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Theorem mdslmd1lem2 23678
Description: Lemma for mdslmd1i 23681. (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 3510 . . . 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 22821 . . . . . . . . . . 11  |-  C  C_  ( R  vH  C )
8 sstr 3300 . . . . . . . . . . 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 3509 . . . . . . 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 3507 . . . . . . . . . 10  |-  ( ( A  C_  C  /\  A  C_  D )  <->  A  C_  ( C  i^i  D ) )
16 mdslmd.4 . . . . . . . . . . . . 13  |-  D  e. 
CH
175, 16chincli 22811 . . . . . . . . . . . 12  |-  ( C  i^i  D )  e. 
CH
1817, 6chub2i 22821 . . . . . . . . . . 11  |-  ( C  i^i  D )  C_  ( R  vH  ( C  i^i  D ) )
19 sstr 3300 . . . . . . . . . . 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 3509 . . . . . 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 3506 . . . . . . . . . . 11  |-  ( ( R  vH  C )  i^i  D )  C_  D
26 sstr 3300 . . . . . . . . . . 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 3300 . . . . . . . . . . . . . 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 3513 . . . . . . . . . . . 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 22808 . . . . . . . . . 10  |-  ( A  vH  B )  e. 
CH
426, 17, 41chlubi 22822 . . . . . . . . 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 22808 . . . . . . . . 9  |-  ( R  vH  C )  e. 
CH
4645, 16chincli 22811 . . . . . . . 8  |-  ( ( R  vH  C )  i^i  D )  e. 
CH
476, 17chjcli 22808 . . . . . . . 8  |-  ( R  vH  ( C  i^i  D ) )  e.  CH
4846, 47, 41chlubi 22822 . . . . . . 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 23669 . . . . . 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 3503 . . . . . . 7  |-  ( ( ( R  vH  C
)  i^i  D )  i^i  B )  =  ( ( ( R  vH  C )  i^i  B
)  i^i  ( D  i^i  B ) )
53 sstr 3300 . . . . . . . . . . . . . 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 3509 . . . . . . . . . . 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 22822 . . . . . . . . . . . 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 23671 . . . . . . . . . 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 3485 . . . . . . 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 2433 . . . . . 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 3509 . . . . . . . . . . 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 23671 . . . . . . . . 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 3503 . . . . . . . . 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 6037 . . . . . . 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 2421 . . . . . 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 3321 . . . . 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 1717    i^i cin 3263    C_ wss 3264   class class class wbr 4154  (class class class)co 6021   CHcch 22281    vH chj 22285    MH cmd 22318    MH* cdmd 22319
This theorem is referenced by:  mdslmd1lem4  23680
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cc 8249  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003  ax-mulf 9004  ax-hilex 22351  ax-hfvadd 22352  ax-hvcom 22353  ax-hvass 22354  ax-hv0cl 22355  ax-hvaddid 22356  ax-hfvmul 22357  ax-hvmulid 22358  ax-hvmulass 22359  ax-hvdistr1 22360  ax-hvdistr2 22361  ax-hvmul0 22362  ax-hfi 22430  ax-his1 22433  ax-his2 22434  ax-his3 22435  ax-his4 22436  ax-hcompl 22553
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-omul 6666  df-er 6842  df-map 6957  df-pm 6958  df-ixp 7001  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-sup 7382  df-oi 7413  df-card 7760  df-acn 7763  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-ioo 10853  df-ico 10855  df-icc 10856  df-fz 10977  df-fzo 11067  df-fl 11130  df-seq 11252  df-exp 11311  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-clim 12210  df-rlim 12211  df-sum 12408  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-starv 13472  df-sca 13473  df-vsca 13474  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-hom 13481  df-cco 13482  df-rest 13578  df-topn 13579  df-topgen 13595  df-pt 13596  df-prds 13599  df-xrs 13654  df-0g 13655  df-gsum 13656  df-qtop 13661  df-imas 13662  df-xps 13664  df-mre 13739  df-mrc 13740  df-acs 13742  df-mnd 14618  df-submnd 14667  df-mulg 14743  df-cntz 15044  df-cmn 15342  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-fbas 16624  df-fg 16625  df-cnfld 16628  df-top 16887  df-bases 16889  df-topon 16890  df-topsp 16891  df-cld 17007  df-ntr 17008  df-cls 17009  df-nei 17086  df-cn 17214  df-cnp 17215  df-lm 17216  df-haus 17302  df-tx 17516  df-hmeo 17709  df-fil 17800  df-fm 17892  df-flim 17893  df-flf 17894  df-xms 18260  df-ms 18261  df-tms 18262  df-cfil 19080  df-cau 19081  df-cmet 19082  df-grpo 21628  df-gid 21629  df-ginv 21630  df-gdiv 21631  df-ablo 21719  df-subgo 21739  df-vc 21874  df-nv 21920  df-va 21923  df-ba 21924  df-sm 21925  df-0v 21926  df-vs 21927  df-nmcv 21928  df-ims 21929  df-dip 22046  df-ssp 22070  df-ph 22163  df-cbn 22214  df-hnorm 22320  df-hba 22321  df-hvsub 22323  df-hlim 22324  df-hcau 22325  df-sh 22558  df-ch 22573  df-oc 22603  df-ch0 22604  df-shs 22659  df-chj 22661  df-md 23632  df-dmd 23633
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