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Theorem mdslmd1lem2 27110
Description: Lemma for mdslmd1i 27113. (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 3705 . . . 4  |-  ( R 
C_  D  ->  ( R  i^i  B )  C_  ( D  i^i  B ) )
21adantl 466 . . 3  |-  ( ( ( C  i^i  D
)  C_  R  /\  R  C_  D )  -> 
( R  i^i  B
)  C_  ( D  i^i  B ) )
32imim1i 58 . 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 758 . . . . . 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 26253 . . . . . . . . . . 11  |-  C  C_  ( R  vH  C )
8 sstr 3494 . . . . . . . . . . 11  |-  ( ( A  C_  C  /\  C  C_  ( R  vH  C ) )  ->  A  C_  ( R  vH  C ) )
97, 8mpan2 671 . . . . . . . . . 10  |-  ( A 
C_  C  ->  A  C_  ( R  vH  C
) )
109ad2antrr 725 . . . . . . . . 9  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) )  ->  A  C_  ( R  vH  C ) )
1110ad2antlr 726 . . . . . . . 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 754 . . . . . . . . 9  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) )  ->  A  C_  D
)
1312ad2antlr 726 . . . . . . . 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 3704 . . . . . . 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 3702 . . . . . . . . . 10  |-  ( ( A  C_  C  /\  A  C_  D )  <->  A  C_  ( C  i^i  D ) )
16 mdslmd.4 . . . . . . . . . . . . 13  |-  D  e. 
CH
175, 16chincli 26243 . . . . . . . . . . . 12  |-  ( C  i^i  D )  e. 
CH
1817, 6chub2i 26253 . . . . . . . . . . 11  |-  ( C  i^i  D )  C_  ( R  vH  ( C  i^i  D ) )
19 sstr 3494 . . . . . . . . . . 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 671 . . . . . . . . . 10  |-  ( A 
C_  ( C  i^i  D )  ->  A  C_  ( R  vH  ( C  i^i  D ) ) )
2115, 20sylbi 195 . . . . . . . . 9  |-  ( ( A  C_  C  /\  A  C_  D )  ->  A  C_  ( R  vH  ( C  i^i  D ) ) )
2221adantr 465 . . . . . . . 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 726 . . . . . . 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 3704 . . . . . 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 3701 . . . . . . . . . . 11  |-  ( ( R  vH  C )  i^i  D )  C_  D
26 sstr 3494 . . . . . . . . . . 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 670 . . . . . . . . . 10  |-  ( D 
C_  ( A  vH  B )  ->  (
( R  vH  C
)  i^i  D )  C_  ( A  vH  B
) )
2827ad2antll 728 . . . . . . . . 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 726 . . . . . . . 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 3494 . . . . . . . . . . . . . 14  |-  ( ( R  C_  D  /\  D  C_  ( A  vH  B ) )  ->  R  C_  ( A  vH  B ) )
3130ancoms 453 . . . . . . . . . . . . 13  |-  ( ( D  C_  ( A  vH  B )  /\  R  C_  D )  ->  R  C_  ( A  vH  B
) )
3231ad2ant2l 745 . . . . . . . . . . . 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 713 . . . . . . . . . . 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 713 . . . . . . . . . 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 3708 . . . . . . . . . . . 12  |-  ( C 
C_  ( A  vH  B )  ->  ( C  i^i  D )  C_  ( A  vH  B ) )
3635ad2antrl 727 . . . . . . . . . . 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 726 . . . . . . . . . 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 532 . . . . . . . . 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 26240 . . . . . . . . . 10  |-  ( A  vH  B )  e. 
CH
426, 17, 41chlubi 26254 . . . . . . . . 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 196 . . . . . . . 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 532 . . . . . . 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 26240 . . . . . . . . 9  |-  ( R  vH  C )  e. 
CH
4645, 16chincli 26243 . . . . . . . 8  |-  ( ( R  vH  C )  i^i  D )  e. 
CH
476, 17chjcli 26240 . . . . . . . 8  |-  ( R  vH  ( C  i^i  D ) )  e.  CH
4846, 47, 41chlubi 26254 . . . . . . 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 196 . . . . . 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 27101 . . . . . 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 1227 . . . . 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 3698 . . . . . . 7  |-  ( ( ( R  vH  C
)  i^i  D )  i^i  B )  =  ( ( ( R  vH  C )  i^i  B
)  i^i  ( D  i^i  B ) )
53 sstr 3494 . . . . . . . . . . . . . 14  |-  ( ( A  C_  ( C  i^i  D )  /\  ( C  i^i  D )  C_  R )  ->  A  C_  R )
5415, 53sylanb 472 . . . . . . . . . . . . 13  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  i^i  D )  C_  R )  ->  A  C_  R )
5554ad2ant2r 746 . . . . . . . . . . . 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 757 . . . . . . . . . . . 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 3704 . . . . . . . . . . 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 759 . . . . . . . . . . . . 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 532 . . . . . . . . . . . 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 26254 . . . . . . . . . . . 12  |-  ( ( R  C_  ( A  vH  B )  /\  C  C_  ( A  vH  B
) )  <->  ( R  vH  C )  C_  ( A  vH  B ) )
6159, 60sylib 196 . . . . . . . . . . 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 532 . . . . . . . . . 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 27103 . . . . . . . . . 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 474 . . . . . . . . 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 648 . . . . . . . 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 3681 . . . . . . 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 2495 . . . . . 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 194 . . . . . . . . . . . . 13  |-  ( ( A  C_  C  /\  A  C_  D )  ->  A  C_  ( C  i^i  D ) )
6968adantr 465 . . . . . . . . . . . 12  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  i^i  D )  C_  R )  ->  A  C_  ( C  i^i  D ) )
7054, 69ssind 3704 . . . . . . . . . . 11  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  i^i  D )  C_  R )  ->  A  C_  ( R  i^i  ( C  i^i  D
) ) )
7131adantll 713 . . . . . . . . . . . . 13  |-  ( ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) )  /\  R  C_  D )  ->  R  C_  ( A  vH  B ) )
7235ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) )  /\  R  C_  D )  -> 
( C  i^i  D
)  C_  ( A  vH  B ) )
7371, 72jca 532 . . . . . . . . . . . 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 196 . . . . . . . . . . 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 566 . . . . . . . . . 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 824 . . . . . . . . 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 27103 . . . . . . . . 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 474 . . . . . . . 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 648 . . . . . . 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 3698 . . . . . . . . 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 6293 . . . . . . 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 2483 . . . . . 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 3515 . . . . 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 256 . . . 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 622 . . 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 26 . 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 32 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 184    /\ wa 369    = wceq 1381    e. wcel 1802    i^i cin 3457    C_ wss 3458   class class class wbr 4433  (class class class)co 6277   CHcch 25711    vH chj 25715    MH cmd 25748    MH* cdmd 25749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cc 8813  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-addf 9569  ax-mulf 9570  ax-hilex 25781  ax-hfvadd 25782  ax-hvcom 25783  ax-hvass 25784  ax-hv0cl 25785  ax-hvaddid 25786  ax-hfvmul 25787  ax-hvmulid 25788  ax-hvmulass 25789  ax-hvdistr1 25790  ax-hvdistr2 25791  ax-hvmul0 25792  ax-hfi 25861  ax-his1 25864  ax-his2 25865  ax-his3 25866  ax-his4 25867  ax-hcompl 25984
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6521  df-om 6682  df-1st 6781  df-2nd 6782  df-supp 6900  df-recs 7040  df-rdg 7074  df-1o 7128  df-2o 7129  df-oadd 7132  df-omul 7133  df-er 7309  df-map 7420  df-pm 7421  df-ixp 7468  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-fsupp 7828  df-fi 7869  df-sup 7899  df-oi 7933  df-card 8318  df-acn 8321  df-cda 8546  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-q 11187  df-rp 11225  df-xneg 11322  df-xadd 11323  df-xmul 11324  df-ioo 11537  df-ico 11539  df-icc 11540  df-fz 11677  df-fzo 11799  df-fl 11903  df-seq 12082  df-exp 12141  df-hash 12380  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-clim 13285  df-rlim 13286  df-sum 13483  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-starv 14584  df-sca 14585  df-vsca 14586  df-ip 14587  df-tset 14588  df-ple 14589  df-ds 14591  df-unif 14592  df-hom 14593  df-cco 14594  df-rest 14692  df-topn 14693  df-0g 14711  df-gsum 14712  df-topgen 14713  df-pt 14714  df-prds 14717  df-xrs 14771  df-qtop 14776  df-imas 14777  df-xps 14779  df-mre 14855  df-mrc 14856  df-acs 14858  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-submnd 15836  df-mulg 15929  df-cntz 16224  df-cmn 16669  df-psmet 18279  df-xmet 18280  df-met 18281  df-bl 18282  df-mopn 18283  df-fbas 18284  df-fg 18285  df-cnfld 18289  df-top 19266  df-bases 19268  df-topon 19269  df-topsp 19270  df-cld 19386  df-ntr 19387  df-cls 19388  df-nei 19465  df-cn 19594  df-cnp 19595  df-lm 19596  df-haus 19682  df-tx 19929  df-hmeo 20122  df-fil 20213  df-fm 20305  df-flim 20306  df-flf 20307  df-xms 20689  df-ms 20690  df-tms 20691  df-cfil 21560  df-cau 21561  df-cmet 21562  df-grpo 25058  df-gid 25059  df-ginv 25060  df-gdiv 25061  df-ablo 25149  df-subgo 25169  df-vc 25304  df-nv 25350  df-va 25353  df-ba 25354  df-sm 25355  df-0v 25356  df-vs 25357  df-nmcv 25358  df-ims 25359  df-dip 25476  df-ssp 25500  df-ph 25593  df-cbn 25644  df-hnorm 25750  df-hba 25751  df-hvsub 25753  df-hlim 25754  df-hcau 25755  df-sh 25989  df-ch 26004  df-oc 26035  df-ch0 26036  df-shs 26091  df-chj 26093  df-md 27064  df-dmd 27065
This theorem is referenced by:  mdslmd1lem4  27112
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