Metamath Proof Explorer

Theorem constr01

Description: 0 and 1 are in all steps of the construction of constructible points. (Contributed by Thierry Arnoux, 25-Jun-2025)

Ref Expression
Hypotheses constr0.1 
|- C = rec ( ( s e. _V |-> { x e. CC | ( E. a e. s E. b e. s E. c e. s E. d e. s E. t e. RR E. r e. RR ( x = ( a + ( t x. ( b - a ) ) ) /\ x = ( c + ( r x. ( d - c ) ) ) /\ ( Im ` ( ( * ` ( b - a ) ) x. ( d - c ) ) ) =/= 0 ) \/ E. a e. s E. b e. s E. c e. s E. e e. s E. f e. s E. t e. RR ( x = ( a + ( t x. ( b - a ) ) ) /\ ( abs ` ( x - c ) ) = ( abs ` ( e - f ) ) ) \/ E. a e. s E. b e. s E. c e. s E. d e. s E. e e. s E. f e. s ( a =/= d /\ ( abs ` ( x - a ) ) = ( abs ` ( b - c ) ) /\ ( abs ` ( x - d ) ) = ( abs ` ( e - f ) ) ) ) } ) , { 0 , 1 } )
constrsscn.1 
|- ( ph -> N e. On )
Assertion constr01 
|- ( ph -> { 0 , 1 } C_ ( C ` N ) )

Proof

Step Hyp Ref Expression
1 constr0.1 
 |-  C = rec ( ( s e. _V |-> { x e. CC | ( E. a e. s E. b e. s E. c e. s E. d e. s E. t e. RR E. r e. RR ( x = ( a + ( t x. ( b - a ) ) ) /\ x = ( c + ( r x. ( d - c ) ) ) /\ ( Im ` ( ( * ` ( b - a ) ) x. ( d - c ) ) ) =/= 0 ) \/ E. a e. s E. b e. s E. c e. s E. e e. s E. f e. s E. t e. RR ( x = ( a + ( t x. ( b - a ) ) ) /\ ( abs ` ( x - c ) ) = ( abs ` ( e - f ) ) ) \/ E. a e. s E. b e. s E. c e. s E. d e. s E. e e. s E. f e. s ( a =/= d /\ ( abs ` ( x - a ) ) = ( abs ` ( b - c ) ) /\ ( abs ` ( x - d ) ) = ( abs ` ( e - f ) ) ) ) } ) , { 0 , 1 } )
2 constrsscn.1 
 |-  ( ph -> N e. On )
3 fveq2 
 |-  ( m = (/) -> ( C ` m ) = ( C ` (/) ) )
4 3 sseq2d 
 |-  ( m = (/) -> ( { 0 , 1 } C_ ( C ` m ) <-> { 0 , 1 } C_ ( C ` (/) ) ) )
5 fveq2 
 |-  ( m = n -> ( C ` m ) = ( C ` n ) )
6 5 sseq2d 
 |-  ( m = n -> ( { 0 , 1 } C_ ( C ` m ) <-> { 0 , 1 } C_ ( C ` n ) ) )
7 fveq2 
 |-  ( m = suc n -> ( C ` m ) = ( C ` suc n ) )
8 7 sseq2d 
 |-  ( m = suc n -> ( { 0 , 1 } C_ ( C ` m ) <-> { 0 , 1 } C_ ( C ` suc n ) ) )
9 fveq2 
 |-  ( m = N -> ( C ` m ) = ( C ` N ) )
10 9 sseq2d 
 |-  ( m = N -> ( { 0 , 1 } C_ ( C ` m ) <-> { 0 , 1 } C_ ( C ` N ) ) )
11 1 constr0 
 |-  ( C ` (/) ) = { 0 , 1 }
12 11 eqimss2i 
 |-  { 0 , 1 } C_ ( C ` (/) )
13 simpr 
 |-  ( ( n e. On /\ { 0 , 1 } C_ ( C ` n ) ) -> { 0 , 1 } C_ ( C ` n ) )
14 simpl 
 |-  ( ( n e. On /\ { 0 , 1 } C_ ( C ` n ) ) -> n e. On )
15 0elpr01 
 |-  0 e. { 0 , 1 }
16 15 a1i 
 |-  ( ( n e. On /\ { 0 , 1 } C_ ( C ` n ) ) -> 0 e. { 0 , 1 } )
17 13 16 sseldd 
 |-  ( ( n e. On /\ { 0 , 1 } C_ ( C ` n ) ) -> 0 e. ( C ` n ) )
18 1 14 17 constrsslem 
 |-  ( ( n e. On /\ { 0 , 1 } C_ ( C ` n ) ) -> ( C ` n ) C_ ( C ` suc n ) )
19 13 18 sstrd 
 |-  ( ( n e. On /\ { 0 , 1 } C_ ( C ` n ) ) -> { 0 , 1 } C_ ( C ` suc n ) )
20 19 ex 
 |-  ( n e. On -> ( { 0 , 1 } C_ ( C ` n ) -> { 0 , 1 } C_ ( C ` suc n ) ) )
21 0ellim 
 |-  ( Lim m -> (/) e. m )
22 fveq2 
 |-  ( o = (/) -> ( C ` o ) = ( C ` (/) ) )
23 22 11 eqtrdi 
 |-  ( o = (/) -> ( C ` o ) = { 0 , 1 } )
24 23 ssiun2s 
 |-  ( (/) e. m -> { 0 , 1 } C_ U_ o e. m ( C ` o ) )
25 21 24 syl 
 |-  ( Lim m -> { 0 , 1 } C_ U_ o e. m ( C ` o ) )
26 vex 
 |-  m e. _V
27 26 a1i 
 |-  ( Lim m -> m e. _V )
28 id 
 |-  ( Lim m -> Lim m )
29 1 27 28 constrlim 
 |-  ( Lim m -> ( C ` m ) = U_ o e. m ( C ` o ) )
30 25 29 sseqtrrd 
 |-  ( Lim m -> { 0 , 1 } C_ ( C ` m ) )
31 30 a1d 
 |-  ( Lim m -> ( A. n e. m { 0 , 1 } C_ ( C ` n ) -> { 0 , 1 } C_ ( C ` m ) ) )
32 4 6 8 10 12 20 31 tfinds 
 |-  ( N e. On -> { 0 , 1 } C_ ( C ` N ) )
33 2 32 syl 
 |-  ( ph -> { 0 , 1 } C_ ( C ` N ) )