) )
26	3, 19, 25	mpbir2and 957	1 |- ( ( W e. Word A /\ ( # ` W ) = 1 ) -> W = <" ( W ` 0 ) "> )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   {csn 4177   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   NNcn 11020  ..^cfzo 12465   #chash 13117  Word cword 13291   <"cs1 13294

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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-s1 13302

This theorem is referenced by:  wrdl1exs1  13393  wrdl1s1  13394  swrds1  13451  revs1  13514  signsvtn0  30647