Theorem eqs1 11316

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 110	. . 3 |- ( ( W e. Word A /\ (♯ ` W ) = 1 ) -> (♯ ` W ) = 1 )
2		0nn0 9511	. . . . . 6 |- 0 e. NN0
3		fvexg 5689	. . . . . 6 |- ( ( W e. Word A /\ 0 e. NN0 ) -> ( W ` 0 ) e. _V )
4	2, 3	mpan2 425	. . . . 5 |- ( W e. Word A -> ( W ` 0 ) e. _V )
5	4	adantr 276	. . . 4 |- ( ( W e. Word A /\ (♯ ` W ) = 1 ) -> ( W ` 0 ) e. _V )
6		s1leng 11312	. . . 4 |- ( ( W ` 0 ) e. _V -> (♯ ` <" ( W ` 0 ) "> ) = 1 )
7	5, 6	syl 14	. . 3 |- ( ( W e. Word A /\ (♯ ` W ) = 1 ) -> (♯ ` <" ( W ` 0 ) "> ) = 1 )
8	1, 7	eqtr4d 2268	. 2 |- ( ( W e. Word A /\ (♯ ` W ) = 1 ) -> (♯ ` W ) = (♯ ` <" ( W ` 0 ) "> ) )
9		s1fv 11314	. . . . . . 7 |- ( ( W ` 0 ) e. _V -> ( <" ( W ` 0 ) "> ` 0 ) = ( W ` 0 ) )
10	4, 9	syl 14	. . . . . 6 |- ( W e. Word A -> ( <" ( W ` 0 ) "> ` 0 ) = ( W ` 0 ) )
11	10	eqcomd 2238	. . . . 5 |- ( W e. Word A -> ( W ` 0 ) = ( <" ( W ` 0 ) "> ` 0 ) )
12		c0ex 8268	. . . . . 6 |- 0 e. _V
13		fveq2 5670	. . . . . . 7 |- ( x = 0 -> ( W ` x ) = ( W ` 0 ) )
14		fveq2 5670	. . . . . . 7 |- ( x = 0 -> ( <" ( W ` 0 ) "> ` x ) = ( <" ( W ` 0 ) "> ` 0 ) )
15	13, 14	eqeq12d 2247	. . . . . 6 |- ( x = 0 -> ( ( W ` x ) = ( <" ( W ` 0 ) "> ` x ) <-> ( W ` 0 ) = ( <" ( W ` 0 ) "> ` 0 ) ) )
16	12, 15	ralsn 3732	. . . . 5 |- ( A. x e. { 0 } ( W ` x ) = ( <" ( W ` 0 ) "> ` x ) <-> ( W ` 0 ) = ( <" ( W ` 0 ) "> ` 0 ) )
17	11, 16	sylibr 134	. . . 4 |- ( W e. Word A -> A. x e. { 0 } ( W ` x ) = ( <" ( W ` 0 ) "> ` x ) )
18	17	adantr 276	. . 3 |- ( ( W e. Word A /\ (♯ ` W ) = 1 ) -> A. x e. { 0 } ( W ` x ) = ( <" ( W ` 0 ) "> ` x ) )
19		oveq2 6058	. . . . . 6 |- ( (♯ ` W ) = 1 -> ( 0..^ (♯ ` W ) ) = ( 0..^ 1 ) )
20		fzo01 10561	. . . . . 6 |- ( 0..^ 1 ) = { 0 }
21	19, 20	eqtrdi 2281	. . . . 5 |- ( (♯ ` W ) = 1 -> ( 0..^ (♯ ` W ) ) = { 0 } )
22	21	raleqdv 2747	. . . 4 |- ( (♯ ` W ) = 1 -> ( A. x e. ( 0..^ (♯ ` W ) ) ( W ` x ) = ( <" ( W ` 0 ) "> ` x ) <-> A. x e. { 0 } ( W ` x ) = ( <" ( W ` 0 ) "> ` x ) ) )
23	22	adantl 277	. . 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 ) ) )
24	18, 23	mpbird 167	. 2 |- ( ( W e. Word A /\ (♯ ` W ) = 1 ) -> A. x e. ( 0..^ (♯ ` W ) ) ( W ` x ) = ( <" ( W ` 0 ) "> ` x ) )
25	4	s1cld 11310	. . . 4 |- ( W e. Word A -> <" ( W ` 0 ) "> e. Word _V )
26		eqwrd 11265	. . . 4 |- ( ( W e. Word A /\ <" ( W ` 0 ) "> e. Word _V ) -> ( W = <" ( W ` 0 ) "> <-> ( (♯ ` W ) = (♯ ` <" ( W ` 0 ) "> ) /\ A. x e. ( 0..^ (♯ ` W ) ) ( W ` x ) = ( <" ( W ` 0 ) "> ` x ) ) ) )
27	25, 26	mpdan 421	. . 3 |- ( W e. Word A -> ( W = <" ( W ` 0 ) "> <-> ( (♯ ` W ) = (♯ ` <" ( W ` 0 ) "> ) /\ A. x e. ( 0..^ (♯ ` W ) ) ( W ` x ) = ( <" ( W ` 0 ) "> ` x ) ) ) )
28	27	adantr 276	. 2 |- ( ( W e. Word A /\ (♯ ` W ) = 1 ) -> ( W = <" ( W ` 0 ) "> <-> ( (♯ ` W ) = (♯ ` <" ( W ` 0 ) "> ) /\ A. x e. ( 0..^ (♯ ` W ) ) ( W ` x ) = ( <" ( W ` 0 ) "> ` x ) ) ) )
29	8, 24, 28	mpbir2and 953	1 |- ( ( W e. Word A /\ (♯ ` W ) = 1 ) -> W = <" ( W ` 0 ) "> )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   A.wral 2520   _Vcvv 2813   {csn 3689   ` cfv 5352  (class class class)co 6050   0cc0 8127   1c1 8128   NN0cn0 9496  ..^cfzo 10476  ♯chash 11138  Word cword 11224   <"cs1 11303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343  df-fzo 10477  df-ihash 11139  df-word 11225  df-s1 11304

This theorem is referenced by:  wrdl1exs1  11317  wrdl1s1  11318  swrds1  11360