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Theorem mdslmd1lem1 26948
Description: Lemma for mdslmd1i 26952. (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 26082 . . . . . 6  |-  ( D  i^i  B )  e. 
CH
5 mdslmd.1 . . . . . 6  |-  A  e. 
CH
61, 4, 5chlej1i 26095 . . . . 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 1174 . . . . . . . . 9  |-  ( A  e.  CH  /\  B  e.  CH  /\  D  e. 
CH )
11 dmdsl3 26938 . . . . . . . . 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 1270 . . . . . . 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 1197 . . . . . 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 3532 . . . . 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 26092 . . . . . . . . . . . 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 3720 . . . . . . . . . . 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 3718 . . . . . . . . . . . . . . . . . . . 20  |-  ( D  i^i  B )  C_  D
27 sstr 3512 . . . . . . . . . . . . . . . . . . . 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 3512 . . . . . . . . . . . . . . . . . . 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 26091 . . . . . . . . . . . . . 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 26079 . . . . . . . . . . . . . 14  |-  ( A  vH  B )  e. 
CH
381, 5, 37chlubi 26093 . . . . . . . . . . . . 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 26079 . . . . . . . . . . . 12  |-  ( R  vH  A )  e. 
CH
44 mdslmd.3 . . . . . . . . . . . 12  |-  C  e. 
CH
4543, 44, 37chlubi 26093 . . . . . . . . . . 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 26942 . . . . . . . . . 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 1226 . . . . . . . . 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 3720 . . . . . . . . . . . . . . 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 3723 . . . . . . . . . . . . . 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 3716 . . . . . . . . . . . . 13  |-  ( ( C  i^i  D )  i^i  B )  =  ( ( C  i^i  B )  i^i  ( D  i^i  B ) )
5654, 55syl6sseq 3550 . . . . . . . . . . . 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 3514 . . . . . . . . . . 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 3719 . . . . . . . . . . . . 13  |-  ( D  i^i  B )  C_  B
60 sstr 3512 . . . . . . . . . . . . 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 1174 . . . . . . . . . . . 12  |-  ( A  e.  CH  /\  B  e.  CH  /\  R  e. 
CH )
64 mdsl3 26939 . . . . . . . . . . . 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 1228 . . . . . . . . . 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 6299 . . . . . . . . 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 2509 . . . . . . . 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 3699 . . . . . . 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 3716 . . . . . . 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 2526 . . . . . 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 3720 . . . . . . . . 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 3726 . . . . . . . . . . . 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 26082 . . . . . . . . . 10  |-  ( C  i^i  D )  e. 
CH
8043, 79, 37chlubi 26093 . . . . . . . . 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 26942 . . . . . . . 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 1226 . . . . . . 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 6302 . . . . . . 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 2509 . . . . . 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 3533 . . . . 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 26091 . . . . . . . . . . 11  |-  ( R  vH  A )  C_  ( ( R  vH  A )  vH  C
)
9222, 91sstri 3513 . . . . . . . . . 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 3720 . . . . . . . . 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 26091 . . . . . . . . 9  |-  ( R  vH  A )  C_  ( ( R  vH  A )  vH  ( C  i^i  D ) )
9722, 96sstri 3513 . . . . . . . 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 3720 . . . . . . 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 3719 . . . . . . . . . . 11  |-  ( ( ( R  vH  A
)  vH  C )  i^i  D )  C_  D
102 sstr 3512 . . . . . . . . . . 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 26079 . . . . . . . . 9  |-  ( ( R  vH  A )  vH  C )  e. 
CH
108107, 2chincli 26082 . . . . . . . 8  |-  ( ( ( R  vH  A
)  vH  C )  i^i  D )  e.  CH
10943, 79chjcli 26079 . . . . . . . 8  |-  ( ( R  vH  A )  vH  ( C  i^i  D ) )  e.  CH
110108, 109, 37chlubi 26093 . . . . . . 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 26940 . . . . . 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 1228 . . . . 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 973    = wceq 1379    e. wcel 1767    i^i cin 3475    C_ wss 3476   class class class wbr 4447  (class class class)co 6284   CHcch 25550    vH chj 25554    MH cmd 25587    MH* cdmd 25588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cc 8815  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572  ax-hilex 25620  ax-hfvadd 25621  ax-hvcom 25622  ax-hvass 25623  ax-hv0cl 25624  ax-hvaddid 25625  ax-hfvmul 25626  ax-hvmulid 25627  ax-hvmulass 25628  ax-hvdistr1 25629  ax-hvdistr2 25630  ax-hvmul0 25631  ax-hfi 25700  ax-his1 25703  ax-his2 25704  ax-his3 25705  ax-his4 25706  ax-hcompl 25823
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-omul 7135  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-acn 8323  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-rlim 13275  df-sum 13472  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-mulg 15870  df-cntz 16160  df-cmn 16606  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-fbas 18215  df-fg 18216  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-nei 19393  df-cn 19522  df-cnp 19523  df-lm 19524  df-haus 19610  df-tx 19826  df-hmeo 20019  df-fil 20110  df-fm 20202  df-flim 20203  df-flf 20204  df-xms 20586  df-ms 20587  df-tms 20588  df-cfil 21457  df-cau 21458  df-cmet 21459  df-grpo 24897  df-gid 24898  df-ginv 24899  df-gdiv 24900  df-ablo 24988  df-subgo 25008  df-vc 25143  df-nv 25189  df-va 25192  df-ba 25193  df-sm 25194  df-0v 25195  df-vs 25196  df-nmcv 25197  df-ims 25198  df-dip 25315  df-ssp 25339  df-ph 25432  df-cbn 25483  df-hnorm 25589  df-hba 25590  df-hvsub 25592  df-hlim 25593  df-hcau 25594  df-sh 25828  df-ch 25843  df-oc 25874  df-ch0 25875  df-shs 25930  df-chj 25932  df-md 26903  df-dmd 26904
This theorem is referenced by:  mdslmd1lem3  26950
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