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Theorem golem1 26866
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 25901 . . . . . . . . . 10  |-  ( _|_ `  A )  e.  CH
3 stcl 26811 . . . . . . . . . 10  |-  ( f  e.  States  ->  ( ( _|_ `  A )  e.  CH  ->  ( f `  ( _|_ `  A ) )  e.  RR ) )
42, 3mpi 17 . . . . . . . . 9  |-  ( f  e.  States  ->  ( f `  ( _|_ `  A ) )  e.  RR )
54recnd 9618 . . . . . . . 8  |-  ( f  e.  States  ->  ( f `  ( _|_ `  A ) )  e.  CC )
6 golem1.2 . . . . . . . . . . 11  |-  B  e. 
CH
76choccli 25901 . . . . . . . . . 10  |-  ( _|_ `  B )  e.  CH
8 stcl 26811 . . . . . . . . . 10  |-  ( f  e.  States  ->  ( ( _|_ `  B )  e.  CH  ->  ( f `  ( _|_ `  B ) )  e.  RR ) )
97, 8mpi 17 . . . . . . . . 9  |-  ( f  e.  States  ->  ( f `  ( _|_ `  B ) )  e.  RR )
109recnd 9618 . . . . . . . 8  |-  ( f  e.  States  ->  ( f `  ( _|_ `  B ) )  e.  CC )
11 golem1.3 . . . . . . . . . . 11  |-  C  e. 
CH
1211choccli 25901 . . . . . . . . . 10  |-  ( _|_ `  C )  e.  CH
13 stcl 26811 . . . . . . . . . 10  |-  ( f  e.  States  ->  ( ( _|_ `  C )  e.  CH  ->  ( f `  ( _|_ `  C ) )  e.  RR ) )
1412, 13mpi 17 . . . . . . . . 9  |-  ( f  e.  States  ->  ( f `  ( _|_ `  C ) )  e.  RR )
1514recnd 9618 . . . . . . . 8  |-  ( f  e.  States  ->  ( f `  ( _|_ `  C ) )  e.  CC )
165, 10, 15addassd 9614 . . . . . . 7  |-  ( f  e.  States  ->  ( ( ( f `  ( _|_ `  A ) )  +  ( f `  ( _|_ `  B ) ) )  +  ( f `
 ( _|_ `  C
) ) )  =  ( ( f `  ( _|_ `  A ) )  +  ( ( f `  ( _|_ `  B ) )  +  ( f `  ( _|_ `  C ) ) ) ) )
1710, 15addcld 9611 . . . . . . . 8  |-  ( f  e.  States  ->  ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( _|_ `  C ) ) )  e.  CC )
185, 17addcomd 9777 . . . . . . 7  |-  ( f  e.  States  ->  ( ( f `
 ( _|_ `  A
) )  +  ( ( f `  ( _|_ `  B ) )  +  ( f `  ( _|_ `  C ) ) ) )  =  ( ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( _|_ `  C ) ) )  +  ( f `  ( _|_ `  A ) ) ) )
1916, 18eqtrd 2508 . . . . . 6  |-  ( f  e.  States  ->  ( ( ( f `  ( _|_ `  A ) )  +  ( f `  ( _|_ `  B ) ) )  +  ( f `
 ( _|_ `  C
) ) )  =  ( ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( _|_ `  C ) ) )  +  ( f `  ( _|_ `  A ) ) ) )
2019oveq1d 6297 . . . . 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 9611 . . . . . 6  |-  ( f  e.  States  ->  ( ( f `
 ( _|_ `  A
) )  +  ( f `  ( _|_ `  B ) ) )  e.  CC )
221, 6chincli 26054 . . . . . . . . 9  |-  ( A  i^i  B )  e. 
CH
23 stcl 26811 . . . . . . . . 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 9618 . . . . . . 7  |-  ( f  e.  States  ->  ( f `  ( A  i^i  B ) )  e.  CC )
266, 11chincli 26054 . . . . . . . . 9  |-  ( B  i^i  C )  e. 
CH
27 stcl 26811 . . . . . . . . 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 9618 . . . . . . 7  |-  ( f  e.  States  ->  ( f `  ( B  i^i  C ) )  e.  CC )
3025, 29addcld 9611 . . . . . 6  |-  ( f  e.  States  ->  ( ( f `
 ( A  i^i  B ) )  +  ( f `  ( B  i^i  C ) ) )  e.  CC )
3111, 1chincli 26054 . . . . . . . 8  |-  ( C  i^i  A )  e. 
CH
32 stcl 26811 . . . . . . . 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 9618 . . . . . 6  |-  ( f  e.  States  ->  ( f `  ( C  i^i  A ) )  e.  CC )
3521, 30, 15, 34add4d 9799 . . . . 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 9799 . . . . 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 2518 . . . 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 9799 . . . . 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 6297 . . . 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 9799 . . . . 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 6297 . . . 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 2518 . . 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 26837 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  A
)  vH  ( A  i^i  B ) ) )  =  ( ( f `
 ( _|_ `  A
) )  +  ( f `  ( A  i^i  B ) ) ) )
446, 11stji1i 26837 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  B
)  vH  ( B  i^i  C ) ) )  =  ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( B  i^i  C ) ) ) )
4543, 44oveq12d 6300 . . . 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 26837 . . . 4  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  C
)  vH  ( C  i^i  A ) ) )  =  ( ( f `
 ( _|_ `  C
) )  +  ( f `  ( C  i^i  A ) ) ) )
4745, 46oveq12d 6300 . . 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 26837 . . . . . 6  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  B
)  vH  ( B  i^i  A ) ) )  =  ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( B  i^i  A ) ) ) )
49 incom 3691 . . . . . . . 8  |-  ( B  i^i  A )  =  ( A  i^i  B
)
5049fveq2i 5867 . . . . . . 7  |-  ( f `
 ( B  i^i  A ) )  =  ( f `  ( A  i^i  B ) )
5150oveq2i 6293 . . . . . 6  |-  ( ( f `  ( _|_ `  B ) )  +  ( f `  ( B  i^i  A ) ) )  =  ( ( f `  ( _|_ `  B ) )  +  ( f `  ( A  i^i  B ) ) )
5248, 51syl6eq 2524 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  B
)  vH  ( B  i^i  A ) ) )  =  ( ( f `
 ( _|_ `  B
) )  +  ( f `  ( A  i^i  B ) ) ) )
5311, 6stji1i 26837 . . . . . 6  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  C
)  vH  ( C  i^i  B ) ) )  =  ( ( f `
 ( _|_ `  C
) )  +  ( f `  ( C  i^i  B ) ) ) )
54 incom 3691 . . . . . . . 8  |-  ( C  i^i  B )  =  ( B  i^i  C
)
5554fveq2i 5867 . . . . . . 7  |-  ( f `
 ( C  i^i  B ) )  =  ( f `  ( B  i^i  C ) )
5655oveq2i 6293 . . . . . 6  |-  ( ( f `  ( _|_ `  C ) )  +  ( f `  ( C  i^i  B ) ) )  =  ( ( f `  ( _|_ `  C ) )  +  ( f `  ( B  i^i  C ) ) )
5753, 56syl6eq 2524 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  C
)  vH  ( C  i^i  B ) ) )  =  ( ( f `
 ( _|_ `  C
) )  +  ( f `  ( B  i^i  C ) ) ) )
5852, 57oveq12d 6300 . . . 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 26837 . . . . 5  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  A
)  vH  ( A  i^i  C ) ) )  =  ( ( f `
 ( _|_ `  A
) )  +  ( f `  ( A  i^i  C ) ) ) )
60 incom 3691 . . . . . . 7  |-  ( A  i^i  C )  =  ( C  i^i  A
)
6160fveq2i 5867 . . . . . 6  |-  ( f `
 ( A  i^i  C ) )  =  ( f `  ( C  i^i  A ) )
6261oveq2i 6293 . . . . 5  |-  ( ( f `  ( _|_ `  A ) )  +  ( f `  ( A  i^i  C ) ) )  =  ( ( f `  ( _|_ `  A ) )  +  ( f `  ( C  i^i  A ) ) )
6359, 62syl6eq 2524 . . . 4  |-  ( f  e.  States  ->  ( f `  ( ( _|_ `  A
)  vH  ( A  i^i  C ) ) )  =  ( ( f `
 ( _|_ `  A
) )  +  ( f `  ( C  i^i  A ) ) ) )
6458, 63oveq12d 6300 . . 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 2518 . 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 5867 . . . 4  |-  ( f `
 F )  =  ( f `  (
( _|_ `  A
)  vH  ( A  i^i  B ) ) )
68 golem1.5 . . . . 5  |-  G  =  ( ( _|_ `  B
)  vH  ( B  i^i  C ) )
6968fveq2i 5867 . . . 4  |-  ( f `
 G )  =  ( f `  (
( _|_ `  B
)  vH  ( B  i^i  C ) ) )
7067, 69oveq12i 6294 . . 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 5867 . . 3  |-  ( f `
 H )  =  ( f `  (
( _|_ `  C
)  vH  ( C  i^i  A ) ) )
7370, 72oveq12i 6294 . 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 5867 . . . 4  |-  ( f `
 D )  =  ( f `  (
( _|_ `  B
)  vH  ( B  i^i  A ) ) )
76 golem1.8 . . . . 5  |-  R  =  ( ( _|_ `  C
)  vH  ( C  i^i  B ) )
7776fveq2i 5867 . . . 4  |-  ( f `
 R )  =  ( f `  (
( _|_ `  C
)  vH  ( C  i^i  B ) ) )
7875, 77oveq12i 6294 . . 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 5867 . . 3  |-  ( f `
 S )  =  ( f `  (
( _|_ `  A
)  vH  ( A  i^i  C ) ) )
8178, 80oveq12i 6294 . 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 2533 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 1379    e. wcel 1767    i^i cin 3475   ` cfv 5586  (class class class)co 6282   RRcr 9487    + caddc 9491   CHcch 25522   _|_cort 25523    vH chj 25526   Statescst 25555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cc 8811  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568  ax-hilex 25592  ax-hfvadd 25593  ax-hvcom 25594  ax-hvass 25595  ax-hv0cl 25596  ax-hvaddid 25597  ax-hfvmul 25598  ax-hvmulid 25599  ax-hvmulass 25600  ax-hvdistr1 25601  ax-hvdistr2 25602  ax-hvmul0 25603  ax-hfi 25672  ax-his1 25675  ax-his2 25676  ax-his3 25677  ax-his4 25678  ax-hcompl 25795
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-omul 7132  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-acn 8319  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12072  df-exp 12131  df-hash 12370  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-clim 13270  df-rlim 13271  df-sum 13468  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-pt 14696  df-prds 14699  df-xrs 14753  df-qtop 14758  df-imas 14759  df-xps 14761  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-fbas 18187  df-fg 18188  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-cn 19494  df-cnp 19495  df-lm 19496  df-haus 19582  df-tx 19798  df-hmeo 19991  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-xms 20558  df-ms 20559  df-tms 20560  df-cfil 21429  df-cau 21430  df-cmet 21431  df-grpo 24869  df-gid 24870  df-ginv 24871  df-gdiv 24872  df-ablo 24960  df-subgo 24980  df-vc 25115  df-nv 25161  df-va 25164  df-ba 25165  df-sm 25166  df-0v 25167  df-vs 25168  df-nmcv 25169  df-ims 25170  df-dip 25287  df-ssp 25311  df-ph 25404  df-cbn 25455  df-hnorm 25561  df-hba 25562  df-hvsub 25564  df-hlim 25565  df-hcau 25566  df-sh 25800  df-ch 25815  df-oc 25846  df-ch0 25847  df-st 26806
This theorem is referenced by:  golem2  26867
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