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Theorem mdslmd1lem1 22905
Description: Lemma for mdslmd1i 22909. (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
mdslmd.1  |-  A  e. 
CH
mdslmd.2  |-  B  e. 
CH
mdslmd.3  |-  C  e. 
CH
mdslmd.4  |-  D  e. 
CH
mdslmd1lem.5  |-  R  e. 
CH
Assertion
Ref Expression
mdslmd1lem1  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( ( ( R  vH  A ) 
C_  D  ->  (
( ( R  vH  A )  vH  C
)  i^i  D )  C_  ( ( R  vH  A )  vH  ( C  i^i  D ) ) )  ->  ( (
( ( C  i^i  B )  i^i  ( D  i^i  B ) ) 
C_  R  /\  R  C_  ( D  i^i  B
) )  ->  (
( R  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) )  C_  ( R  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B
) ) ) ) ) )

Proof of Theorem mdslmd1lem1
StepHypRef Expression
1 mdslmd1lem.5 . . . . . 6  |-  R  e. 
CH
2 mdslmd.4 . . . . . . 7  |-  D  e. 
CH
3 mdslmd.2 . . . . . . 7  |-  B  e. 
CH
42, 3chincli 22039 . . . . . 6  |-  ( D  i^i  B )  e. 
CH
5 mdslmd.1 . . . . . 6  |-  A  e. 
CH
61, 4, 5chlej1i 22052 . . . . 5  |-  ( R 
C_  ( D  i^i  B )  ->  ( R  vH  A )  C_  (
( D  i^i  B
)  vH  A )
)
7 simpr 447 . . . . . . . 8  |-  ( ( A  MH  B  /\  B  MH*  A )  ->  B  MH*  A )
8 simpr 447 . . . . . . . 8  |-  ( ( A  C_  C  /\  A  C_  D )  ->  A  C_  D )
9 simpr 447 . . . . . . . 8  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  ->  D  C_  ( A  vH  B
) )
105, 3, 23pm3.2i 1130 . . . . . . . . 9  |-  ( A  e.  CH  /\  B  e.  CH  /\  D  e. 
CH )
11 dmdsl3 22895 . . . . . . . . 9  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  D  e.  CH )  /\  ( B  MH*  A  /\  A  C_  D  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( D  i^i  B )  vH  A )  =  D )
1210, 11mpan 651 . . . . . . . 8  |-  ( ( B  MH*  A  /\  A  C_  D  /\  D  C_  ( A  vH  B
) )  ->  (
( D  i^i  B
)  vH  A )  =  D )
137, 8, 9, 12syl3an 1224 . . . . . . 7  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( D  i^i  B )  vH  A )  =  D )
14133expb 1152 . . . . . 6  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( ( D  i^i  B )  vH  A )  =  D )
1514sseq2d 3206 . . . . 5  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( ( R  vH  A )  C_  ( ( D  i^i  B )  vH  A )  <-> 
( R  vH  A
)  C_  D )
)
166, 15syl5ib 210 . . . 4  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( R  C_  ( D  i^i  B )  ->  ( R  vH  A )  C_  D
) )
1716adantld 453 . . 3  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( ( ( ( C  i^i  B
)  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) )  ->  ( R  vH  A )  C_  D
) )
1817imim1d 69 . 2  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( ( ( R  vH  A ) 
C_  D  ->  (
( ( R  vH  A )  vH  C
)  i^i  D )  C_  ( ( R  vH  A )  vH  ( C  i^i  D ) ) )  ->  ( (
( ( C  i^i  B )  i^i  ( D  i^i  B ) ) 
C_  R  /\  R  C_  ( D  i^i  B
) )  ->  (
( ( R  vH  A )  vH  C
)  i^i  D )  C_  ( ( R  vH  A )  vH  ( C  i^i  D ) ) ) ) )
19 simpll 730 . . . . . . . . . 10  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( A  MH  B  /\  B  MH*  A ) )
20 simpll 730 . . . . . . . . . . . . 13  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) )  ->  A  C_  C
)
2120ad2antlr 707 . . . . . . . . . . . 12  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  A  C_  C
)
225, 1chub2i 22049 . . . . . . . . . . . 12  |-  A  C_  ( R  vH  A )
2321, 22jctil 523 . . . . . . . . . . 11  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( A  C_  ( R  vH  A
)  /\  A  C_  C
) )
24 ssin 3391 . . . . . . . . . . 11  |-  ( ( A  C_  ( R  vH  A )  /\  A  C_  C )  <->  A  C_  (
( R  vH  A
)  i^i  C )
)
2523, 24sylib 188 . . . . . . . . . 10  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  A  C_  (
( R  vH  A
)  i^i  C )
)
26 inss1 3389 . . . . . . . . . . . . . . . . . . . 20  |-  ( D  i^i  B )  C_  D
27 sstr 3187 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( R  C_  ( D  i^i  B )  /\  ( D  i^i  B )  C_  D )  ->  R  C_  D )
2826, 27mpan2 652 . . . . . . . . . . . . . . . . . . 19  |-  ( R 
C_  ( D  i^i  B )  ->  R  C_  D
)
29 sstr 3187 . . . . . . . . . . . . . . . . . . 19  |-  ( ( R  C_  D  /\  D  C_  ( A  vH  B ) )  ->  R  C_  ( A  vH  B ) )
3028, 29sylan 457 . . . . . . . . . . . . . . . . . 18  |-  ( ( R  C_  ( D  i^i  B )  /\  D  C_  ( A  vH  B
) )  ->  R  C_  ( A  vH  B
) )
3130ancoms 439 . . . . . . . . . . . . . . . . 17  |-  ( ( D  C_  ( A  vH  B )  /\  R  C_  ( D  i^i  B
) )  ->  R  C_  ( A  vH  B
) )
3231adantll 694 . . . . . . . . . . . . . . . 16  |-  ( ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) )  /\  R  C_  ( D  i^i  B ) )  ->  R  C_  ( A  vH  B
) )
3332adantll 694 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  /\  R  C_  ( D  i^i  B ) )  ->  R  C_  ( A  vH  B ) )
3433ad2ant2l 726 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  R  C_  ( A  vH  B ) )
355, 3chub1i 22048 . . . . . . . . . . . . . 14  |-  A  C_  ( A  vH  B )
3634, 35jctir 524 . . . . . . . . . . . . 13  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( R  C_  ( A  vH  B
)  /\  A  C_  ( A  vH  B ) ) )
375, 3chjcli 22036 . . . . . . . . . . . . . 14  |-  ( A  vH  B )  e. 
CH
381, 5, 37chlubi 22050 . . . . . . . . . . . . 13  |-  ( ( R  C_  ( A  vH  B )  /\  A  C_  ( A  vH  B
) )  <->  ( R  vH  A )  C_  ( A  vH  B ) )
3936, 38sylib 188 . . . . . . . . . . . 12  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( R  vH  A )  C_  ( A  vH  B ) )
40 simprrl 740 . . . . . . . . . . . . 13  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  C  C_  ( A  vH  B ) )
4140adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  C  C_  ( A  vH  B ) )
4239, 41jca 518 . . . . . . . . . . 11  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( ( R  vH  A )  C_  ( A  vH  B )  /\  C  C_  ( A  vH  B ) ) )
431, 5chjcli 22036 . . . . . . . . . . . 12  |-  ( R  vH  A )  e. 
CH
44 mdslmd.3 . . . . . . . . . . . 12  |-  C  e. 
CH
4543, 44, 37chlubi 22050 . . . . . . . . . . 11  |-  ( ( ( R  vH  A
)  C_  ( A  vH  B )  /\  C  C_  ( A  vH  B
) )  <->  ( ( R  vH  A )  vH  C )  C_  ( A  vH  B ) )
4642, 45sylib 188 . . . . . . . . . 10  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( ( R  vH  A )  vH  C )  C_  ( A  vH  B ) )
475, 3, 43, 44mdslj1i 22899 . . . . . . . . . 10  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( ( R  vH  A )  i^i  C
)  /\  ( ( R  vH  A )  vH  C )  C_  ( A  vH  B ) ) )  ->  ( (
( R  vH  A
)  vH  C )  i^i  B )  =  ( ( ( R  vH  A )  i^i  B
)  vH  ( C  i^i  B ) ) )
4819, 25, 46, 47syl12anc 1180 . . . . . . . . 9  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( (
( R  vH  A
)  vH  C )  i^i  B )  =  ( ( ( R  vH  A )  i^i  B
)  vH  ( C  i^i  B ) ) )
49 simplll 734 . . . . . . . . . . 11  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  A  MH  B )
50 simplrl 736 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( A  C_  C  /\  A  C_  D ) )
51 ssin 3391 . . . . . . . . . . . . . . 15  |-  ( ( A  C_  C  /\  A  C_  D )  <->  A  C_  ( C  i^i  D ) )
5250, 51sylib 188 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  A  C_  ( C  i^i  D ) )
53 ssrin 3394 . . . . . . . . . . . . . 14  |-  ( A 
C_  ( C  i^i  D )  ->  ( A  i^i  B )  C_  (
( C  i^i  D
)  i^i  B )
)
5452, 53syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( A  i^i  B )  C_  (
( C  i^i  D
)  i^i  B )
)
55 inindir 3387 . . . . . . . . . . . . 13  |-  ( ( C  i^i  D )  i^i  B )  =  ( ( C  i^i  B )  i^i  ( D  i^i  B ) )
5654, 55syl6sseq 3224 . . . . . . . . . . . 12  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( A  i^i  B )  C_  (
( C  i^i  B
)  i^i  ( D  i^i  B ) ) )
57 simprl 732 . . . . . . . . . . . 12  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( ( C  i^i  B )  i^i  ( D  i^i  B
) )  C_  R
)
5856, 57sstrd 3189 . . . . . . . . . . 11  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( A  i^i  B )  C_  R
)
59 inss2 3390 . . . . . . . . . . . . 13  |-  ( D  i^i  B )  C_  B
60 sstr 3187 . . . . . . . . . . . . 13  |-  ( ( R  C_  ( D  i^i  B )  /\  ( D  i^i  B )  C_  B )  ->  R  C_  B )
6159, 60mpan2 652 . . . . . . . . . . . 12  |-  ( R 
C_  ( D  i^i  B )  ->  R  C_  B
)
6261ad2antll 709 . . . . . . . . . . 11  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  R  C_  B
)
635, 3, 13pm3.2i 1130 . . . . . . . . . . . 12  |-  ( A  e.  CH  /\  B  e.  CH  /\  R  e. 
CH )
64 mdsl3 22896 . . . . . . . . . . . 12  |-  ( ( ( A  e.  CH  /\  B  e.  CH  /\  R  e.  CH )  /\  ( A  MH  B  /\  ( A  i^i  B
)  C_  R  /\  R  C_  B ) )  ->  ( ( R  vH  A )  i^i 
B )  =  R )
6563, 64mpan 651 . . . . . . . . . . 11  |-  ( ( A  MH  B  /\  ( A  i^i  B ) 
C_  R  /\  R  C_  B )  ->  (
( R  vH  A
)  i^i  B )  =  R )
6649, 58, 62, 65syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( ( R  vH  A )  i^i 
B )  =  R )
6766oveq1d 5873 . . . . . . . . 9  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( (
( R  vH  A
)  i^i  B )  vH  ( C  i^i  B
) )  =  ( R  vH  ( C  i^i  B ) ) )
6848, 67eqtr2d 2316 . . . . . . . 8  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( R  vH  ( C  i^i  B
) )  =  ( ( ( R  vH  A )  vH  C
)  i^i  B )
)
6968ineq1d 3369 . . . . . . 7  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( ( R  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) )  =  ( ( ( ( R  vH  A
)  vH  C )  i^i  B )  i^i  ( D  i^i  B ) ) )
70 inindir 3387 . . . . . . 7  |-  ( ( ( ( R  vH  A )  vH  C
)  i^i  D )  i^i  B )  =  ( ( ( ( R  vH  A )  vH  C )  i^i  B
)  i^i  ( D  i^i  B ) )
7169, 70syl6eqr 2333 . . . . . 6  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( ( R  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) )  =  ( ( ( ( R  vH  A
)  vH  C )  i^i  D )  i^i  B
) )
7252, 22jctil 523 . . . . . . . . 9  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( A  C_  ( R  vH  A
)  /\  A  C_  ( C  i^i  D ) ) )
73 ssin 3391 . . . . . . . . 9  |-  ( ( A  C_  ( R  vH  A )  /\  A  C_  ( C  i^i  D
) )  <->  A  C_  (
( R  vH  A
)  i^i  ( C  i^i  D ) ) )
7472, 73sylib 188 . . . . . . . 8  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  A  C_  (
( R  vH  A
)  i^i  ( C  i^i  D ) ) )
75 ssinss1 3397 . . . . . . . . . . . 12  |-  ( C 
C_  ( A  vH  B )  ->  ( C  i^i  D )  C_  ( A  vH  B ) )
7675ad2antrl 708 . . . . . . . . . . 11  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) )  ->  ( C  i^i  D )  C_  ( A  vH  B ) )
7776ad2antlr 707 . . . . . . . . . 10  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( C  i^i  D )  C_  ( A  vH  B ) )
7839, 77jca 518 . . . . . . . . 9  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( ( R  vH  A )  C_  ( A  vH  B )  /\  ( C  i^i  D )  C_  ( A  vH  B ) ) )
7944, 2chincli 22039 . . . . . . . . . 10  |-  ( C  i^i  D )  e. 
CH
8043, 79, 37chlubi 22050 . . . . . . . . 9  |-  ( ( ( R  vH  A
)  C_  ( A  vH  B )  /\  ( C  i^i  D )  C_  ( A  vH  B ) )  <->  ( ( R  vH  A )  vH  ( C  i^i  D ) )  C_  ( A  vH  B ) )
8178, 80sylib 188 . . . . . . . 8  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( ( R  vH  A )  vH  ( C  i^i  D ) )  C_  ( A  vH  B ) )
825, 3, 43, 79mdslj1i 22899 . . . . . . . 8  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( ( R  vH  A )  i^i  ( C  i^i  D ) )  /\  ( ( R  vH  A )  vH  ( C  i^i  D ) )  C_  ( A  vH  B ) ) )  ->  ( ( ( R  vH  A )  vH  ( C  i^i  D ) )  i^i  B
)  =  ( ( ( R  vH  A
)  i^i  B )  vH  ( ( C  i^i  D )  i^i  B ) ) )
8319, 74, 81, 82syl12anc 1180 . . . . . . 7  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( (
( R  vH  A
)  vH  ( C  i^i  D ) )  i^i 
B )  =  ( ( ( R  vH  A )  i^i  B
)  vH  ( ( C  i^i  D )  i^i 
B ) ) )
8455a1i 10 . . . . . . . 8  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( ( C  i^i  D )  i^i 
B )  =  ( ( C  i^i  B
)  i^i  ( D  i^i  B ) ) )
8566, 84oveq12d 5876 . . . . . . 7  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( (
( R  vH  A
)  i^i  B )  vH  ( ( C  i^i  D )  i^i  B ) )  =  ( R  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B
) ) ) )
8683, 85eqtr2d 2316 . . . . . 6  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( R  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) )  =  ( ( ( R  vH  A
)  vH  ( C  i^i  D ) )  i^i 
B ) )
8771, 86sseq12d 3207 . . . . 5  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( (
( R  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) )  C_  ( R  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B
) ) )  <->  ( (
( ( R  vH  A )  vH  C
)  i^i  D )  i^i  B )  C_  (
( ( R  vH  A )  vH  ( C  i^i  D ) )  i^i  B ) ) )
88 simpllr 735 . . . . . 6  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  B  MH*  A )
89 simplr 731 . . . . . . . . . . 11  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) )  ->  A  C_  D
)
9089ad2antlr 707 . . . . . . . . . 10  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  A  C_  D
)
9143, 44chub1i 22048 . . . . . . . . . . 11  |-  ( R  vH  A )  C_  ( ( R  vH  A )  vH  C
)
9222, 91sstri 3188 . . . . . . . . . 10  |-  A  C_  ( ( R  vH  A )  vH  C
)
9390, 92jctil 523 . . . . . . . . 9  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( A  C_  ( ( R  vH  A )  vH  C
)  /\  A  C_  D
) )
94 ssin 3391 . . . . . . . . 9  |-  ( ( A  C_  ( ( R  vH  A )  vH  C )  /\  A  C_  D )  <->  A  C_  (
( ( R  vH  A )  vH  C
)  i^i  D )
)
9593, 94sylib 188 . . . . . . . 8  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  A  C_  (
( ( R  vH  A )  vH  C
)  i^i  D )
)
9643, 79chub1i 22048 . . . . . . . . 9  |-  ( R  vH  A )  C_  ( ( R  vH  A )  vH  ( C  i^i  D ) )
9722, 96sstri 3188 . . . . . . . 8  |-  A  C_  ( ( R  vH  A )  vH  ( C  i^i  D ) )
9895, 97jctir 524 . . . . . . 7  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( A  C_  ( ( ( R  vH  A )  vH  C )  i^i  D
)  /\  A  C_  (
( R  vH  A
)  vH  ( C  i^i  D ) ) ) )
99 ssin 3391 . . . . . . 7  |-  ( ( A  C_  ( (
( R  vH  A
)  vH  C )  i^i  D )  /\  A  C_  ( ( R  vH  A )  vH  ( C  i^i  D ) ) )  <->  A  C_  ( ( ( ( R  vH  A )  vH  C
)  i^i  D )  i^i  ( ( R  vH  A )  vH  ( C  i^i  D ) ) ) )
10098, 99sylib 188 . . . . . 6  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  A  C_  (
( ( ( R  vH  A )  vH  C )  i^i  D
)  i^i  ( ( R  vH  A )  vH  ( C  i^i  D ) ) ) )
101 inss2 3390 . . . . . . . . . . 11  |-  ( ( ( R  vH  A
)  vH  C )  i^i  D )  C_  D
102 sstr 3187 . . . . . . . . . . 11  |-  ( ( ( ( ( R  vH  A )  vH  C )  i^i  D
)  C_  D  /\  D  C_  ( A  vH  B ) )  -> 
( ( ( R  vH  A )  vH  C )  i^i  D
)  C_  ( A  vH  B ) )
103101, 102mpan 651 . . . . . . . . . 10  |-  ( D 
C_  ( A  vH  B )  ->  (
( ( R  vH  A )  vH  C
)  i^i  D )  C_  ( A  vH  B
) )
104103ad2antll 709 . . . . . . . . 9  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) )  ->  ( (
( R  vH  A
)  vH  C )  i^i  D )  C_  ( A  vH  B ) )
105104ad2antlr 707 . . . . . . . 8  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( (
( R  vH  A
)  vH  C )  i^i  D )  C_  ( A  vH  B ) )
106105, 81jca 518 . . . . . . 7  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( (
( ( R  vH  A )  vH  C
)  i^i  D )  C_  ( A  vH  B
)  /\  ( ( R  vH  A )  vH  ( C  i^i  D ) )  C_  ( A  vH  B ) ) )
10743, 44chjcli 22036 . . . . . . . . 9  |-  ( ( R  vH  A )  vH  C )  e. 
CH
108107, 2chincli 22039 . . . . . . . 8  |-  ( ( ( R  vH  A
)  vH  C )  i^i  D )  e.  CH
10943, 79chjcli 22036 . . . . . . . 8  |-  ( ( R  vH  A )  vH  ( C  i^i  D ) )  e.  CH
110108, 109, 37chlubi 22050 . . . . . . 7  |-  ( ( ( ( ( R  vH  A )  vH  C )  i^i  D
)  C_  ( A  vH  B )  /\  (
( R  vH  A
)  vH  ( C  i^i  D ) )  C_  ( A  vH  B ) )  <->  ( ( ( ( R  vH  A
)  vH  C )  i^i  D )  vH  (
( R  vH  A
)  vH  ( C  i^i  D ) ) ) 
C_  ( A  vH  B ) )
111106, 110sylib 188 . . . . . 6  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( (
( ( R  vH  A )  vH  C
)  i^i  D )  vH  ( ( R  vH  A )  vH  ( C  i^i  D ) ) )  C_  ( A  vH  B ) )
1125, 3, 108, 109mdslle1i 22897 . . . . . 6  |-  ( ( B  MH*  A  /\  A  C_  ( ( ( ( R  vH  A
)  vH  C )  i^i  D )  i^i  (
( R  vH  A
)  vH  ( C  i^i  D ) ) )  /\  ( ( ( ( R  vH  A
)  vH  C )  i^i  D )  vH  (
( R  vH  A
)  vH  ( C  i^i  D ) ) ) 
C_  ( A  vH  B ) )  -> 
( ( ( ( R  vH  A )  vH  C )  i^i 
D )  C_  (
( R  vH  A
)  vH  ( C  i^i  D ) )  <->  ( (
( ( R  vH  A )  vH  C
)  i^i  D )  i^i  B )  C_  (
( ( R  vH  A )  vH  ( C  i^i  D ) )  i^i  B ) ) )
11388, 100, 111, 112syl3anc 1182 . . . . 5  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( (
( ( R  vH  A )  vH  C
)  i^i  D )  C_  ( ( R  vH  A )  vH  ( C  i^i  D ) )  <-> 
( ( ( ( R  vH  A )  vH  C )  i^i 
D )  i^i  B
)  C_  ( (
( R  vH  A
)  vH  ( C  i^i  D ) )  i^i 
B ) ) )
11487, 113bitr4d 247 . . . 4  |-  ( ( ( ( A  MH  B  /\  B  MH*  A
)  /\  ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  /\  ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  ( (
( R  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) )  C_  ( R  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B
) ) )  <->  ( (
( R  vH  A
)  vH  C )  i^i  D )  C_  (
( R  vH  A
)  vH  ( C  i^i  D ) ) ) )
115114exbiri 605 . . 3  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( ( ( ( C  i^i  B
)  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) )  ->  ( ( ( ( R  vH  A
)  vH  C )  i^i  D )  C_  (
( R  vH  A
)  vH  ( C  i^i  D ) )  -> 
( ( R  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
C_  ( R  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) ) ) )
116115a2d 23 . 2  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( ( ( ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) 
C_  R  /\  R  C_  ( D  i^i  B
) )  ->  (
( ( R  vH  A )  vH  C
)  i^i  D )  C_  ( ( R  vH  A )  vH  ( C  i^i  D ) ) )  ->  ( (
( ( C  i^i  B )  i^i  ( D  i^i  B ) ) 
C_  R  /\  R  C_  ( D  i^i  B
) )  ->  (
( R  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) )  C_  ( R  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B
) ) ) ) ) )
11718, 116syld 40 1  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( ( ( R  vH  A ) 
C_  D  ->  (
( ( R  vH  A )  vH  C
)  i^i  D )  C_  ( ( R  vH  A )  vH  ( C  i^i  D ) ) )  ->  ( (
( ( C  i^i  B )  i^i  ( D  i^i  B ) ) 
C_  R  /\  R  C_  ( D  i^i  B
) )  ->  (
( R  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) )  C_  ( R  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B
) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   class class class wbr 4023  (class class class)co 5858   CHcch 21509    vH chj 21513    MH cmd 21546    MH* cdmd 21547
This theorem is referenced by:  mdslmd1lem3  22907
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cc 8061  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817  ax-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvmulass 21587  ax-hvdistr1 21588  ax-hvdistr2 21589  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663  ax-his4 21664  ax-hcompl 21781
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-cn 16957  df-cnp 16958  df-lm 16959  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cfil 18681  df-cau 18682  df-cmet 18683  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-subgo 20969  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156  df-ims 21157  df-dip 21274  df-ssp 21298  df-ph 21391  df-cbn 21442  df-hnorm 21548  df-hba 21549  df-hvsub 21551  df-hlim 21552  df-hcau 21553  df-sh 21786  df-ch 21801  df-oc 21831  df-ch0 21832  df-shs 21887  df-chj 21889  df-md 22860  df-dmd 22861
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