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Theorem golem1 23776
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 22811 . . . . . . . . . 10  |-  ( _|_ `  A )  e.  CH
3 stcl 23721 . . . . . . . . . 10  |-  ( f  e.  States  ->  ( ( _|_ `  A )  e.  CH  ->  ( f `  ( _|_ `  A ) )  e.  RR ) )
42, 3mpi 17 . . . . . . . . 9  |-  ( f  e.  States  ->  ( f `  ( _|_ `  A ) )  e.  RR )
54recnd 9116 . . . . . . . 8  |-  ( f  e.  States  ->  ( f `  ( _|_ `  A ) )  e.  CC )
6 golem1.2 . . . . . . . . . . 11  |-  B  e. 
CH
76choccli 22811 . . . . . . . . . 10  |-  ( _|_ `  B )  e.  CH
8 stcl 23721 . . . . . . . . . 10  |-  ( f  e.  States  ->  ( ( _|_ `  B )  e.  CH  ->  ( f `  ( _|_ `  B ) )  e.  RR ) )
97, 8mpi 17 . . . . . . . . 9  |-  ( f  e.  States  ->  ( f `  ( _|_ `  B ) )  e.  RR )
109recnd 9116 . . . . . . . 8  |-  ( f  e.  States  ->  ( f `  ( _|_ `  B ) )  e.  CC )
11 golem1.3 . . . . . . . . . . 11  |-  C  e. 
CH
1211choccli 22811 . . . . . . . . . 10  |-  ( _|_ `  C )  e.  CH
13 stcl 23721 . . . . . . . . . 10  |-  ( f  e.  States  ->  ( ( _|_ `  C )  e.  CH  ->  ( f `  ( _|_ `  C ) )  e.  RR ) )
1412, 13mpi 17 . . . . . . . . 9  |-  ( f  e.  States  ->  ( f `  ( _|_ `  C ) )  e.  RR )
1514recnd 9116 . . . . . . . 8  |-  ( f  e.  States  ->  ( f `  ( _|_ `  C ) )  e.  CC )
165, 10, 15addassd 9112 . . . . . . 7  |-  ( f  e.  States  ->  ( ( ( f `  ( _|_ `  A ) )  +  ( f `  ( _|_ `  B ) ) )  +  ( f `
 ( _|_ `  C
) ) )  =  ( ( f `  ( _|_ `  A ) )  +  ( ( f `  ( _|_ `  B ) )  +  ( f `  ( _|_ `  C ) ) ) ) )
1710, 15addcld 9109 . . . . . . . 8  |-  ( f  e.  States  ->  ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( _|_ `  C ) ) )  e.  CC )
185, 17addcomd 9270 . . . . . . 7  |-  ( f  e.  States  ->  ( ( f `
 ( _|_ `  A
) )  +  ( ( f `  ( _|_ `  B ) )  +  ( f `  ( _|_ `  C ) ) ) )  =  ( ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( _|_ `  C ) ) )  +  ( f `  ( _|_ `  A ) ) ) )
1916, 18eqtrd 2470 . . . . . 6  |-  ( f  e.  States  ->  ( ( ( f `  ( _|_ `  A ) )  +  ( f `  ( _|_ `  B ) ) )  +  ( f `
 ( _|_ `  C
) ) )  =  ( ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( _|_ `  C ) ) )  +  ( f `  ( _|_ `  A ) ) ) )
2019oveq1d 6098 . . . . 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 9109 . . . . . 6  |-  ( f  e.  States  ->  ( ( f `
 ( _|_ `  A
) )  +  ( f `  ( _|_ `  B ) ) )  e.  CC )
221, 6chincli 22964 . . . . . . . . 9  |-  ( A  i^i  B )  e. 
CH
23 stcl 23721 . . . . . . . . 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 9116 . . . . . . 7  |-  ( f  e.  States  ->  ( f `  ( A  i^i  B ) )  e.  CC )
266, 11chincli 22964 . . . . . . . . 9  |-  ( B  i^i  C )  e. 
CH
27 stcl 23721 . . . . . . . . 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 9116 . . . . . . 7  |-  ( f  e.  States  ->  ( f `  ( B  i^i  C ) )  e.  CC )
3025, 29addcld 9109 . . . . . 6  |-  ( f  e.  States  ->  ( ( f `
 ( A  i^i  B ) )  +  ( f `  ( B  i^i  C ) ) )  e.  CC )
3111, 1chincli 22964 . . . . . . . 8  |-  ( C  i^i  A )  e. 
CH
32 stcl 23721 . . . . . . . 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 9116 . . . . . 6  |-  ( f  e.  States  ->  ( f `  ( C  i^i  A ) )  e.  CC )
3521, 30, 15, 34add4d 9291 . . . . 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 9291 . . . . 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 2480 . . . 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 9291 . . . . 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 6098 . . . 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 9291 . . . . 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 6098 . . . 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 2480 . . 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 23747 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  A
)  vH  ( A  i^i  B ) ) )  =  ( ( f `
 ( _|_ `  A
) )  +  ( f `  ( A  i^i  B ) ) ) )
446, 11stji1i 23747 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  B
)  vH  ( B  i^i  C ) ) )  =  ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( B  i^i  C ) ) ) )
4543, 44oveq12d 6101 . . . 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 23747 . . . 4  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  C
)  vH  ( C  i^i  A ) ) )  =  ( ( f `
 ( _|_ `  C
) )  +  ( f `  ( C  i^i  A ) ) ) )
4745, 46oveq12d 6101 . . 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 23747 . . . . . 6  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  B
)  vH  ( B  i^i  A ) ) )  =  ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( B  i^i  A ) ) ) )
49 incom 3535 . . . . . . . 8  |-  ( B  i^i  A )  =  ( A  i^i  B
)
5049fveq2i 5733 . . . . . . 7  |-  ( f `
 ( B  i^i  A ) )  =  ( f `  ( A  i^i  B ) )
5150oveq2i 6094 . . . . . 6  |-  ( ( f `  ( _|_ `  B ) )  +  ( f `  ( B  i^i  A ) ) )  =  ( ( f `  ( _|_ `  B ) )  +  ( f `  ( A  i^i  B ) ) )
5248, 51syl6eq 2486 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  B
)  vH  ( B  i^i  A ) ) )  =  ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( A  i^i  B ) ) ) )
5311, 6stji1i 23747 . . . . . 6  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  C
)  vH  ( C  i^i  B ) ) )  =  ( ( f `
 ( _|_ `  C
) )  +  ( f `  ( C  i^i  B ) ) ) )
54 incom 3535 . . . . . . . 8  |-  ( C  i^i  B )  =  ( B  i^i  C
)
5554fveq2i 5733 . . . . . . 7  |-  ( f `
 ( C  i^i  B ) )  =  ( f `  ( B  i^i  C ) )
5655oveq2i 6094 . . . . . 6  |-  ( ( f `  ( _|_ `  C ) )  +  ( f `  ( C  i^i  B ) ) )  =  ( ( f `  ( _|_ `  C ) )  +  ( f `  ( B  i^i  C ) ) )
5753, 56syl6eq 2486 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  C
)  vH  ( C  i^i  B ) ) )  =  ( ( f `
 ( _|_ `  C
) )  +  ( f `  ( B  i^i  C ) ) ) )
5852, 57oveq12d 6101 . . . 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 23747 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  A
)  vH  ( A  i^i  C ) ) )  =  ( ( f `
 ( _|_ `  A
) )  +  ( f `  ( A  i^i  C ) ) ) )
60 incom 3535 . . . . . . 7  |-  ( A  i^i  C )  =  ( C  i^i  A
)
6160fveq2i 5733 . . . . . 6  |-  ( f `
 ( A  i^i  C ) )  =  ( f `  ( C  i^i  A ) )
6261oveq2i 6094 . . . . 5  |-  ( ( f `  ( _|_ `  A ) )  +  ( f `  ( A  i^i  C ) ) )  =  ( ( f `  ( _|_ `  A ) )  +  ( f `  ( C  i^i  A ) ) )
6359, 62syl6eq 2486 . . . 4  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  A
)  vH  ( A  i^i  C ) ) )  =  ( ( f `
 ( _|_ `  A
) )  +  ( f `  ( C  i^i  A ) ) ) )
6458, 63oveq12d 6101 . . 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 2480 . 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 5733 . . . 4  |-  ( f `
 F )  =  ( f `  (
( _|_ `  A
)  vH  ( A  i^i  B ) ) )
68 golem1.5 . . . . 5  |-  G  =  ( ( _|_ `  B
)  vH  ( B  i^i  C ) )
6968fveq2i 5733 . . . 4  |-  ( f `
 G )  =  ( f `  (
( _|_ `  B
)  vH  ( B  i^i  C ) ) )
7067, 69oveq12i 6095 . . 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 5733 . . 3  |-  ( f `
 H )  =  ( f `  (
( _|_ `  C
)  vH  ( C  i^i  A ) ) )
7370, 72oveq12i 6095 . 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 5733 . . . 4  |-  ( f `
 D )  =  ( f `  (
( _|_ `  B
)  vH  ( B  i^i  A ) ) )
76 golem1.8 . . . . 5  |-  R  =  ( ( _|_ `  C
)  vH  ( C  i^i  B ) )
7776fveq2i 5733 . . . 4  |-  ( f `
 R )  =  ( f `  (
( _|_ `  C
)  vH  ( C  i^i  B ) ) )
7875, 77oveq12i 6095 . . 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 5733 . . 3  |-  ( f `
 S )  =  ( f `  (
( _|_ `  A
)  vH  ( A  i^i  C ) ) )
8178, 80oveq12i 6095 . 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 2495 1  |-  ( f  e.  States  ->  ( ( ( f `  F )  +  ( f `  G ) )  +  ( f `  H
) )  =  ( ( ( f `  D )  +  ( f `  R ) )  +  ( f `
 S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    i^i cin 3321   ` cfv 5456  (class class class)co 6083   RRcr 8991    + caddc 8995   CHcch 22434   _|_cort 22435    vH chj 22438   Statescst 22467
This theorem is referenced by:  golem2  23777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cc 8317  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070  ax-addf 9071  ax-mulf 9072  ax-hilex 22504  ax-hfvadd 22505  ax-hvcom 22506  ax-hvass 22507  ax-hv0cl 22508  ax-hvaddid 22509  ax-hfvmul 22510  ax-hvmulid 22511  ax-hvmulass 22512  ax-hvdistr1 22513  ax-hvdistr2 22514  ax-hvmul0 22515  ax-hfi 22583  ax-his1 22586  ax-his2 22587  ax-his3 22588  ax-his4 22589  ax-hcompl 22706
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-omul 6731  df-er 6907  df-map 7022  df-pm 7023  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-fi 7418  df-sup 7448  df-oi 7481  df-card 7828  df-acn 7831  df-cda 8050  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-q 10577  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-ioo 10922  df-ico 10924  df-icc 10925  df-fz 11046  df-fzo 11138  df-fl 11204  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-clim 12284  df-rlim 12285  df-sum 12482  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-hom 13555  df-cco 13556  df-rest 13652  df-topn 13653  df-topgen 13669  df-pt 13670  df-prds 13673  df-xrs 13728  df-0g 13729  df-gsum 13730  df-qtop 13735  df-imas 13736  df-xps 13738  df-mre 13813  df-mrc 13814  df-acs 13816  df-mnd 14692  df-submnd 14741  df-mulg 14817  df-cntz 15118  df-cmn 15416  df-psmet 16696  df-xmet 16697  df-met 16698  df-bl 16699  df-mopn 16700  df-fbas 16701  df-fg 16702  df-cnfld 16706  df-top 16965  df-bases 16967  df-topon 16968  df-topsp 16969  df-cld 17085  df-ntr 17086  df-cls 17087  df-nei 17164  df-cn 17293  df-cnp 17294  df-lm 17295  df-haus 17381  df-tx 17596  df-hmeo 17789  df-fil 17880  df-fm 17972  df-flim 17973  df-flf 17974  df-xms 18352  df-ms 18353  df-tms 18354  df-cfil 19210  df-cau 19211  df-cmet 19212  df-grpo 21781  df-gid 21782  df-ginv 21783  df-gdiv 21784  df-ablo 21872  df-subgo 21892  df-vc 22027  df-nv 22073  df-va 22076  df-ba 22077  df-sm 22078  df-0v 22079  df-vs 22080  df-nmcv 22081  df-ims 22082  df-dip 22199  df-ssp 22223  df-ph 22316  df-cbn 22367  df-hnorm 22473  df-hba 22474  df-hvsub 22476  df-hlim 22477  df-hcau 22478  df-sh 22711  df-ch 22726  df-oc 22756  df-ch0 22757  df-st 23716
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