HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  mdslmd1lem1 Structured version   Unicode version

Theorem mdslmd1lem1 25880
Description: Lemma for mdslmd1i 25884. (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 25014 . . . . . 6  |-  ( D  i^i  B )  e. 
CH
5 mdslmd.1 . . . . . 6  |-  A  e. 
CH
61, 4, 5chlej1i 25027 . . . . 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 25870 . . . . . . . . 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 1261 . . . . . . 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 1189 . . . . . 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 3491 . . . . 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 25024 . . . . . . . . . . . 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 3679 . . . . . . . . . . 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 3677 . . . . . . . . . . . . . . . . . . . 20  |-  ( D  i^i  B )  C_  D
27 sstr 3471 . . . . . . . . . . . . . . . . . . . 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 3471 . . . . . . . . . . . . . . . . . . 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 25023 . . . . . . . . . . . . . 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 25011 . . . . . . . . . . . . . 14  |-  ( A  vH  B )  e. 
CH
381, 5, 37chlubi 25025 . . . . . . . . . . . . 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 25011 . . . . . . . . . . . 12  |-  ( R  vH  A )  e. 
CH
44 mdslmd.3 . . . . . . . . . . . 12  |-  C  e. 
CH
4543, 44, 37chlubi 25025 . . . . . . . . . . 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 25874 . . . . . . . . . 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 1217 . . . . . . . . 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 3679 . . . . . . . . . . . . . . 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 3682 . . . . . . . . . . . . . 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 3675 . . . . . . . . . . . . 13  |-  ( ( C  i^i  D )  i^i  B )  =  ( ( C  i^i  B )  i^i  ( D  i^i  B ) )
5654, 55syl6sseq 3509 . . . . . . . . . . . 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 3473 . . . . . . . . . . 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 3678 . . . . . . . . . . . . 13  |-  ( D  i^i  B )  C_  B
60 sstr 3471 . . . . . . . . . . . . 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 25871 . . . . . . . . . . . 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 1219 . . . . . . . . . 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 6214 . . . . . . . . 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 2496 . . . . . . . 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 3658 . . . . . . 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 3675 . . . . . . 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 2513 . . . . . 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 3679 . . . . . . . . 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 3685 . . . . . . . . . . . 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 25014 . . . . . . . . . 10  |-  ( C  i^i  D )  e. 
CH
8043, 79, 37chlubi 25025 . . . . . . . . 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 25874 . . . . . . . 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 1217 . . . . . . 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 6217 . . . . . . 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 2496 . . . . . 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 3492 . . . . 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 25023 . . . . . . . . . . 11  |-  ( R  vH  A )  C_  ( ( R  vH  A )  vH  C
)
9222, 91sstri 3472 . . . . . . . . . 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 3679 . . . . . . . . 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 25023 . . . . . . . . 9  |-  ( R  vH  A )  C_  ( ( R  vH  A )  vH  ( C  i^i  D ) )
9722, 96sstri 3472 . . . . . . . 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 3679 . . . . . . 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 3678 . . . . . . . . . . 11  |-  ( ( ( R  vH  A
)  vH  C )  i^i  D )  C_  D
102 sstr 3471 . . . . . . . . . . 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 25011 . . . . . . . . 9  |-  ( ( R  vH  A )  vH  C )  e. 
CH
108107, 2chincli 25014 . . . . . . . 8  |-  ( ( ( R  vH  A
)  vH  C )  i^i  D )  e.  CH
10943, 79chjcli 25011 . . . . . . . 8  |-  ( ( R  vH  A )  vH  ( C  i^i  D ) )  e.  CH
110108, 109, 37chlubi 25025 . . . . . . 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 25872 . . . . . 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 1219 . . . . 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 1370    e. wcel 1758    i^i cin 3434    C_ wss 3435   class class class wbr 4399  (class class class)co 6199   CHcch 24482    vH chj 24486    MH cmd 24519    MH* cdmd 24520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cc 8714  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470  ax-addf 9471  ax-mulf 9472  ax-hilex 24552  ax-hfvadd 24553  ax-hvcom 24554  ax-hvass 24555  ax-hv0cl 24556  ax-hvaddid 24557  ax-hfvmul 24558  ax-hvmulid 24559  ax-hvmulass 24560  ax-hvdistr1 24561  ax-hvdistr2 24562  ax-hvmul0 24563  ax-hfi 24632  ax-his1 24635  ax-his2 24636  ax-his3 24637  ax-his4 24638  ax-hcompl 24755
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-omul 7034  df-er 7210  df-map 7325  df-pm 7326  df-ixp 7373  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-fi 7771  df-sup 7801  df-oi 7834  df-card 8219  df-acn 8222  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ioo 11414  df-ico 11416  df-icc 11417  df-fz 11554  df-fzo 11665  df-fl 11758  df-seq 11923  df-exp 11982  df-hash 12220  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-clim 13083  df-rlim 13084  df-sum 13281  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-starv 14371  df-sca 14372  df-vsca 14373  df-ip 14374  df-tset 14375  df-ple 14376  df-ds 14378  df-unif 14379  df-hom 14380  df-cco 14381  df-rest 14479  df-topn 14480  df-0g 14498  df-gsum 14499  df-topgen 14500  df-pt 14501  df-prds 14504  df-xrs 14558  df-qtop 14563  df-imas 14564  df-xps 14566  df-mre 14642  df-mrc 14643  df-acs 14645  df-mnd 15533  df-submnd 15583  df-mulg 15666  df-cntz 15953  df-cmn 16399  df-psmet 17933  df-xmet 17934  df-met 17935  df-bl 17936  df-mopn 17937  df-fbas 17938  df-fg 17939  df-cnfld 17943  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-cld 18754  df-ntr 18755  df-cls 18756  df-nei 18833  df-cn 18962  df-cnp 18963  df-lm 18964  df-haus 19050  df-tx 19266  df-hmeo 19459  df-fil 19550  df-fm 19642  df-flim 19643  df-flf 19644  df-xms 20026  df-ms 20027  df-tms 20028  df-cfil 20897  df-cau 20898  df-cmet 20899  df-grpo 23829  df-gid 23830  df-ginv 23831  df-gdiv 23832  df-ablo 23920  df-subgo 23940  df-vc 24075  df-nv 24121  df-va 24124  df-ba 24125  df-sm 24126  df-0v 24127  df-vs 24128  df-nmcv 24129  df-ims 24130  df-dip 24247  df-ssp 24271  df-ph 24364  df-cbn 24415  df-hnorm 24521  df-hba 24522  df-hvsub 24524  df-hlim 24525  df-hcau 24526  df-sh 24760  df-ch 24775  df-oc 24806  df-ch0 24807  df-shs 24862  df-chj 24864  df-md 25835  df-dmd 25836
This theorem is referenced by:  mdslmd1lem3  25882
  Copyright terms: Public domain W3C validator