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Theorem golem1 25626
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 24661 . . . . . . . . . 10  |-  ( _|_ `  A )  e.  CH
3 stcl 25571 . . . . . . . . . 10  |-  ( f  e.  States  ->  ( ( _|_ `  A )  e.  CH  ->  ( f `  ( _|_ `  A ) )  e.  RR ) )
42, 3mpi 17 . . . . . . . . 9  |-  ( f  e.  States  ->  ( f `  ( _|_ `  A ) )  e.  RR )
54recnd 9404 . . . . . . . 8  |-  ( f  e.  States  ->  ( f `  ( _|_ `  A ) )  e.  CC )
6 golem1.2 . . . . . . . . . . 11  |-  B  e. 
CH
76choccli 24661 . . . . . . . . . 10  |-  ( _|_ `  B )  e.  CH
8 stcl 25571 . . . . . . . . . 10  |-  ( f  e.  States  ->  ( ( _|_ `  B )  e.  CH  ->  ( f `  ( _|_ `  B ) )  e.  RR ) )
97, 8mpi 17 . . . . . . . . 9  |-  ( f  e.  States  ->  ( f `  ( _|_ `  B ) )  e.  RR )
109recnd 9404 . . . . . . . 8  |-  ( f  e.  States  ->  ( f `  ( _|_ `  B ) )  e.  CC )
11 golem1.3 . . . . . . . . . . 11  |-  C  e. 
CH
1211choccli 24661 . . . . . . . . . 10  |-  ( _|_ `  C )  e.  CH
13 stcl 25571 . . . . . . . . . 10  |-  ( f  e.  States  ->  ( ( _|_ `  C )  e.  CH  ->  ( f `  ( _|_ `  C ) )  e.  RR ) )
1412, 13mpi 17 . . . . . . . . 9  |-  ( f  e.  States  ->  ( f `  ( _|_ `  C ) )  e.  RR )
1514recnd 9404 . . . . . . . 8  |-  ( f  e.  States  ->  ( f `  ( _|_ `  C ) )  e.  CC )
165, 10, 15addassd 9400 . . . . . . 7  |-  ( f  e.  States  ->  ( ( ( f `  ( _|_ `  A ) )  +  ( f `  ( _|_ `  B ) ) )  +  ( f `
 ( _|_ `  C
) ) )  =  ( ( f `  ( _|_ `  A ) )  +  ( ( f `  ( _|_ `  B ) )  +  ( f `  ( _|_ `  C ) ) ) ) )
1710, 15addcld 9397 . . . . . . . 8  |-  ( f  e.  States  ->  ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( _|_ `  C ) ) )  e.  CC )
185, 17addcomd 9563 . . . . . . 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 6101 . . . . 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 9397 . . . . . 6  |-  ( f  e.  States  ->  ( ( f `
 ( _|_ `  A
) )  +  ( f `  ( _|_ `  B ) ) )  e.  CC )
221, 6chincli 24814 . . . . . . . . 9  |-  ( A  i^i  B )  e. 
CH
23 stcl 25571 . . . . . . . . 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 9404 . . . . . . 7  |-  ( f  e.  States  ->  ( f `  ( A  i^i  B ) )  e.  CC )
266, 11chincli 24814 . . . . . . . . 9  |-  ( B  i^i  C )  e. 
CH
27 stcl 25571 . . . . . . . . 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 9404 . . . . . . 7  |-  ( f  e.  States  ->  ( f `  ( B  i^i  C ) )  e.  CC )
3025, 29addcld 9397 . . . . . 6  |-  ( f  e.  States  ->  ( ( f `
 ( A  i^i  B ) )  +  ( f `  ( B  i^i  C ) ) )  e.  CC )
3111, 1chincli 24814 . . . . . . . 8  |-  ( C  i^i  A )  e. 
CH
32 stcl 25571 . . . . . . . 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 9404 . . . . . 6  |-  ( f  e.  States  ->  ( f `  ( C  i^i  A ) )  e.  CC )
3521, 30, 15, 34add4d 9585 . . . . 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 9585 . . . . 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 9585 . . . . 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 6101 . . . 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 9585 . . . . 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 6101 . . . 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 25597 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  A
)  vH  ( A  i^i  B ) ) )  =  ( ( f `
 ( _|_ `  A
) )  +  ( f `  ( A  i^i  B ) ) ) )
446, 11stji1i 25597 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  B
)  vH  ( B  i^i  C ) ) )  =  ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( B  i^i  C ) ) ) )
4543, 44oveq12d 6104 . . . 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 25597 . . . 4  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  C
)  vH  ( C  i^i  A ) ) )  =  ( ( f `
 ( _|_ `  C
) )  +  ( f `  ( C  i^i  A ) ) ) )
4745, 46oveq12d 6104 . . 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 25597 . . . . . 6  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  B
)  vH  ( B  i^i  A ) ) )  =  ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( B  i^i  A ) ) ) )
49 incom 3538 . . . . . . . 8  |-  ( B  i^i  A )  =  ( A  i^i  B
)
5049fveq2i 5689 . . . . . . 7  |-  ( f `
 ( B  i^i  A ) )  =  ( f `  ( A  i^i  B ) )
5150oveq2i 6097 . . . . . 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 25597 . . . . . 6  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  C
)  vH  ( C  i^i  B ) ) )  =  ( ( f `
 ( _|_ `  C
) )  +  ( f `  ( C  i^i  B ) ) ) )
54 incom 3538 . . . . . . . 8  |-  ( C  i^i  B )  =  ( B  i^i  C
)
5554fveq2i 5689 . . . . . . 7  |-  ( f `
 ( C  i^i  B ) )  =  ( f `  ( B  i^i  C ) )
5655oveq2i 6097 . . . . . 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 6104 . . . 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 25597 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  A
)  vH  ( A  i^i  C ) ) )  =  ( ( f `
 ( _|_ `  A
) )  +  ( f `  ( A  i^i  C ) ) ) )
60 incom 3538 . . . . . . 7  |-  ( A  i^i  C )  =  ( C  i^i  A
)
6160fveq2i 5689 . . . . . 6  |-  ( f `
 ( A  i^i  C ) )  =  ( f `  ( C  i^i  A ) )
6261oveq2i 6097 . . . . 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 6104 . . 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 5689 . . . 4  |-  ( f `
 F )  =  ( f `  (
( _|_ `  A
)  vH  ( A  i^i  B ) ) )
68 golem1.5 . . . . 5  |-  G  =  ( ( _|_ `  B
)  vH  ( B  i^i  C ) )
6968fveq2i 5689 . . . 4  |-  ( f `
 G )  =  ( f `  (
( _|_ `  B
)  vH  ( B  i^i  C ) ) )
7067, 69oveq12i 6098 . . 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 5689 . . 3  |-  ( f `
 H )  =  ( f `  (
( _|_ `  C
)  vH  ( C  i^i  A ) ) )
7370, 72oveq12i 6098 . 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 5689 . . . 4  |-  ( f `
 D )  =  ( f `  (
( _|_ `  B
)  vH  ( B  i^i  A ) ) )
76 golem1.8 . . . . 5  |-  R  =  ( ( _|_ `  C
)  vH  ( C  i^i  B ) )
7776fveq2i 5689 . . . 4  |-  ( f `
 R )  =  ( f `  (
( _|_ `  C
)  vH  ( C  i^i  B ) ) )
7875, 77oveq12i 6098 . . 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 5689 . . 3  |-  ( f `
 S )  =  ( f `  (
( _|_ `  A
)  vH  ( A  i^i  C ) ) )
8178, 80oveq12i 6098 . 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 setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756    i^i cin 3322   ` cfv 5413  (class class class)co 6086   RRcr 9273    + caddc 9277   CHcch 24282   _|_cort 24283    vH chj 24286   Statescst 24315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cc 8596  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354  ax-hilex 24352  ax-hfvadd 24353  ax-hvcom 24354  ax-hvass 24355  ax-hv0cl 24356  ax-hvaddid 24357  ax-hfvmul 24358  ax-hvmulid 24359  ax-hvmulass 24360  ax-hvdistr1 24361  ax-hvdistr2 24362  ax-hvmul0 24363  ax-hfi 24432  ax-his1 24435  ax-his2 24436  ax-his3 24437  ax-his4 24438  ax-hcompl 24555
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-omul 6917  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-acn 8104  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959  df-sum 13156  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-fbas 17789  df-fg 17790  df-cnfld 17794  df-top 18478  df-bases 18480  df-topon 18481  df-topsp 18482  df-cld 18598  df-ntr 18599  df-cls 18600  df-nei 18677  df-cn 18806  df-cnp 18807  df-lm 18808  df-haus 18894  df-tx 19110  df-hmeo 19303  df-fil 19394  df-fm 19486  df-flim 19487  df-flf 19488  df-xms 19870  df-ms 19871  df-tms 19872  df-cfil 20741  df-cau 20742  df-cmet 20743  df-grpo 23629  df-gid 23630  df-ginv 23631  df-gdiv 23632  df-ablo 23720  df-subgo 23740  df-vc 23875  df-nv 23921  df-va 23924  df-ba 23925  df-sm 23926  df-0v 23927  df-vs 23928  df-nmcv 23929  df-ims 23930  df-dip 24047  df-ssp 24071  df-ph 24164  df-cbn 24215  df-hnorm 24321  df-hba 24322  df-hvsub 24324  df-hlim 24325  df-hcau 24326  df-sh 24560  df-ch 24575  df-oc 24606  df-ch0 24607  df-st 25566
This theorem is referenced by:  golem2  25627
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