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Theorem mdslmd1lem1 25697
Description: Lemma for mdslmd1i 25701. (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 24831 . . . . . 6  |-  ( D  i^i  B )  e. 
CH
5 mdslmd.1 . . . . . 6  |-  A  e. 
CH
61, 4, 5chlej1i 24844 . . . . 5  |-  ( R 
C_  ( D  i^i  B )  ->  ( R  vH  A )  C_  (
( D  i^i  B
)  vH  A )
)
7 simpr 461 . . . . . . . 8  |-  ( ( A  MH  B  /\  B  MH*  A )  ->  B  MH*  A )
8 simpr 461 . . . . . . . 8  |-  ( ( A  C_  C  /\  A  C_  D )  ->  A  C_  D )
9 simpr 461 . . . . . . . 8  |-  ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B
) )  ->  D  C_  ( A  vH  B
) )
105, 3, 23pm3.2i 1166 . . . . . . . . 9  |-  ( A  e.  CH  /\  B  e.  CH  /\  D  e. 
CH )
11 dmdsl3 25687 . . . . . . . . 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 670 . . . . . . . 8  |-  ( ( B  MH*  A  /\  A  C_  D  /\  D  C_  ( A  vH  B
) )  ->  (
( D  i^i  B
)  vH  A )  =  D )
137, 8, 9, 12syl3an 1260 . . . . . . 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 1188 . . . . . 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 3379 . . . . 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 219 . . . 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 467 . . 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 75 . 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 753 . . . . . . . . . 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 753 . . . . . . . . . . . . 13  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) )  ->  A  C_  C
)
2120ad2antlr 726 . . . . . . . . . . . 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 24841 . . . . . . . . . . . 12  |-  A  C_  ( R  vH  A )
2321, 22jctil 537 . . . . . . . . . . 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 3567 . . . . . . . . . . 11  |-  ( ( A  C_  ( R  vH  A )  /\  A  C_  C )  <->  A  C_  (
( R  vH  A
)  i^i  C )
)
2523, 24sylib 196 . . . . . . . . . 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 3565 . . . . . . . . . . . . . . . . . . . 20  |-  ( D  i^i  B )  C_  D
27 sstr 3359 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( R  C_  ( D  i^i  B )  /\  ( D  i^i  B )  C_  D )  ->  R  C_  D )
2826, 27mpan2 671 . . . . . . . . . . . . . . . . . . 19  |-  ( R 
C_  ( D  i^i  B )  ->  R  C_  D
)
29 sstr 3359 . . . . . . . . . . . . . . . . . . 19  |-  ( ( R  C_  D  /\  D  C_  ( A  vH  B ) )  ->  R  C_  ( A  vH  B ) )
3028, 29sylan 471 . . . . . . . . . . . . . . . . . 18  |-  ( ( R  C_  ( D  i^i  B )  /\  D  C_  ( A  vH  B
) )  ->  R  C_  ( A  vH  B
) )
3130ancoms 453 . . . . . . . . . . . . . . . . 17  |-  ( ( D  C_  ( A  vH  B )  /\  R  C_  ( D  i^i  B
) )  ->  R  C_  ( A  vH  B
) )
3231adantll 713 . . . . . . . . . . . . . . . 16  |-  ( ( ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) )  /\  R  C_  ( D  i^i  B ) )  ->  R  C_  ( A  vH  B
) )
3332adantll 713 . . . . . . . . . . . . . . 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 745 . . . . . . . . . . . . . 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 24840 . . . . . . . . . . . . . 14  |-  A  C_  ( A  vH  B )
3634, 35jctir 538 . . . . . . . . . . . . 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 24828 . . . . . . . . . . . . . 14  |-  ( A  vH  B )  e. 
CH
381, 5, 37chlubi 24842 . . . . . . . . . . . . 13  |-  ( ( R  C_  ( A  vH  B )  /\  A  C_  ( A  vH  B
) )  <->  ( R  vH  A )  C_  ( A  vH  B ) )
3936, 38sylib 196 . . . . . . . . . . . 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 763 . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 532 . . . . . . . . . . 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 24828 . . . . . . . . . . . 12  |-  ( R  vH  A )  e. 
CH
44 mdslmd.3 . . . . . . . . . . . 12  |-  C  e. 
CH
4543, 44, 37chlubi 24842 . . . . . . . . . . 11  |-  ( ( ( R  vH  A
)  C_  ( A  vH  B )  /\  C  C_  ( A  vH  B
) )  <->  ( ( R  vH  A )  vH  C )  C_  ( A  vH  B ) )
4642, 45sylib 196 . . . . . . . . . 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 25691 . . . . . . . . . 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 1216 . . . . . . . . 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 757 . . . . . . . . . . 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 759 . . . . . . . . . . . . . . 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 3567 . . . . . . . . . . . . . . 15  |-  ( ( A  C_  C  /\  A  C_  D )  <->  A  C_  ( C  i^i  D ) )
5250, 51sylib 196 . . . . . . . . . . . . . 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 3570 . . . . . . . . . . . . . 14  |-  ( A 
C_  ( C  i^i  D )  ->  ( A  i^i  B )  C_  (
( C  i^i  D
)  i^i  B )
)
5452, 53syl 16 . . . . . . . . . . . . 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 3563 . . . . . . . . . . . . 13  |-  ( ( C  i^i  D )  i^i  B )  =  ( ( C  i^i  B )  i^i  ( D  i^i  B ) )
5654, 55syl6sseq 3397 . . . . . . . . . . . 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 755 . . . . . . . . . . . 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 3361 . . . . . . . . . . 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 3566 . . . . . . . . . . . . 13  |-  ( D  i^i  B )  C_  B
60 sstr 3359 . . . . . . . . . . . . 13  |-  ( ( R  C_  ( D  i^i  B )  /\  ( D  i^i  B )  C_  B )  ->  R  C_  B )
6159, 60mpan2 671 . . . . . . . . . . . 12  |-  ( R 
C_  ( D  i^i  B )  ->  R  C_  B
)
6261ad2antll 728 . . . . . . . . . . 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 1166 . . . . . . . . . . . 12  |-  ( A  e.  CH  /\  B  e.  CH  /\  R  e. 
CH )
64 mdsl3 25688 . . . . . . . . . . . 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 670 . . . . . . . . . . 11  |-  ( ( A  MH  B  /\  ( A  i^i  B ) 
C_  R  /\  R  C_  B )  ->  (
( R  vH  A
)  i^i  B )  =  R )
6649, 58, 62, 65syl3anc 1218 . . . . . . . . . 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 6101 . . . . . . . . 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 2471 . . . . . . . 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 3546 . . . . . . 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 3563 . . . . . . 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 2488 . . . . . 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 537 . . . . . . . . 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 3567 . . . . . . . . 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 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  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 3573 . . . . . . . . . . . 12  |-  ( C 
C_  ( A  vH  B )  ->  ( C  i^i  D )  C_  ( A  vH  B ) )
7675ad2antrl 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 ) )
7776ad2antlr 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  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 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  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 24831 . . . . . . . . . 10  |-  ( C  i^i  D )  e. 
CH
8043, 79, 37chlubi 24842 . . . . . . . . 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 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  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 25691 . . . . . . . 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 1216 . . . . . . 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 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  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 6104 . . . . . . 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 2471 . . . . . 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 3380 . . . . 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 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  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  B  MH*  A )
89 simplr 754 . . . . . . . . . . 11  |-  ( ( ( A  C_  C  /\  A  C_  D )  /\  ( C  C_  ( A  vH  B )  /\  D  C_  ( A  vH  B ) ) )  ->  A  C_  D
)
9089ad2antlr 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  B )  i^i  ( D  i^i  B ) )  C_  R  /\  R  C_  ( D  i^i  B ) ) )  ->  A  C_  D
)
9143, 44chub1i 24840 . . . . . . . . . . 11  |-  ( R  vH  A )  C_  ( ( R  vH  A )  vH  C
)
9222, 91sstri 3360 . . . . . . . . . 10  |-  A  C_  ( ( R  vH  A )  vH  C
)
9390, 92jctil 537 . . . . . . . . 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 3567 . . . . . . . . 9  |-  ( ( A  C_  ( ( R  vH  A )  vH  C )  /\  A  C_  D )  <->  A  C_  (
( ( R  vH  A )  vH  C
)  i^i  D )
)
9593, 94sylib 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  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 24840 . . . . . . . . 9  |-  ( R  vH  A )  C_  ( ( R  vH  A )  vH  ( C  i^i  D ) )
9722, 96sstri 3360 . . . . . . . 8  |-  A  C_  ( ( R  vH  A )  vH  ( C  i^i  D ) )
9895, 97jctir 538 . . . . . . 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 3567 . . . . . . 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 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  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 3566 . . . . . . . . . . 11  |-  ( ( ( R  vH  A
)  vH  C )  i^i  D )  C_  D
102 sstr 3359 . . . . . . . . . . 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 670 . . . . . . . . . 10  |-  ( D 
C_  ( A  vH  B )  ->  (
( ( R  vH  A )  vH  C
)  i^i  D )  C_  ( A  vH  B
) )
104103ad2antll 728 . . . . . . . . 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 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  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 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  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 24828 . . . . . . . . 9  |-  ( ( R  vH  A )  vH  C )  e. 
CH
108107, 2chincli 24831 . . . . . . . 8  |-  ( ( ( R  vH  A
)  vH  C )  i^i  D )  e.  CH
10943, 79chjcli 24828 . . . . . . . 8  |-  ( ( R  vH  A )  vH  ( C  i^i  D ) )  e.  CH
110108, 109, 37chlubi 24842 . . . . . . 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 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  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 25689 . . . . . 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 1218 . . . . 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 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  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 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  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 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  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 44 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    i^i cin 3322    C_ wss 3323   class class class wbr 4287  (class class class)co 6086   CHcch 24299    vH chj 24303    MH cmd 24336    MH* cdmd 24337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cc 8596  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354  ax-hilex 24369  ax-hfvadd 24370  ax-hvcom 24371  ax-hvass 24372  ax-hv0cl 24373  ax-hvaddid 24374  ax-hfvmul 24375  ax-hvmulid 24376  ax-hvmulass 24377  ax-hvdistr1 24378  ax-hvdistr2 24379  ax-hvmul0 24380  ax-hfi 24449  ax-his1 24452  ax-his2 24453  ax-his3 24454  ax-his4 24455  ax-hcompl 24572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  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-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-omul 6917  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-acn 8104  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959  df-sum 13156  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-fbas 17794  df-fg 17795  df-cnfld 17799  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-cld 18603  df-ntr 18604  df-cls 18605  df-nei 18682  df-cn 18811  df-cnp 18812  df-lm 18813  df-haus 18899  df-tx 19115  df-hmeo 19308  df-fil 19399  df-fm 19491  df-flim 19492  df-flf 19493  df-xms 19875  df-ms 19876  df-tms 19877  df-cfil 20746  df-cau 20747  df-cmet 20748  df-grpo 23646  df-gid 23647  df-ginv 23648  df-gdiv 23649  df-ablo 23737  df-subgo 23757  df-vc 23892  df-nv 23938  df-va 23941  df-ba 23942  df-sm 23943  df-0v 23944  df-vs 23945  df-nmcv 23946  df-ims 23947  df-dip 24064  df-ssp 24088  df-ph 24181  df-cbn 24232  df-hnorm 24338  df-hba 24339  df-hvsub 24341  df-hlim 24342  df-hcau 24343  df-sh 24577  df-ch 24592  df-oc 24623  df-ch0 24624  df-shs 24679  df-chj 24681  df-md 25652  df-dmd 25653
This theorem is referenced by:  mdslmd1lem3  25699
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