HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  mdslmd1lem2 Structured version   Visualization version   Unicode version

Theorem mdslmd1lem2 27979
Description: Lemma for mdslmd1i 27982. (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
mdslmd1lem2  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( ( ( R  i^i  B ) 
C_  ( D  i^i  B )  ->  ( (
( R  i^i  B
)  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B
) )  C_  (
( R  i^i  B
)  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B
) ) ) )  ->  ( ( ( C  i^i  D ) 
C_  R  /\  R  C_  D )  ->  (
( R  vH  C
)  i^i  D )  C_  ( R  vH  ( C  i^i  D ) ) ) ) )

Proof of Theorem mdslmd1lem2
StepHypRef Expression
1 ssrin 3657 . . . 4  |-  ( R 
C_  D  ->  ( R  i^i  B )  C_  ( D  i^i  B ) )
21adantl 468 . . 3  |-  ( ( ( C  i^i  D
)  C_  R  /\  R  C_  D )  -> 
( R  i^i  B
)  C_  ( D  i^i  B ) )
32imim1i 60 . 2  |-  ( ( ( R  i^i  B
)  C_  ( D  i^i  B )  ->  (
( ( R  i^i  B )  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) )  C_  (
( R  i^i  B
)  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B
) ) ) )  ->  ( ( ( C  i^i  D ) 
C_  R  /\  R  C_  D )  ->  (
( ( R  i^i  B )  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) )  C_  (
( R  i^i  B
)  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B
) ) ) ) )
4 simpllr 769 . . . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  B  MH*  A )
5 mdslmd.3 . . . . . . . . . . . 12  |-  C  e. 
CH
6 mdslmd1lem.5 . . . . . . . . . . . 12  |-  R  e. 
CH
75, 6chub2i 27123 . . . . . . . . . . 11  |-  C  C_  ( R  vH  C )
8 sstr 3440 . . . . . . . . . . 11  |-  ( ( A  C_  C  /\  C  C_  ( R  vH  C ) )  ->  A  C_  ( R  vH  C ) )
97, 8mpan2 677 . . . . . . . . . 10  |-  ( A 
C_  C  ->  A  C_  ( R  vH  C
) )
109ad2antrr 732 . . . . . . . . 9  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) )  ->  A  C_  ( R  vH  C ) )
1110ad2antlr 733 . . . . . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  A  C_  ( R  vH  C
) )
12 simplr 762 . . . . . . . . 9  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) )  ->  A  C_  D
)
1312ad2antlr 733 . . . . . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  A  C_  D )
1411, 13ssind 3656 . . . . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  A  C_  ( ( R  vH  C )  i^i  D
) )
15 ssin 3654 . . . . . . . . . 10  |-  ( ( A  C_  C  /\  A  C_  D )  <->  A  C_  ( C  i^i  D ) )
16 mdslmd.4 . . . . . . . . . . . . 13  |-  D  e. 
CH
175, 16chincli 27113 . . . . . . . . . . . 12  |-  ( C  i^i  D )  e. 
CH
1817, 6chub2i 27123 . . . . . . . . . . 11  |-  ( C  i^i  D )  C_  ( R  vH  ( C  i^i  D ) )
19 sstr 3440 . . . . . . . . . . 11  |-  ( ( A  C_  ( C  i^i  D )  /\  ( C  i^i  D )  C_  ( R  vH  ( C  i^i  D ) ) )  ->  A  C_  ( R  vH  ( C  i^i  D ) ) )
2018, 19mpan2 677 . . . . . . . . . 10  |-  ( A 
C_  ( C  i^i  D )  ->  A  C_  ( R  vH  ( C  i^i  D ) ) )
2115, 20sylbi 199 . . . . . . . . 9  |-  ( ( A  C_  C  /\  A  C_  D )  ->  A  C_  ( R  vH  ( C  i^i  D ) ) )
2221adantr 467 . . . . . . . 8  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) )  ->  A  C_  ( R  vH  ( C  i^i  D ) ) )
2322ad2antlr 733 . . . . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  A  C_  ( R  vH  ( C  i^i  D ) ) )
2414, 23ssind 3656 . . . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  A  C_  ( ( ( R  vH  C )  i^i 
D )  i^i  ( R  vH  ( C  i^i  D ) ) ) )
25 inss2 3653 . . . . . . . . . . 11  |-  ( ( R  vH  C )  i^i  D )  C_  D
26 sstr 3440 . . . . . . . . . . 11  |-  ( ( ( ( R  vH  C )  i^i  D
)  C_  D  /\  D  C_  ( A  vH  B ) )  -> 
( ( R  vH  C )  i^i  D
)  C_  ( A  vH  B ) )
2725, 26mpan 676 . . . . . . . . . 10  |-  ( D 
C_  ( A  vH  B )  ->  (
( R  vH  C
)  i^i  D )  C_  ( A  vH  B
) )
2827ad2antll 735 . . . . . . . . 9  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) )  ->  ( ( R  vH  C )  i^i 
D )  C_  ( A  vH  B ) )
2928ad2antlr 733 . . . . . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  (
( R  vH  C
)  i^i  D )  C_  ( A  vH  B
) )
30 sstr 3440 . . . . . . . . . . . . . 14  |-  ( ( R  C_  D  /\  D  C_  ( A  vH  B ) )  ->  R  C_  ( A  vH  B ) )
3130ancoms 455 . . . . . . . . . . . . 13  |-  ( ( D  C_  ( A  vH  B )  /\  R  C_  D )  ->  R  C_  ( A  vH  B
) )
3231ad2ant2l 752 . . . . . . . . . . . 12  |-  ( ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) )  /\  ( ( C  i^i  D )  C_  R  /\  R  C_  D ) )  ->  R  C_  ( A  vH  B ) )
3332adantll 720 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  /\  ( ( C  i^i  D ) 
C_  R  /\  R  C_  D ) )  ->  R  C_  ( A  vH  B ) )
3433adantll 720 . . . . . . . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  R  C_  ( A  vH  B
) )
35 ssinss1 3660 . . . . . . . . . . . 12  |-  ( C 
C_  ( A  vH  B )  ->  ( C  i^i  D )  C_  ( A  vH  B ) )
3635ad2antrl 734 . . . . . . . . . . 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 ) )
3736ad2antlr 733 . . . . . . . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  ( C  i^i  D )  C_  ( A  vH  B ) )
3834, 37jca 535 . . . . . . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  ( R  C_  ( A  vH  B )  /\  ( C  i^i  D )  C_  ( A  vH  B ) ) )
39 mdslmd.1 . . . . . . . . . . 11  |-  A  e. 
CH
40 mdslmd.2 . . . . . . . . . . 11  |-  B  e. 
CH
4139, 40chjcli 27110 . . . . . . . . . 10  |-  ( A  vH  B )  e. 
CH
426, 17, 41chlubi 27124 . . . . . . . . 9  |-  ( ( R  C_  ( A  vH  B )  /\  ( C  i^i  D )  C_  ( A  vH  B ) )  <->  ( R  vH  ( C  i^i  D ) )  C_  ( A  vH  B ) )
4338, 42sylib 200 . . . . . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  ( R  vH  ( C  i^i  D ) )  C_  ( A  vH  B ) )
4429, 43jca 535 . . . . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  (
( ( R  vH  C )  i^i  D
)  C_  ( A  vH  B )  /\  ( R  vH  ( C  i^i  D ) )  C_  ( A  vH  B ) ) )
456, 5chjcli 27110 . . . . . . . . 9  |-  ( R  vH  C )  e. 
CH
4645, 16chincli 27113 . . . . . . . 8  |-  ( ( R  vH  C )  i^i  D )  e. 
CH
476, 17chjcli 27110 . . . . . . . 8  |-  ( R  vH  ( C  i^i  D ) )  e.  CH
4846, 47, 41chlubi 27124 . . . . . . 7  |-  ( ( ( ( R  vH  C )  i^i  D
)  C_  ( A  vH  B )  /\  ( R  vH  ( C  i^i  D ) )  C_  ( A  vH  B ) )  <-> 
( ( ( R  vH  C )  i^i 
D )  vH  ( R  vH  ( C  i^i  D ) ) )  C_  ( A  vH  B ) )
4944, 48sylib 200 . . . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  (
( ( R  vH  C )  i^i  D
)  vH  ( R  vH  ( C  i^i  D
) ) )  C_  ( A  vH  B ) )
5039, 40, 46, 47mdslle1i 27970 . . . . . 6  |-  ( ( B  MH*  A  /\  A  C_  ( ( ( R  vH  C )  i^i  D )  i^i  ( R  vH  ( C  i^i  D ) ) )  /\  ( ( ( R  vH  C
)  i^i  D )  vH  ( R  vH  ( C  i^i  D ) ) )  C_  ( A  vH  B ) )  -> 
( ( ( R  vH  C )  i^i 
D )  C_  ( R  vH  ( C  i^i  D ) )  <->  ( (
( R  vH  C
)  i^i  D )  i^i  B )  C_  (
( R  vH  ( C  i^i  D ) )  i^i  B ) ) )
514, 24, 49, 50syl3anc 1268 . . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  (
( ( R  vH  C )  i^i  D
)  C_  ( R  vH  ( C  i^i  D
) )  <->  ( (
( R  vH  C
)  i^i  D )  i^i  B )  C_  (
( R  vH  ( C  i^i  D ) )  i^i  B ) ) )
52 inindir 3650 . . . . . . 7  |-  ( ( ( R  vH  C
)  i^i  D )  i^i  B )  =  ( ( ( R  vH  C )  i^i  B
)  i^i  ( D  i^i  B ) )
53 sstr 3440 . . . . . . . . . . . . . 14  |-  ( ( A  C_  ( C  i^i  D )  /\  ( C  i^i  D )  C_  R )  ->  A  C_  R )
5415, 53sylanb 475 . . . . . . . . . . . . 13  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  i^i  D )  C_  R )  ->  A  C_  R )
5554ad2ant2r 753 . . . . . . . . . . . 12  |-  ( ( ( ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  /\  ( ( C  i^i  D ) 
C_  R  /\  R  C_  D ) )  ->  A  C_  R )
56 simplll 768 . . . . . . . . . . . 12  |-  ( ( ( ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  /\  ( ( C  i^i  D ) 
C_  R  /\  R  C_  D ) )  ->  A  C_  C )
5755, 56ssind 3656 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  /\  ( ( C  i^i  D ) 
C_  R  /\  R  C_  D ) )  ->  A  C_  ( R  i^i  C ) )
58 simplrl 770 . . . . . . . . . . . . 13  |-  ( ( ( ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  /\  ( ( C  i^i  D ) 
C_  R  /\  R  C_  D ) )  ->  C  C_  ( A  vH  B ) )
5933, 58jca 535 . . . . . . . . . . . 12  |-  ( ( ( ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  /\  ( ( C  i^i  D ) 
C_  R  /\  R  C_  D ) )  -> 
( R  C_  ( A  vH  B )  /\  C  C_  ( A  vH  B ) ) )
606, 5, 41chlubi 27124 . . . . . . . . . . . 12  |-  ( ( R  C_  ( A  vH  B )  /\  C  C_  ( A  vH  B
) )  <->  ( R  vH  C )  C_  ( A  vH  B ) )
6159, 60sylib 200 . . . . . . . . . . 11  |-  ( ( ( ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  /\  ( ( C  i^i  D ) 
C_  R  /\  R  C_  D ) )  -> 
( R  vH  C
)  C_  ( A  vH  B ) )
6257, 61jca 535 . . . . . . . . . 10  |-  ( ( ( ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  /\  ( ( C  i^i  D ) 
C_  R  /\  R  C_  D ) )  -> 
( A  C_  ( R  i^i  C )  /\  ( R  vH  C ) 
C_  ( A  vH  B ) ) )
6339, 40, 6, 5mdslj1i 27972 . . . . . . . . . 10  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( R  i^i  C )  /\  ( R  vH  C )  C_  ( A  vH  B ) ) )  ->  ( ( R  vH  C )  i^i 
B )  =  ( ( R  i^i  B
)  vH  ( C  i^i  B ) ) )
6462, 63sylan2 477 . . . . . . . . 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  D )  C_  R  /\  R  C_  D
) ) )  -> 
( ( R  vH  C )  i^i  B
)  =  ( ( R  i^i  B )  vH  ( C  i^i  B ) ) )
6564anassrs 654 . . . . . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  (
( R  vH  C
)  i^i  B )  =  ( ( R  i^i  B )  vH  ( C  i^i  B ) ) )
6665ineq1d 3633 . . . . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  (
( ( R  vH  C )  i^i  B
)  i^i  ( D  i^i  B ) )  =  ( ( ( R  i^i  B )  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) )
6752, 66syl5req 2498 . . . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  (
( ( R  i^i  B )  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) )  =  ( ( ( R  vH  C )  i^i  D
)  i^i  B )
)
6815biimpi 198 . . . . . . . . . . . . 13  |-  ( ( A  C_  C  /\  A  C_  D )  ->  A  C_  ( C  i^i  D ) )
6968adantr 467 . . . . . . . . . . . 12  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  i^i  D )  C_  R )  ->  A  C_  ( C  i^i  D ) )
7054, 69ssind 3656 . . . . . . . . . . 11  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  i^i  D )  C_  R )  ->  A  C_  ( R  i^i  ( C  i^i  D
) ) )
7131adantll 720 . . . . . . . . . . . . 13  |-  ( ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) )  /\  R  C_  D )  ->  R  C_  ( A  vH  B ) )
7235ad2antrr 732 . . . . . . . . . . . . 13  |-  ( ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) )  /\  R  C_  D )  -> 
( C  i^i  D
)  C_  ( A  vH  B ) )
7371, 72jca 535 . . . . . . . . . . . 12  |-  ( ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) )  /\  R  C_  D )  -> 
( R  C_  ( A  vH  B )  /\  ( C  i^i  D ) 
C_  ( A  vH  B ) ) )
7473, 42sylib 200 . . . . . . . . . . 11  |-  ( ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) )  /\  R  C_  D )  -> 
( R  vH  ( C  i^i  D ) ) 
C_  ( A  vH  B ) )
7570, 74anim12i 570 . . . . . . . . . 10  |-  ( ( ( ( A  C_  C  /\  A  C_  D
)  /\  ( C  i^i  D )  C_  R
)  /\  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  /\  R  C_  D ) )  -> 
( A  C_  ( R  i^i  ( C  i^i  D ) )  /\  ( R  vH  ( C  i^i  D ) )  C_  ( A  vH  B ) ) )
7675an4s 835 . . . . . . . . 9  |-  ( ( ( ( A  C_  C  /\  A  C_  D
)  /\  ( C  C_  ( A  vH  B
)  /\  D  C_  ( A  vH  B ) ) )  /\  ( ( C  i^i  D ) 
C_  R  /\  R  C_  D ) )  -> 
( A  C_  ( R  i^i  ( C  i^i  D ) )  /\  ( R  vH  ( C  i^i  D ) )  C_  ( A  vH  B ) ) )
7739, 40, 6, 17mdslj1i 27972 . . . . . . . . 9  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( A  C_  ( R  i^i  ( C  i^i  D ) )  /\  ( R  vH  ( C  i^i  D ) )  C_  ( A  vH  B ) ) )  ->  ( ( R  vH  ( C  i^i  D ) )  i^i  B
)  =  ( ( R  i^i  B )  vH  ( ( C  i^i  D )  i^i 
B ) ) )
7876, 77sylan2 477 . . . . . . . 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  D )  C_  R  /\  R  C_  D
) ) )  -> 
( ( R  vH  ( C  i^i  D ) )  i^i  B )  =  ( ( R  i^i  B )  vH  ( ( C  i^i  D )  i^i  B ) ) )
7978anassrs 654 . . . . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  (
( R  vH  ( C  i^i  D ) )  i^i  B )  =  ( ( R  i^i  B )  vH  ( ( C  i^i  D )  i^i  B ) ) )
80 inindir 3650 . . . . . . . . 9  |-  ( ( C  i^i  D )  i^i  B )  =  ( ( C  i^i  B )  i^i  ( D  i^i  B ) )
8180a1i 11 . . . . . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  (
( C  i^i  D
)  i^i  B )  =  ( ( C  i^i  B )  i^i  ( D  i^i  B
) ) )
8281oveq2d 6306 . . . . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  (
( R  i^i  B
)  vH  ( ( C  i^i  D )  i^i 
B ) )  =  ( ( R  i^i  B )  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) )
8379, 82eqtr2d 2486 . . . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  (
( R  i^i  B
)  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B
) ) )  =  ( ( R  vH  ( C  i^i  D ) )  i^i  B ) )
8467, 83sseq12d 3461 . . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  (
( ( ( R  i^i  B )  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
C_  ( ( R  i^i  B )  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) )  <->  ( ( ( R  vH  C )  i^i  D )  i^i 
B )  C_  (
( R  vH  ( C  i^i  D ) )  i^i  B ) ) )
8551, 84bitr4d 260 . . . 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  D )  C_  R  /\  R  C_  D
) )  ->  (
( ( R  vH  C )  i^i  D
)  C_  ( R  vH  ( C  i^i  D
) )  <->  ( (
( R  i^i  B
)  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B
) )  C_  (
( R  i^i  B
)  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B
) ) ) ) )
8685exbiri 628 . . 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  D ) 
C_  R  /\  R  C_  D )  ->  (
( ( ( R  i^i  B )  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
C_  ( ( R  i^i  B )  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) )  ->  ( ( R  vH  C )  i^i 
D )  C_  ( R  vH  ( C  i^i  D ) ) ) ) )
8786a2d 29 . 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  D
)  C_  R  /\  R  C_  D )  -> 
( ( ( R  i^i  B )  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B ) ) 
C_  ( ( R  i^i  B )  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B ) ) ) )  ->  (
( ( C  i^i  D )  C_  R  /\  R  C_  D )  -> 
( ( R  vH  C )  i^i  D
)  C_  ( R  vH  ( C  i^i  D
) ) ) ) )
883, 87syl5 33 1  |-  ( ( ( A  MH  B  /\  B  MH*  A )  /\  ( ( A 
C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) ) ) )  ->  ( ( ( R  i^i  B ) 
C_  ( D  i^i  B )  ->  ( (
( R  i^i  B
)  vH  ( C  i^i  B ) )  i^i  ( D  i^i  B
) )  C_  (
( R  i^i  B
)  vH  ( ( C  i^i  B )  i^i  ( D  i^i  B
) ) ) )  ->  ( ( ( C  i^i  D ) 
C_  R  /\  R  C_  D )  ->  (
( R  vH  C
)  i^i  D )  C_  ( R  vH  ( C  i^i  D ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887    i^i cin 3403    C_ wss 3404   class class class wbr 4402  (class class class)co 6290   CHcch 26582    vH chj 26586    MH cmd 26619    MH* cdmd 26620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cc 8865  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619  ax-hilex 26652  ax-hfvadd 26653  ax-hvcom 26654  ax-hvass 26655  ax-hv0cl 26656  ax-hvaddid 26657  ax-hfvmul 26658  ax-hvmulid 26659  ax-hvmulass 26660  ax-hvdistr1 26661  ax-hvdistr2 26662  ax-hvmul0 26663  ax-hfi 26732  ax-his1 26735  ax-his2 26736  ax-his3 26737  ax-his4 26738  ax-hcompl 26855
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-2o 7183  df-oadd 7186  df-omul 7187  df-er 7363  df-map 7474  df-pm 7475  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-fi 7925  df-sup 7956  df-inf 7957  df-oi 8025  df-card 8373  df-acn 8376  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12028  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-rlim 13553  df-sum 13753  df-struct 15123  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-mulr 15204  df-starv 15205  df-sca 15206  df-vsca 15207  df-ip 15208  df-tset 15209  df-ple 15210  df-ds 15212  df-unif 15213  df-hom 15214  df-cco 15215  df-rest 15321  df-topn 15322  df-0g 15340  df-gsum 15341  df-topgen 15342  df-pt 15343  df-prds 15346  df-xrs 15400  df-qtop 15406  df-imas 15407  df-xps 15410  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-mulg 16676  df-cntz 16971  df-cmn 17432  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-fbas 18967  df-fg 18968  df-cnfld 18971  df-top 19921  df-bases 19922  df-topon 19923  df-topsp 19924  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-cn 20243  df-cnp 20244  df-lm 20245  df-haus 20331  df-tx 20577  df-hmeo 20770  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-xms 21335  df-ms 21336  df-tms 21337  df-cfil 22225  df-cau 22226  df-cmet 22227  df-grpo 25919  df-gid 25920  df-ginv 25921  df-gdiv 25922  df-ablo 26010  df-subgo 26030  df-vc 26165  df-nv 26211  df-va 26214  df-ba 26215  df-sm 26216  df-0v 26217  df-vs 26218  df-nmcv 26219  df-ims 26220  df-dip 26337  df-ssp 26361  df-ph 26454  df-cbn 26505  df-hnorm 26621  df-hba 26622  df-hvsub 26624  df-hlim 26625  df-hcau 26626  df-sh 26860  df-ch 26874  df-oc 26905  df-ch0 26906  df-shs 26961  df-chj 26963  df-md 27933  df-dmd 27934
This theorem is referenced by:  mdslmd1lem4  27981
  Copyright terms: Public domain W3C validator