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Theorem golem1 27603
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 26639 . . . . . . . . . 10  |-  ( _|_ `  A )  e.  CH
3 stcl 27548 . . . . . . . . . 10  |-  ( f  e.  States  ->  ( ( _|_ `  A )  e.  CH  ->  ( f `  ( _|_ `  A ) )  e.  RR ) )
42, 3mpi 20 . . . . . . . . 9  |-  ( f  e.  States  ->  ( f `  ( _|_ `  A ) )  e.  RR )
54recnd 9652 . . . . . . . 8  |-  ( f  e.  States  ->  ( f `  ( _|_ `  A ) )  e.  CC )
6 golem1.2 . . . . . . . . . . 11  |-  B  e. 
CH
76choccli 26639 . . . . . . . . . 10  |-  ( _|_ `  B )  e.  CH
8 stcl 27548 . . . . . . . . . 10  |-  ( f  e.  States  ->  ( ( _|_ `  B )  e.  CH  ->  ( f `  ( _|_ `  B ) )  e.  RR ) )
97, 8mpi 20 . . . . . . . . 9  |-  ( f  e.  States  ->  ( f `  ( _|_ `  B ) )  e.  RR )
109recnd 9652 . . . . . . . 8  |-  ( f  e.  States  ->  ( f `  ( _|_ `  B ) )  e.  CC )
11 golem1.3 . . . . . . . . . . 11  |-  C  e. 
CH
1211choccli 26639 . . . . . . . . . 10  |-  ( _|_ `  C )  e.  CH
13 stcl 27548 . . . . . . . . . 10  |-  ( f  e.  States  ->  ( ( _|_ `  C )  e.  CH  ->  ( f `  ( _|_ `  C ) )  e.  RR ) )
1412, 13mpi 20 . . . . . . . . 9  |-  ( f  e.  States  ->  ( f `  ( _|_ `  C ) )  e.  RR )
1514recnd 9652 . . . . . . . 8  |-  ( f  e.  States  ->  ( f `  ( _|_ `  C ) )  e.  CC )
165, 10, 15addassd 9648 . . . . . . 7  |-  ( f  e.  States  ->  ( ( ( f `  ( _|_ `  A ) )  +  ( f `  ( _|_ `  B ) ) )  +  ( f `
 ( _|_ `  C
) ) )  =  ( ( f `  ( _|_ `  A ) )  +  ( ( f `  ( _|_ `  B ) )  +  ( f `  ( _|_ `  C ) ) ) ) )
1710, 15addcld 9645 . . . . . . . 8  |-  ( f  e.  States  ->  ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( _|_ `  C ) ) )  e.  CC )
185, 17addcomd 9816 . . . . . . 7  |-  ( f  e.  States  ->  ( ( f `
 ( _|_ `  A
) )  +  ( ( f `  ( _|_ `  B ) )  +  ( f `  ( _|_ `  C ) ) ) )  =  ( ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( _|_ `  C ) ) )  +  ( f `  ( _|_ `  A ) ) ) )
1916, 18eqtrd 2443 . . . . . 6  |-  ( f  e.  States  ->  ( ( ( f `  ( _|_ `  A ) )  +  ( f `  ( _|_ `  B ) ) )  +  ( f `
 ( _|_ `  C
) ) )  =  ( ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( _|_ `  C ) ) )  +  ( f `  ( _|_ `  A ) ) ) )
2019oveq1d 6293 . . . . 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 9645 . . . . . 6  |-  ( f  e.  States  ->  ( ( f `
 ( _|_ `  A
) )  +  ( f `  ( _|_ `  B ) ) )  e.  CC )
221, 6chincli 26792 . . . . . . . . 9  |-  ( A  i^i  B )  e. 
CH
23 stcl 27548 . . . . . . . . 9  |-  ( f  e.  States  ->  ( ( A  i^i  B )  e. 
CH  ->  ( f `  ( A  i^i  B ) )  e.  RR ) )
2422, 23mpi 20 . . . . . . . 8  |-  ( f  e.  States  ->  ( f `  ( A  i^i  B ) )  e.  RR )
2524recnd 9652 . . . . . . 7  |-  ( f  e.  States  ->  ( f `  ( A  i^i  B ) )  e.  CC )
266, 11chincli 26792 . . . . . . . . 9  |-  ( B  i^i  C )  e. 
CH
27 stcl 27548 . . . . . . . . 9  |-  ( f  e.  States  ->  ( ( B  i^i  C )  e. 
CH  ->  ( f `  ( B  i^i  C ) )  e.  RR ) )
2826, 27mpi 20 . . . . . . . 8  |-  ( f  e.  States  ->  ( f `  ( B  i^i  C ) )  e.  RR )
2928recnd 9652 . . . . . . 7  |-  ( f  e.  States  ->  ( f `  ( B  i^i  C ) )  e.  CC )
3025, 29addcld 9645 . . . . . 6  |-  ( f  e.  States  ->  ( ( f `
 ( A  i^i  B ) )  +  ( f `  ( B  i^i  C ) ) )  e.  CC )
3111, 1chincli 26792 . . . . . . . 8  |-  ( C  i^i  A )  e. 
CH
32 stcl 27548 . . . . . . . 8  |-  ( f  e.  States  ->  ( ( C  i^i  A )  e. 
CH  ->  ( f `  ( C  i^i  A ) )  e.  RR ) )
3331, 32mpi 20 . . . . . . 7  |-  ( f  e.  States  ->  ( f `  ( C  i^i  A ) )  e.  RR )
3433recnd 9652 . . . . . 6  |-  ( f  e.  States  ->  ( f `  ( C  i^i  A ) )  e.  CC )
3521, 30, 15, 34add4d 9839 . . . . 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 9839 . . . . 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 2453 . . . 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 9839 . . . . 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 6293 . . . 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 9839 . . . . 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 6293 . . . 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 2453 . . 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 27574 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  A
)  vH  ( A  i^i  B ) ) )  =  ( ( f `
 ( _|_ `  A
) )  +  ( f `  ( A  i^i  B ) ) ) )
446, 11stji1i 27574 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  B
)  vH  ( B  i^i  C ) ) )  =  ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( B  i^i  C ) ) ) )
4543, 44oveq12d 6296 . . . 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 27574 . . . 4  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  C
)  vH  ( C  i^i  A ) ) )  =  ( ( f `
 ( _|_ `  C
) )  +  ( f `  ( C  i^i  A ) ) ) )
4745, 46oveq12d 6296 . . 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 27574 . . . . . 6  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  B
)  vH  ( B  i^i  A ) ) )  =  ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( B  i^i  A ) ) ) )
49 incom 3632 . . . . . . . 8  |-  ( B  i^i  A )  =  ( A  i^i  B
)
5049fveq2i 5852 . . . . . . 7  |-  ( f `
 ( B  i^i  A ) )  =  ( f `  ( A  i^i  B ) )
5150oveq2i 6289 . . . . . 6  |-  ( ( f `  ( _|_ `  B ) )  +  ( f `  ( B  i^i  A ) ) )  =  ( ( f `  ( _|_ `  B ) )  +  ( f `  ( A  i^i  B ) ) )
5248, 51syl6eq 2459 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  B
)  vH  ( B  i^i  A ) ) )  =  ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( A  i^i  B ) ) ) )
5311, 6stji1i 27574 . . . . . 6  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  C
)  vH  ( C  i^i  B ) ) )  =  ( ( f `
 ( _|_ `  C
) )  +  ( f `  ( C  i^i  B ) ) ) )
54 incom 3632 . . . . . . . 8  |-  ( C  i^i  B )  =  ( B  i^i  C
)
5554fveq2i 5852 . . . . . . 7  |-  ( f `
 ( C  i^i  B ) )  =  ( f `  ( B  i^i  C ) )
5655oveq2i 6289 . . . . . 6  |-  ( ( f `  ( _|_ `  C ) )  +  ( f `  ( C  i^i  B ) ) )  =  ( ( f `  ( _|_ `  C ) )  +  ( f `  ( B  i^i  C ) ) )
5753, 56syl6eq 2459 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  C
)  vH  ( C  i^i  B ) ) )  =  ( ( f `
 ( _|_ `  C
) )  +  ( f `  ( B  i^i  C ) ) ) )
5852, 57oveq12d 6296 . . . 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 27574 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  A
)  vH  ( A  i^i  C ) ) )  =  ( ( f `
 ( _|_ `  A
) )  +  ( f `  ( A  i^i  C ) ) ) )
60 incom 3632 . . . . . . 7  |-  ( A  i^i  C )  =  ( C  i^i  A
)
6160fveq2i 5852 . . . . . 6  |-  ( f `
 ( A  i^i  C ) )  =  ( f `  ( C  i^i  A ) )
6261oveq2i 6289 . . . . 5  |-  ( ( f `  ( _|_ `  A ) )  +  ( f `  ( A  i^i  C ) ) )  =  ( ( f `  ( _|_ `  A ) )  +  ( f `  ( C  i^i  A ) ) )
6359, 62syl6eq 2459 . . . 4  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  A
)  vH  ( A  i^i  C ) ) )  =  ( ( f `
 ( _|_ `  A
) )  +  ( f `  ( C  i^i  A ) ) ) )
6458, 63oveq12d 6296 . . 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 2453 . 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 5852 . . . 4  |-  ( f `
 F )  =  ( f `  (
( _|_ `  A
)  vH  ( A  i^i  B ) ) )
68 golem1.5 . . . . 5  |-  G  =  ( ( _|_ `  B
)  vH  ( B  i^i  C ) )
6968fveq2i 5852 . . . 4  |-  ( f `
 G )  =  ( f `  (
( _|_ `  B
)  vH  ( B  i^i  C ) ) )
7067, 69oveq12i 6290 . . 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 5852 . . 3  |-  ( f `
 H )  =  ( f `  (
( _|_ `  C
)  vH  ( C  i^i  A ) ) )
7370, 72oveq12i 6290 . 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 5852 . . . 4  |-  ( f `
 D )  =  ( f `  (
( _|_ `  B
)  vH  ( B  i^i  A ) ) )
76 golem1.8 . . . . 5  |-  R  =  ( ( _|_ `  C
)  vH  ( C  i^i  B ) )
7776fveq2i 5852 . . . 4  |-  ( f `
 R )  =  ( f `  (
( _|_ `  C
)  vH  ( C  i^i  B ) ) )
7875, 77oveq12i 6290 . . 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 5852 . . 3  |-  ( f `
 S )  =  ( f `  (
( _|_ `  A
)  vH  ( A  i^i  C ) ) )
8178, 80oveq12i 6290 . 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 2468 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 1405    e. wcel 1842    i^i cin 3413   ` cfv 5569  (class class class)co 6278   RRcr 9521    + caddc 9525   CHcch 26260   _|_cort 26261    vH chj 26264   Statescst 26293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cc 8847  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602  ax-hilex 26330  ax-hfvadd 26331  ax-hvcom 26332  ax-hvass 26333  ax-hv0cl 26334  ax-hvaddid 26335  ax-hfvmul 26336  ax-hvmulid 26337  ax-hvmulass 26338  ax-hvdistr1 26339  ax-hvdistr2 26340  ax-hvmul0 26341  ax-hfi 26410  ax-his1 26413  ax-his2 26414  ax-his3 26415  ax-his4 26416  ax-hcompl 26533
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-omul 7172  df-er 7348  df-map 7459  df-pm 7460  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-fi 7905  df-sup 7935  df-oi 7969  df-card 8352  df-acn 8355  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ioo 11586  df-ico 11588  df-icc 11589  df-fz 11727  df-fzo 11855  df-fl 11966  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-rlim 13461  df-sum 13658  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-hom 14933  df-cco 14934  df-rest 15037  df-topn 15038  df-0g 15056  df-gsum 15057  df-topgen 15058  df-pt 15059  df-prds 15062  df-xrs 15116  df-qtop 15121  df-imas 15122  df-xps 15124  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-fbas 18736  df-fg 18737  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cld 19812  df-ntr 19813  df-cls 19814  df-nei 19892  df-cn 20021  df-cnp 20022  df-lm 20023  df-haus 20109  df-tx 20355  df-hmeo 20548  df-fil 20639  df-fm 20731  df-flim 20732  df-flf 20733  df-xms 21115  df-ms 21116  df-tms 21117  df-cfil 21986  df-cau 21987  df-cmet 21988  df-grpo 25607  df-gid 25608  df-ginv 25609  df-gdiv 25610  df-ablo 25698  df-subgo 25718  df-vc 25853  df-nv 25899  df-va 25902  df-ba 25903  df-sm 25904  df-0v 25905  df-vs 25906  df-nmcv 25907  df-ims 25908  df-dip 26025  df-ssp 26049  df-ph 26142  df-cbn 26193  df-hnorm 26299  df-hba 26300  df-hvsub 26302  df-hlim 26303  df-hcau 26304  df-sh 26538  df-ch 26553  df-oc 26584  df-ch0 26585  df-st 27543
This theorem is referenced by:  golem2  27604
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