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Theorem mdslmd1lem2 23782
Description: Lemma for mdslmd1i 23785. (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 3526 . . . 4  |-  ( R 
C_  D  ->  ( R  i^i  B )  C_  ( D  i^i  B ) )
21adantl 453 . . 3  |-  ( ( ( C  i^i  D
)  C_  R  /\  R  C_  D )  -> 
( R  i^i  B
)  C_  ( D  i^i  B ) )
32imim1i 56 . 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 736 . . . . . 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 22925 . . . . . . . . . . 11  |-  C  C_  ( R  vH  C )
8 sstr 3316 . . . . . . . . . . 11  |-  ( ( A  C_  C  /\  C  C_  ( R  vH  C ) )  ->  A  C_  ( R  vH  C ) )
97, 8mpan2 653 . . . . . . . . . 10  |-  ( A 
C_  C  ->  A  C_  ( R  vH  C
) )
109ad2antrr 707 . . . . . . . . 9  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) )  ->  A  C_  ( R  vH  C ) )
1110ad2antlr 708 . . . . . . . 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 732 . . . . . . . . 9  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) )  ->  A  C_  D
)
1312ad2antlr 708 . . . . . . . 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 3525 . . . . . . 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 3523 . . . . . . . . . 10  |-  ( ( A  C_  C  /\  A  C_  D )  <->  A  C_  ( C  i^i  D ) )
16 mdslmd.4 . . . . . . . . . . . . 13  |-  D  e. 
CH
175, 16chincli 22915 . . . . . . . . . . . 12  |-  ( C  i^i  D )  e. 
CH
1817, 6chub2i 22925 . . . . . . . . . . 11  |-  ( C  i^i  D )  C_  ( R  vH  ( C  i^i  D ) )
19 sstr 3316 . . . . . . . . . . 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 653 . . . . . . . . . 10  |-  ( A 
C_  ( C  i^i  D )  ->  A  C_  ( R  vH  ( C  i^i  D ) ) )
2115, 20sylbi 188 . . . . . . . . 9  |-  ( ( A  C_  C  /\  A  C_  D )  ->  A  C_  ( R  vH  ( C  i^i  D ) ) )
2221adantr 452 . . . . . . . 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 708 . . . . . . 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 3525 . . . . . 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 3522 . . . . . . . . . . 11  |-  ( ( R  vH  C )  i^i  D )  C_  D
26 sstr 3316 . . . . . . . . . . 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 652 . . . . . . . . . 10  |-  ( D 
C_  ( A  vH  B )  ->  (
( R  vH  C
)  i^i  D )  C_  ( A  vH  B
) )
2827ad2antll 710 . . . . . . . . 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 708 . . . . . . . 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 3316 . . . . . . . . . . . . . 14  |-  ( ( R  C_  D  /\  D  C_  ( A  vH  B ) )  ->  R  C_  ( A  vH  B ) )
3130ancoms 440 . . . . . . . . . . . . 13  |-  ( ( D  C_  ( A  vH  B )  /\  R  C_  D )  ->  R  C_  ( A  vH  B
) )
3231ad2ant2l 727 . . . . . . . . . . . 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 695 . . . . . . . . . . 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 695 . . . . . . . . . 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 3529 . . . . . . . . . . . 12  |-  ( C 
C_  ( A  vH  B )  ->  ( C  i^i  D )  C_  ( A  vH  B ) )
3635ad2antrl 709 . . . . . . . . . . 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 708 . . . . . . . . . 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 519 . . . . . . . . 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 22912 . . . . . . . . . 10  |-  ( A  vH  B )  e. 
CH
426, 17, 41chlubi 22926 . . . . . . . . 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 189 . . . . . . . 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 519 . . . . . . 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 22912 . . . . . . . . 9  |-  ( R  vH  C )  e. 
CH
4645, 16chincli 22915 . . . . . . . 8  |-  ( ( R  vH  C )  i^i  D )  e. 
CH
476, 17chjcli 22912 . . . . . . . 8  |-  ( R  vH  ( C  i^i  D ) )  e.  CH
4846, 47, 41chlubi 22926 . . . . . . 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 189 . . . . . 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 23773 . . . . . 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 1184 . . . . 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 3519 . . . . . . 7  |-  ( ( ( R  vH  C
)  i^i  D )  i^i  B )  =  ( ( ( R  vH  C )  i^i  B
)  i^i  ( D  i^i  B ) )
53 sstr 3316 . . . . . . . . . . . . . 14  |-  ( ( A  C_  ( C  i^i  D )  /\  ( C  i^i  D )  C_  R )  ->  A  C_  R )
5415, 53sylanb 459 . . . . . . . . . . . . 13  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  i^i  D )  C_  R )  ->  A  C_  R )
5554ad2ant2r 728 . . . . . . . . . . . 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 735 . . . . . . . . . . . 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 3525 . . . . . . . . . . 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 737 . . . . . . . . . . . . 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 519 . . . . . . . . . . . 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 22926 . . . . . . . . . . . 12  |-  ( ( R  C_  ( A  vH  B )  /\  C  C_  ( A  vH  B
) )  <->  ( R  vH  C )  C_  ( A  vH  B ) )
6159, 60sylib 189 . . . . . . . . . . 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 519 . . . . . . . . . 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 23775 . . . . . . . . . 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 461 . . . . . . . . 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 630 . . . . . . . 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 3501 . . . . . . 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 2449 . . . . . 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 187 . . . . . . . . . . . . 13  |-  ( ( A  C_  C  /\  A  C_  D )  ->  A  C_  ( C  i^i  D ) )
6968adantr 452 . . . . . . . . . . . 12  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  i^i  D )  C_  R )  ->  A  C_  ( C  i^i  D ) )
7054, 69ssind 3525 . . . . . . . . . . 11  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  i^i  D )  C_  R )  ->  A  C_  ( R  i^i  ( C  i^i  D
) ) )
7131adantll 695 . . . . . . . . . . . . 13  |-  ( ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) )  /\  R  C_  D )  ->  R  C_  ( A  vH  B ) )
7235ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) )  /\  R  C_  D )  -> 
( C  i^i  D
)  C_  ( A  vH  B ) )
7371, 72jca 519 . . . . . . . . . . . 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 189 . . . . . . . . . . 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 550 . . . . . . . . . 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 800 . . . . . . . . 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 23775 . . . . . . . . 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 461 . . . . . . . 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 630 . . . . . . 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 3519 . . . . . . . . 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 6056 . . . . . . 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 2437 . . . . . 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 3337 . . . . 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 248 . . . 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 606 . . 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 24 . 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 30 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    i^i cin 3279    C_ wss 3280   class class class wbr 4172  (class class class)co 6040   CHcch 22385    vH chj 22389    MH cmd 22422    MH* cdmd 22423
This theorem is referenced by:  mdslmd1lem4  23784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cc 8271  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026  ax-hilex 22455  ax-hfvadd 22456  ax-hvcom 22457  ax-hvass 22458  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  ax-hvmulass 22463  ax-hvdistr1 22464  ax-hvdistr2 22465  ax-hvmul0 22466  ax-hfi 22534  ax-his1 22537  ax-his2 22538  ax-his3 22539  ax-his4 22540  ax-hcompl 22657
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-acn 7785  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-cn 17245  df-cnp 17246  df-lm 17247  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cfil 19161  df-cau 19162  df-cmet 19163  df-grpo 21732  df-gid 21733  df-ginv 21734  df-gdiv 21735  df-ablo 21823  df-subgo 21843  df-vc 21978  df-nv 22024  df-va 22027  df-ba 22028  df-sm 22029  df-0v 22030  df-vs 22031  df-nmcv 22032  df-ims 22033  df-dip 22150  df-ssp 22174  df-ph 22267  df-cbn 22318  df-hnorm 22424  df-hba 22425  df-hvsub 22427  df-hlim 22428  df-hcau 22429  df-sh 22662  df-ch 22677  df-oc 22707  df-ch0 22708  df-shs 22763  df-chj 22765  df-md 23736  df-dmd 23737
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