Metamath Proof Explorer

Theorem upwordsing

Description: Singleton is an increasing sequence for any compatible range. (Contributed by Ender Ting, 21-Nov-2024)

Ref Expression
Hypothesis upwordsing.1 
|- A e. S
Assertion upwordsing 
|- <" A "> e. UpWord S

Proof

Step Hyp Ref Expression
1 upwordsing.1 
 |-  A e. S
2 s1cl 
 |-  ( A e. S -> <" A "> e. Word S )
3 elab6g 
 |-  ( <" A "> e. Word S -> ( <" A "> e. { w | ( w e. Word S /\ A. k e. ( 0 ..^ ( ( # ` w ) - 1 ) ) ( w ` k ) < ( w ` ( k + 1 ) ) ) } <-> A. w ( w = <" A "> -> ( w e. Word S /\ A. k e. ( 0 ..^ ( ( # ` w ) - 1 ) ) ( w ` k ) < ( w ` ( k + 1 ) ) ) ) ) )
4 1 2 3 mp2b 
 |-  ( <" A "> e. { w | ( w e. Word S /\ A. k e. ( 0 ..^ ( ( # ` w ) - 1 ) ) ( w ` k ) < ( w ` ( k + 1 ) ) ) } <-> A. w ( w = <" A "> -> ( w e. Word S /\ A. k e. ( 0 ..^ ( ( # ` w ) - 1 ) ) ( w ` k ) < ( w ` ( k + 1 ) ) ) ) )
5 eleq1a 
 |-  ( <" A "> e. Word S -> ( w = <" A "> -> w e. Word S ) )
6 1 2 5 mp2b 
 |-  ( w = <" A "> -> w e. Word S )
7 fveq2 
 |-  ( w = <" A "> -> ( # ` w ) = ( # ` <" A "> ) )
8 7 oveq1d 
 |-  ( w = <" A "> -> ( ( # ` w ) - 1 ) = ( ( # ` <" A "> ) - 1 ) )
9 s1len 
 |-  ( # ` <" A "> ) = 1
10 9 oveq1i 
 |-  ( ( # ` <" A "> ) - 1 ) = ( 1 - 1 )
11 1m1e0 
 |-  ( 1 - 1 ) = 0
12 10 11 eqtri 
 |-  ( ( # ` <" A "> ) - 1 ) = 0
13 8 12 eqtrdi 
 |-  ( w = <" A "> -> ( ( # ` w ) - 1 ) = 0 )
14 13 oveq2d 
 |-  ( w = <" A "> -> ( 0 ..^ ( ( # ` w ) - 1 ) ) = ( 0 ..^ 0 ) )
15 fzo0 
 |-  ( 0 ..^ 0 ) = (/)
16 14 15 eqtrdi 
 |-  ( w = <" A "> -> ( 0 ..^ ( ( # ` w ) - 1 ) ) = (/) )
17 rzal 
 |-  ( ( 0 ..^ ( ( # ` w ) - 1 ) ) = (/) -> A. k e. ( 0 ..^ ( ( # ` w ) - 1 ) ) ( w ` k ) < ( w ` ( k + 1 ) ) )
18 16 17 syl 
 |-  ( w = <" A "> -> A. k e. ( 0 ..^ ( ( # ` w ) - 1 ) ) ( w ` k ) < ( w ` ( k + 1 ) ) )
19 6 18 jca 
 |-  ( w = <" A "> -> ( w e. Word S /\ A. k e. ( 0 ..^ ( ( # ` w ) - 1 ) ) ( w ` k ) < ( w ` ( k + 1 ) ) ) )
20 4 19 mpgbir 
 |-  <" A "> e. { w | ( w e. Word S /\ A. k e. ( 0 ..^ ( ( # ` w ) - 1 ) ) ( w ` k ) < ( w ` ( k + 1 ) ) ) }
21 df-upword 
 |-  UpWord S = { w | ( w e. Word S /\ A. k e. ( 0 ..^ ( ( # ` w ) - 1 ) ) ( w ` k ) < ( w ` ( k + 1 ) ) ) }
22 20 21 eleqtrri 
 |-  <" A "> e. UpWord S