HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  golem1 Structured version   Unicode version

Theorem golem1 25826
Description: Lemma for Godowski's equation. (Contributed by NM, 10-Nov-2002.) (New usage is discouraged.)
Hypotheses
Ref Expression
golem1.1  |-  A  e. 
CH
golem1.2  |-  B  e. 
CH
golem1.3  |-  C  e. 
CH
golem1.4  |-  F  =  ( ( _|_ `  A
)  vH  ( A  i^i  B ) )
golem1.5  |-  G  =  ( ( _|_ `  B
)  vH  ( B  i^i  C ) )
golem1.6  |-  H  =  ( ( _|_ `  C
)  vH  ( C  i^i  A ) )
golem1.7  |-  D  =  ( ( _|_ `  B
)  vH  ( B  i^i  A ) )
golem1.8  |-  R  =  ( ( _|_ `  C
)  vH  ( C  i^i  B ) )
golem1.9  |-  S  =  ( ( _|_ `  A
)  vH  ( A  i^i  C ) )
Assertion
Ref Expression
golem1  |-  ( f  e.  States  ->  ( ( ( f `  F )  +  ( f `  G ) )  +  ( f `  H
) )  =  ( ( ( f `  D )  +  ( f `  R ) )  +  ( f `
 S ) ) )

Proof of Theorem golem1
StepHypRef Expression
1 golem1.1 . . . . . . . . . . 11  |-  A  e. 
CH
21choccli 24861 . . . . . . . . . 10  |-  ( _|_ `  A )  e.  CH
3 stcl 25771 . . . . . . . . . 10  |-  ( f  e.  States  ->  ( ( _|_ `  A )  e.  CH  ->  ( f `  ( _|_ `  A ) )  e.  RR ) )
42, 3mpi 17 . . . . . . . . 9  |-  ( f  e.  States  ->  ( f `  ( _|_ `  A ) )  e.  RR )
54recnd 9522 . . . . . . . 8  |-  ( f  e.  States  ->  ( f `  ( _|_ `  A ) )  e.  CC )
6 golem1.2 . . . . . . . . . . 11  |-  B  e. 
CH
76choccli 24861 . . . . . . . . . 10  |-  ( _|_ `  B )  e.  CH
8 stcl 25771 . . . . . . . . . 10  |-  ( f  e.  States  ->  ( ( _|_ `  B )  e.  CH  ->  ( f `  ( _|_ `  B ) )  e.  RR ) )
97, 8mpi 17 . . . . . . . . 9  |-  ( f  e.  States  ->  ( f `  ( _|_ `  B ) )  e.  RR )
109recnd 9522 . . . . . . . 8  |-  ( f  e.  States  ->  ( f `  ( _|_ `  B ) )  e.  CC )
11 golem1.3 . . . . . . . . . . 11  |-  C  e. 
CH
1211choccli 24861 . . . . . . . . . 10  |-  ( _|_ `  C )  e.  CH
13 stcl 25771 . . . . . . . . . 10  |-  ( f  e.  States  ->  ( ( _|_ `  C )  e.  CH  ->  ( f `  ( _|_ `  C ) )  e.  RR ) )
1412, 13mpi 17 . . . . . . . . 9  |-  ( f  e.  States  ->  ( f `  ( _|_ `  C ) )  e.  RR )
1514recnd 9522 . . . . . . . 8  |-  ( f  e.  States  ->  ( f `  ( _|_ `  C ) )  e.  CC )
165, 10, 15addassd 9518 . . . . . . 7  |-  ( f  e.  States  ->  ( ( ( f `  ( _|_ `  A ) )  +  ( f `  ( _|_ `  B ) ) )  +  ( f `
 ( _|_ `  C
) ) )  =  ( ( f `  ( _|_ `  A ) )  +  ( ( f `  ( _|_ `  B ) )  +  ( f `  ( _|_ `  C ) ) ) ) )
1710, 15addcld 9515 . . . . . . . 8  |-  ( f  e.  States  ->  ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( _|_ `  C ) ) )  e.  CC )
185, 17addcomd 9681 . . . . . . 7  |-  ( f  e.  States  ->  ( ( f `
 ( _|_ `  A
) )  +  ( ( f `  ( _|_ `  B ) )  +  ( f `  ( _|_ `  C ) ) ) )  =  ( ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( _|_ `  C ) ) )  +  ( f `  ( _|_ `  A ) ) ) )
1916, 18eqtrd 2495 . . . . . 6  |-  ( f  e.  States  ->  ( ( ( f `  ( _|_ `  A ) )  +  ( f `  ( _|_ `  B ) ) )  +  ( f `
 ( _|_ `  C
) ) )  =  ( ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( _|_ `  C ) ) )  +  ( f `  ( _|_ `  A ) ) ) )
2019oveq1d 6214 . . . . 5  |-  ( f  e.  States  ->  ( ( ( ( f `  ( _|_ `  A ) )  +  ( f `  ( _|_ `  B ) ) )  +  ( f `  ( _|_ `  C ) ) )  +  ( ( ( f `  ( A  i^i  B ) )  +  ( f `  ( B  i^i  C ) ) )  +  ( f `  ( C  i^i  A ) ) ) )  =  ( ( ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( _|_ `  C ) ) )  +  ( f `  ( _|_ `  A ) ) )  +  ( ( ( f `  ( A  i^i  B ) )  +  ( f `
 ( B  i^i  C ) ) )  +  ( f `  ( C  i^i  A ) ) ) ) )
215, 10addcld 9515 . . . . . 6  |-  ( f  e.  States  ->  ( ( f `
 ( _|_ `  A
) )  +  ( f `  ( _|_ `  B ) ) )  e.  CC )
221, 6chincli 25014 . . . . . . . . 9  |-  ( A  i^i  B )  e. 
CH
23 stcl 25771 . . . . . . . . 9  |-  ( f  e.  States  ->  ( ( A  i^i  B )  e. 
CH  ->  ( f `  ( A  i^i  B ) )  e.  RR ) )
2422, 23mpi 17 . . . . . . . 8  |-  ( f  e.  States  ->  ( f `  ( A  i^i  B ) )  e.  RR )
2524recnd 9522 . . . . . . 7  |-  ( f  e.  States  ->  ( f `  ( A  i^i  B ) )  e.  CC )
266, 11chincli 25014 . . . . . . . . 9  |-  ( B  i^i  C )  e. 
CH
27 stcl 25771 . . . . . . . . 9  |-  ( f  e.  States  ->  ( ( B  i^i  C )  e. 
CH  ->  ( f `  ( B  i^i  C ) )  e.  RR ) )
2826, 27mpi 17 . . . . . . . 8  |-  ( f  e.  States  ->  ( f `  ( B  i^i  C ) )  e.  RR )
2928recnd 9522 . . . . . . 7  |-  ( f  e.  States  ->  ( f `  ( B  i^i  C ) )  e.  CC )
3025, 29addcld 9515 . . . . . 6  |-  ( f  e.  States  ->  ( ( f `
 ( A  i^i  B ) )  +  ( f `  ( B  i^i  C ) ) )  e.  CC )
3111, 1chincli 25014 . . . . . . . 8  |-  ( C  i^i  A )  e. 
CH
32 stcl 25771 . . . . . . . 8  |-  ( f  e.  States  ->  ( ( C  i^i  A )  e. 
CH  ->  ( f `  ( C  i^i  A ) )  e.  RR ) )
3331, 32mpi 17 . . . . . . 7  |-  ( f  e.  States  ->  ( f `  ( C  i^i  A ) )  e.  RR )
3433recnd 9522 . . . . . 6  |-  ( f  e.  States  ->  ( f `  ( C  i^i  A ) )  e.  CC )
3521, 30, 15, 34add4d 9703 . . . . 5  |-  ( f  e.  States  ->  ( ( ( ( f `  ( _|_ `  A ) )  +  ( f `  ( _|_ `  B ) ) )  +  ( ( f `  ( A  i^i  B ) )  +  ( f `  ( B  i^i  C ) ) ) )  +  ( ( f `  ( _|_ `  C ) )  +  ( f `
 ( C  i^i  A ) ) ) )  =  ( ( ( ( f `  ( _|_ `  A ) )  +  ( f `  ( _|_ `  B ) ) )  +  ( f `  ( _|_ `  C ) ) )  +  ( ( ( f `  ( A  i^i  B ) )  +  ( f `  ( B  i^i  C ) ) )  +  ( f `  ( C  i^i  A ) ) ) ) )
3617, 30, 5, 34add4d 9703 . . . . 5  |-  ( f  e.  States  ->  ( ( ( ( f `  ( _|_ `  B ) )  +  ( f `  ( _|_ `  C ) ) )  +  ( ( f `  ( A  i^i  B ) )  +  ( f `  ( B  i^i  C ) ) ) )  +  ( ( f `  ( _|_ `  A ) )  +  ( f `
 ( C  i^i  A ) ) ) )  =  ( ( ( ( f `  ( _|_ `  B ) )  +  ( f `  ( _|_ `  C ) ) )  +  ( f `  ( _|_ `  A ) ) )  +  ( ( ( f `  ( A  i^i  B ) )  +  ( f `  ( B  i^i  C ) ) )  +  ( f `  ( C  i^i  A ) ) ) ) )
3720, 35, 363eqtr4d 2505 . . . 4  |-  ( f  e.  States  ->  ( ( ( ( f `  ( _|_ `  A ) )  +  ( f `  ( _|_ `  B ) ) )  +  ( ( f `  ( A  i^i  B ) )  +  ( f `  ( B  i^i  C ) ) ) )  +  ( ( f `  ( _|_ `  C ) )  +  ( f `
 ( C  i^i  A ) ) ) )  =  ( ( ( ( f `  ( _|_ `  B ) )  +  ( f `  ( _|_ `  C ) ) )  +  ( ( f `  ( A  i^i  B ) )  +  ( f `  ( B  i^i  C ) ) ) )  +  ( ( f `  ( _|_ `  A ) )  +  ( f `
 ( C  i^i  A ) ) ) ) )
385, 25, 10, 29add4d 9703 . . . . 5  |-  ( f  e.  States  ->  ( ( ( f `  ( _|_ `  A ) )  +  ( f `  ( A  i^i  B ) ) )  +  ( ( f `  ( _|_ `  B ) )  +  ( f `  ( B  i^i  C ) ) ) )  =  ( ( ( f `  ( _|_ `  A ) )  +  ( f `
 ( _|_ `  B
) ) )  +  ( ( f `  ( A  i^i  B ) )  +  ( f `
 ( B  i^i  C ) ) ) ) )
3938oveq1d 6214 . . . 4  |-  ( f  e.  States  ->  ( ( ( ( f `  ( _|_ `  A ) )  +  ( f `  ( A  i^i  B ) ) )  +  ( ( f `  ( _|_ `  B ) )  +  ( f `  ( B  i^i  C ) ) ) )  +  ( ( f `  ( _|_ `  C ) )  +  ( f `
 ( C  i^i  A ) ) ) )  =  ( ( ( ( f `  ( _|_ `  A ) )  +  ( f `  ( _|_ `  B ) ) )  +  ( ( f `  ( A  i^i  B ) )  +  ( f `  ( B  i^i  C ) ) ) )  +  ( ( f `  ( _|_ `  C ) )  +  ( f `
 ( C  i^i  A ) ) ) ) )
4010, 25, 15, 29add4d 9703 . . . . 5  |-  ( f  e.  States  ->  ( ( ( f `  ( _|_ `  B ) )  +  ( f `  ( A  i^i  B ) ) )  +  ( ( f `  ( _|_ `  C ) )  +  ( f `  ( B  i^i  C ) ) ) )  =  ( ( ( f `  ( _|_ `  B ) )  +  ( f `
 ( _|_ `  C
) ) )  +  ( ( f `  ( A  i^i  B ) )  +  ( f `
 ( B  i^i  C ) ) ) ) )
4140oveq1d 6214 . . . 4  |-  ( f  e.  States  ->  ( ( ( ( f `  ( _|_ `  B ) )  +  ( f `  ( A  i^i  B ) ) )  +  ( ( f `  ( _|_ `  C ) )  +  ( f `  ( B  i^i  C ) ) ) )  +  ( ( f `  ( _|_ `  A ) )  +  ( f `
 ( C  i^i  A ) ) ) )  =  ( ( ( ( f `  ( _|_ `  B ) )  +  ( f `  ( _|_ `  C ) ) )  +  ( ( f `  ( A  i^i  B ) )  +  ( f `  ( B  i^i  C ) ) ) )  +  ( ( f `  ( _|_ `  A ) )  +  ( f `
 ( C  i^i  A ) ) ) ) )
4237, 39, 413eqtr4d 2505 . . 3  |-  ( f  e.  States  ->  ( ( ( ( f `  ( _|_ `  A ) )  +  ( f `  ( A  i^i  B ) ) )  +  ( ( f `  ( _|_ `  B ) )  +  ( f `  ( B  i^i  C ) ) ) )  +  ( ( f `  ( _|_ `  C ) )  +  ( f `
 ( C  i^i  A ) ) ) )  =  ( ( ( ( f `  ( _|_ `  B ) )  +  ( f `  ( A  i^i  B ) ) )  +  ( ( f `  ( _|_ `  C ) )  +  ( f `  ( B  i^i  C ) ) ) )  +  ( ( f `  ( _|_ `  A ) )  +  ( f `
 ( C  i^i  A ) ) ) ) )
431, 6stji1i 25797 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  A
)  vH  ( A  i^i  B ) ) )  =  ( ( f `
 ( _|_ `  A
) )  +  ( f `  ( A  i^i  B ) ) ) )
446, 11stji1i 25797 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  B
)  vH  ( B  i^i  C ) ) )  =  ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( B  i^i  C ) ) ) )
4543, 44oveq12d 6217 . . . 4  |-  ( f  e.  States  ->  ( ( f `
 ( ( _|_ `  A )  vH  ( A  i^i  B ) ) )  +  ( f `
 ( ( _|_ `  B )  vH  ( B  i^i  C ) ) ) )  =  ( ( ( f `  ( _|_ `  A ) )  +  ( f `
 ( A  i^i  B ) ) )  +  ( ( f `  ( _|_ `  B ) )  +  ( f `
 ( B  i^i  C ) ) ) ) )
4611, 1stji1i 25797 . . . 4  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  C
)  vH  ( C  i^i  A ) ) )  =  ( ( f `
 ( _|_ `  C
) )  +  ( f `  ( C  i^i  A ) ) ) )
4745, 46oveq12d 6217 . . 3  |-  ( f  e.  States  ->  ( ( ( f `  ( ( _|_ `  A )  vH  ( A  i^i  B ) ) )  +  ( f `  (
( _|_ `  B
)  vH  ( B  i^i  C ) ) ) )  +  ( f `
 ( ( _|_ `  C )  vH  ( C  i^i  A ) ) ) )  =  ( ( ( ( f `
 ( _|_ `  A
) )  +  ( f `  ( A  i^i  B ) ) )  +  ( ( f `  ( _|_ `  B ) )  +  ( f `  ( B  i^i  C ) ) ) )  +  ( ( f `  ( _|_ `  C ) )  +  ( f `  ( C  i^i  A ) ) ) ) )
486, 1stji1i 25797 . . . . . 6  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  B
)  vH  ( B  i^i  A ) ) )  =  ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( B  i^i  A ) ) ) )
49 incom 3650 . . . . . . . 8  |-  ( B  i^i  A )  =  ( A  i^i  B
)
5049fveq2i 5801 . . . . . . 7  |-  ( f `
 ( B  i^i  A ) )  =  ( f `  ( A  i^i  B ) )
5150oveq2i 6210 . . . . . 6  |-  ( ( f `  ( _|_ `  B ) )  +  ( f `  ( B  i^i  A ) ) )  =  ( ( f `  ( _|_ `  B ) )  +  ( f `  ( A  i^i  B ) ) )
5248, 51syl6eq 2511 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  B
)  vH  ( B  i^i  A ) ) )  =  ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( A  i^i  B ) ) ) )
5311, 6stji1i 25797 . . . . . 6  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  C
)  vH  ( C  i^i  B ) ) )  =  ( ( f `
 ( _|_ `  C
) )  +  ( f `  ( C  i^i  B ) ) ) )
54 incom 3650 . . . . . . . 8  |-  ( C  i^i  B )  =  ( B  i^i  C
)
5554fveq2i 5801 . . . . . . 7  |-  ( f `
 ( C  i^i  B ) )  =  ( f `  ( B  i^i  C ) )
5655oveq2i 6210 . . . . . 6  |-  ( ( f `  ( _|_ `  C ) )  +  ( f `  ( C  i^i  B ) ) )  =  ( ( f `  ( _|_ `  C ) )  +  ( f `  ( B  i^i  C ) ) )
5753, 56syl6eq 2511 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  C
)  vH  ( C  i^i  B ) ) )  =  ( ( f `
 ( _|_ `  C
) )  +  ( f `  ( B  i^i  C ) ) ) )
5852, 57oveq12d 6217 . . . 4  |-  ( f  e.  States  ->  ( ( f `
 ( ( _|_ `  B )  vH  ( B  i^i  A ) ) )  +  ( f `
 ( ( _|_ `  C )  vH  ( C  i^i  B ) ) ) )  =  ( ( ( f `  ( _|_ `  B ) )  +  ( f `
 ( A  i^i  B ) ) )  +  ( ( f `  ( _|_ `  C ) )  +  ( f `
 ( B  i^i  C ) ) ) ) )
591, 11stji1i 25797 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  A
)  vH  ( A  i^i  C ) ) )  =  ( ( f `
 ( _|_ `  A
) )  +  ( f `  ( A  i^i  C ) ) ) )
60 incom 3650 . . . . . . 7  |-  ( A  i^i  C )  =  ( C  i^i  A
)
6160fveq2i 5801 . . . . . 6  |-  ( f `
 ( A  i^i  C ) )  =  ( f `  ( C  i^i  A ) )
6261oveq2i 6210 . . . . 5  |-  ( ( f `  ( _|_ `  A ) )  +  ( f `  ( A  i^i  C ) ) )  =  ( ( f `  ( _|_ `  A ) )  +  ( f `  ( C  i^i  A ) ) )
6359, 62syl6eq 2511 . . . 4  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  A
)  vH  ( A  i^i  C ) ) )  =  ( ( f `
 ( _|_ `  A
) )  +  ( f `  ( C  i^i  A ) ) ) )
6458, 63oveq12d 6217 . . 3  |-  ( f  e.  States  ->  ( ( ( f `  ( ( _|_ `  B )  vH  ( B  i^i  A ) ) )  +  ( f `  (
( _|_ `  C
)  vH  ( C  i^i  B ) ) ) )  +  ( f `
 ( ( _|_ `  A )  vH  ( A  i^i  C ) ) ) )  =  ( ( ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( A  i^i  B ) ) )  +  ( ( f `  ( _|_ `  C ) )  +  ( f `  ( B  i^i  C ) ) ) )  +  ( ( f `  ( _|_ `  A ) )  +  ( f `  ( C  i^i  A ) ) ) ) )
6542, 47, 643eqtr4d 2505 . 2  |-  ( f  e.  States  ->  ( ( ( f `  ( ( _|_ `  A )  vH  ( A  i^i  B ) ) )  +  ( f `  (
( _|_ `  B
)  vH  ( B  i^i  C ) ) ) )  +  ( f `
 ( ( _|_ `  C )  vH  ( C  i^i  A ) ) ) )  =  ( ( ( f `  ( ( _|_ `  B
)  vH  ( B  i^i  A ) ) )  +  ( f `  ( ( _|_ `  C
)  vH  ( C  i^i  B ) ) ) )  +  ( f `
 ( ( _|_ `  A )  vH  ( A  i^i  C ) ) ) ) )
66 golem1.4 . . . . 5  |-  F  =  ( ( _|_ `  A
)  vH  ( A  i^i  B ) )
6766fveq2i 5801 . . . 4  |-  ( f `
 F )  =  ( f `  (
( _|_ `  A
)  vH  ( A  i^i  B ) ) )
68 golem1.5 . . . . 5  |-  G  =  ( ( _|_ `  B
)  vH  ( B  i^i  C ) )
6968fveq2i 5801 . . . 4  |-  ( f `
 G )  =  ( f `  (
( _|_ `  B
)  vH  ( B  i^i  C ) ) )
7067, 69oveq12i 6211 . . 3  |-  ( ( f `  F )  +  ( f `  G ) )  =  ( ( f `  ( ( _|_ `  A
)  vH  ( A  i^i  B ) ) )  +  ( f `  ( ( _|_ `  B
)  vH  ( B  i^i  C ) ) ) )
71 golem1.6 . . . 4  |-  H  =  ( ( _|_ `  C
)  vH  ( C  i^i  A ) )
7271fveq2i 5801 . . 3  |-  ( f `
 H )  =  ( f `  (
( _|_ `  C
)  vH  ( C  i^i  A ) ) )
7370, 72oveq12i 6211 . 2  |-  ( ( ( f `  F
)  +  ( f `
 G ) )  +  ( f `  H ) )  =  ( ( ( f `
 ( ( _|_ `  A )  vH  ( A  i^i  B ) ) )  +  ( f `
 ( ( _|_ `  B )  vH  ( B  i^i  C ) ) ) )  +  ( f `  ( ( _|_ `  C )  vH  ( C  i^i  A ) ) ) )
74 golem1.7 . . . . 5  |-  D  =  ( ( _|_ `  B
)  vH  ( B  i^i  A ) )
7574fveq2i 5801 . . . 4  |-  ( f `
 D )  =  ( f `  (
( _|_ `  B
)  vH  ( B  i^i  A ) ) )
76 golem1.8 . . . . 5  |-  R  =  ( ( _|_ `  C
)  vH  ( C  i^i  B ) )
7776fveq2i 5801 . . . 4  |-  ( f `
 R )  =  ( f `  (
( _|_ `  C
)  vH  ( C  i^i  B ) ) )
7875, 77oveq12i 6211 . . 3  |-  ( ( f `  D )  +  ( f `  R ) )  =  ( ( f `  ( ( _|_ `  B
)  vH  ( B  i^i  A ) ) )  +  ( f `  ( ( _|_ `  C
)  vH  ( C  i^i  B ) ) ) )
79 golem1.9 . . . 4  |-  S  =  ( ( _|_ `  A
)  vH  ( A  i^i  C ) )
8079fveq2i 5801 . . 3  |-  ( f `
 S )  =  ( f `  (
( _|_ `  A
)  vH  ( A  i^i  C ) ) )
8178, 80oveq12i 6211 . 2  |-  ( ( ( f `  D
)  +  ( f `
 R ) )  +  ( f `  S ) )  =  ( ( ( f `
 ( ( _|_ `  B )  vH  ( B  i^i  A ) ) )  +  ( f `
 ( ( _|_ `  C )  vH  ( C  i^i  B ) ) ) )  +  ( f `  ( ( _|_ `  A )  vH  ( A  i^i  C ) ) ) )
8265, 73, 813eqtr4g 2520 1  |-  ( f  e.  States  ->  ( ( ( f `  F )  +  ( f `  G ) )  +  ( f `  H
) )  =  ( ( ( f `  D )  +  ( f `  R ) )  +  ( f `
 S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    i^i cin 3434   ` cfv 5525  (class class class)co 6199   RRcr 9391    + caddc 9395   CHcch 24482   _|_cort 24483    vH chj 24486   Statescst 24515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cc 8714  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470  ax-addf 9471  ax-mulf 9472  ax-hilex 24552  ax-hfvadd 24553  ax-hvcom 24554  ax-hvass 24555  ax-hv0cl 24556  ax-hvaddid 24557  ax-hfvmul 24558  ax-hvmulid 24559  ax-hvmulass 24560  ax-hvdistr1 24561  ax-hvdistr2 24562  ax-hvmul0 24563  ax-hfi 24632  ax-his1 24635  ax-his2 24636  ax-his3 24637  ax-his4 24638  ax-hcompl 24755
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-omul 7034  df-er 7210  df-map 7325  df-pm 7326  df-ixp 7373  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-fi 7771  df-sup 7801  df-oi 7834  df-card 8219  df-acn 8222  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ioo 11414  df-ico 11416  df-icc 11417  df-fz 11554  df-fzo 11665  df-fl 11758  df-seq 11923  df-exp 11982  df-hash 12220  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-clim 13083  df-rlim 13084  df-sum 13281  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-starv 14371  df-sca 14372  df-vsca 14373  df-ip 14374  df-tset 14375  df-ple 14376  df-ds 14378  df-unif 14379  df-hom 14380  df-cco 14381  df-rest 14479  df-topn 14480  df-0g 14498  df-gsum 14499  df-topgen 14500  df-pt 14501  df-prds 14504  df-xrs 14558  df-qtop 14563  df-imas 14564  df-xps 14566  df-mre 14642  df-mrc 14643  df-acs 14645  df-mnd 15533  df-submnd 15583  df-mulg 15666  df-cntz 15953  df-cmn 16399  df-psmet 17933  df-xmet 17934  df-met 17935  df-bl 17936  df-mopn 17937  df-fbas 17938  df-fg 17939  df-cnfld 17943  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-cld 18754  df-ntr 18755  df-cls 18756  df-nei 18833  df-cn 18962  df-cnp 18963  df-lm 18964  df-haus 19050  df-tx 19266  df-hmeo 19459  df-fil 19550  df-fm 19642  df-flim 19643  df-flf 19644  df-xms 20026  df-ms 20027  df-tms 20028  df-cfil 20897  df-cau 20898  df-cmet 20899  df-grpo 23829  df-gid 23830  df-ginv 23831  df-gdiv 23832  df-ablo 23920  df-subgo 23940  df-vc 24075  df-nv 24121  df-va 24124  df-ba 24125  df-sm 24126  df-0v 24127  df-vs 24128  df-nmcv 24129  df-ims 24130  df-dip 24247  df-ssp 24271  df-ph 24364  df-cbn 24415  df-hnorm 24521  df-hba 24522  df-hvsub 24524  df-hlim 24525  df-hcau 24526  df-sh 24760  df-ch 24775  df-oc 24806  df-ch0 24807  df-st 25766
This theorem is referenced by:  golem2  25827
  Copyright terms: Public domain W3C validator