Theorem eqs1 11341

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 9528	. . . . . 6 |- 0 e. NN0
3		fvexg 5694	. . . . . 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 11337	. . . 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 2270	. 2 |- ( ( W e. Word A /\ (♯ ` W ) = 1 ) -> (♯ ` W ) = (♯ ` <" ( W ` 0 ) "> ) )
9		s1fv 11339	. . . . . . 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 2240	. . . . 5 |- ( W e. Word A -> ( W ` 0 ) = ( <" ( W ` 0 ) "> ` 0 ) )
12		c0ex 8284	. . . . . 6 |- 0 e. _V
13		fveq2 5675	. . . . . . 7 |- ( x = 0 -> ( W ` x ) = ( W ` 0 ) )
14		fveq2 5675	. . . . . . 7 |- ( x = 0 -> ( <" ( W ` 0 ) "> ` x ) = ( <" ( W ` 0 ) "> ` 0 ) )
15	13, 14	eqeq12d 2249	. . . . . 6 |- ( x = 0 -> ( ( W ` x ) = ( <" ( W ` 0 ) "> ` x ) <-> ( W ` 0 ) = ( <" ( W ` 0 ) "> ` 0 ) ) )
16	12, 15	ralsn 3737	. . . . 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 6066	. . . . . 6 |- ( (♯ ` W ) = 1 -> ( 0..^ (♯ ` W ) ) = ( 0..^ 1 ) )
20		fzo01 10583	. . . . . 6 |- ( 0..^ 1 ) = { 0 }
21	19, 20	eqtrdi 2283	. . . . 5 |- ( (♯ ` W ) = 1 -> ( 0..^ (♯ ` W ) ) = { 0 } )
22	21	raleqdv 2749	. . . 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 11335	. . . 4 |- ( W e. Word A -> <" ( W ` 0 ) "> e. Word _V )
26		eqwrd 11290	. . . 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 2205   A.wral 2522   _Vcvv 2815   {csn 3694   ` cfv 5357  (class class class)co 6058   0cc0 8143   1c1 8144   NN0cn0 9513  ..^cfzo 10498  ♯chash 11163  Word cword 11249   <"cs1 11328

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-s1 11329

This theorem is referenced by:  wrdl1exs1  11342  wrdl1s1  11343  swrds1  11385