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Theorem 3oalem2 25069
Description: Lemma for 3OA (weak) orthoarguesian law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
3oalem1.1  |-  B  e. 
CH
3oalem1.2  |-  C  e. 
CH
3oalem1.3  |-  R  e. 
CH
3oalem1.4  |-  S  e. 
CH
Assertion
Ref Expression
3oalem2  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  v  e.  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) ) )
Distinct variable groups:    x, y,
z, w, v, B   
x, C, y, z, w, v    x, R, y, z, w, v   
x, S, y, z, w, v

Proof of Theorem 3oalem2
StepHypRef Expression
1 simplll 757 . . 3  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  x  e.  B
)
2 simpllr 758 . . . 4  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  y  e.  R
)
3 3oalem1.1 . . . . . . 7  |-  B  e. 
CH
4 3oalem1.2 . . . . . . 7  |-  C  e. 
CH
5 3oalem1.3 . . . . . . 7  |-  R  e. 
CH
6 3oalem1.4 . . . . . . 7  |-  S  e. 
CH
73, 4, 5, 63oalem1 25068 . . . . . 6  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  v  e.  ~H )  /\  ( z  e. 
~H  /\  w  e.  ~H ) ) )
8 hvaddsub12 24443 . . . . . . . . . 10  |-  ( ( y  e.  ~H  /\  w  e.  ~H  /\  w  e.  ~H )  ->  (
y  +h  ( w  -h  w ) )  =  ( w  +h  ( y  -h  w
) ) )
983anidm23 1277 . . . . . . . . 9  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( y  +h  (
w  -h  w ) )  =  ( w  +h  ( y  -h  w ) ) )
10 hvsubid 24431 . . . . . . . . . . 11  |-  ( w  e.  ~H  ->  (
w  -h  w )  =  0h )
1110oveq2d 6110 . . . . . . . . . 10  |-  ( w  e.  ~H  ->  (
y  +h  ( w  -h  w ) )  =  ( y  +h 
0h ) )
12 ax-hvaddid 24409 . . . . . . . . . 10  |-  ( y  e.  ~H  ->  (
y  +h  0h )  =  y )
1311, 12sylan9eqr 2497 . . . . . . . . 9  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( y  +h  (
w  -h  w ) )  =  y )
149, 13eqtr3d 2477 . . . . . . . 8  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( w  +h  (
y  -h  w ) )  =  y )
1514ad2ant2l 745 . . . . . . 7  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( w  +h  ( y  -h  w
) )  =  y )
1615adantlr 714 . . . . . 6  |-  ( ( ( ( x  e. 
~H  /\  y  e.  ~H )  /\  v  e.  ~H )  /\  (
z  e.  ~H  /\  w  e.  ~H )
)  ->  ( w  +h  ( y  -h  w
) )  =  y )
177, 16syl 16 . . . . 5  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( w  +h  ( y  -h  w
) )  =  y )
18 simprlr 762 . . . . . 6  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  w  e.  S
)
19 eqtr2 2461 . . . . . . . . . . 11  |-  ( ( v  =  ( x  +h  y )  /\  v  =  ( z  +h  w ) )  -> 
( x  +h  y
)  =  ( z  +h  w ) )
2019oveq1d 6109 . . . . . . . . . 10  |-  ( ( v  =  ( x  +h  y )  /\  v  =  ( z  +h  w ) )  -> 
( ( x  +h  y )  -h  (
x  +h  w ) )  =  ( ( z  +h  w )  -h  ( x  +h  w ) ) )
2120ad2ant2l 745 . . . . . . . . 9  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( ( x  +h  y )  -h  ( x  +h  w
) )  =  ( ( z  +h  w
)  -h  ( x  +h  w ) ) )
22 simpl 457 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  x  e.  ~H )
2322anim1i 568 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( x  e. 
~H  /\  w  e.  ~H ) )
24 hvsub4 24442 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  ( x  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
x  +h  y )  -h  ( x  +h  w ) )  =  ( ( x  -h  x )  +h  (
y  -h  w ) ) )
2523, 24syldan 470 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( ( x  +h  y )  -h  ( x  +h  w
) )  =  ( ( x  -h  x
)  +h  ( y  -h  w ) ) )
26 hvsubid 24431 . . . . . . . . . . . . . 14  |-  ( x  e.  ~H  ->  (
x  -h  x )  =  0h )
2726ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( x  -h  x )  =  0h )
2827oveq1d 6109 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( ( x  -h  x )  +h  ( y  -h  w
) )  =  ( 0h  +h  ( y  -h  w ) ) )
29 hvsubcl 24422 . . . . . . . . . . . . . 14  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( y  -h  w
)  e.  ~H )
30 hvaddid2 24428 . . . . . . . . . . . . . 14  |-  ( ( y  -h  w )  e.  ~H  ->  ( 0h  +h  ( y  -h  w ) )  =  ( y  -h  w
) )
3129, 30syl 16 . . . . . . . . . . . . 13  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( 0h  +h  (
y  -h  w ) )  =  ( y  -h  w ) )
3231adantll 713 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( 0h  +h  ( y  -h  w
) )  =  ( y  -h  w ) )
3325, 28, 323eqtrd 2479 . . . . . . . . . . 11  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( ( x  +h  y )  -h  ( x  +h  w
) )  =  ( y  -h  w ) )
3433ad2ant2rl 748 . . . . . . . . . 10  |-  ( ( ( ( x  e. 
~H  /\  y  e.  ~H )  /\  v  e.  ~H )  /\  (
z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
x  +h  y )  -h  ( x  +h  w ) )  =  ( y  -h  w
) )
357, 34syl 16 . . . . . . . . 9  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( ( x  +h  y )  -h  ( x  +h  w
) )  =  ( y  -h  w ) )
36 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( z  e.  ~H  /\  w  e. 
~H ) )
37 simpr 461 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  ~H  /\  w  e.  ~H )  ->  w  e.  ~H )
3837anim2i 569 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( x  e.  ~H  /\  w  e. 
~H ) )
39 hvsub4 24442 . . . . . . . . . . . . . 14  |-  ( ( ( z  e.  ~H  /\  w  e.  ~H )  /\  ( x  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( x  +h  w ) )  =  ( ( z  -h  x )  +h  (
w  -h  w ) ) )
4036, 38, 39syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( x  +h  w ) )  =  ( ( z  -h  x )  +h  (
w  -h  w ) ) )
4110ad2antll 728 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( w  -h  w )  =  0h )
4241oveq2d 6110 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  -h  x )  +h  ( w  -h  w ) )  =  ( ( z  -h  x )  +h  0h ) )
43 hvsubcl 24422 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  ~H  /\  x  e.  ~H )  ->  ( z  -h  x
)  e.  ~H )
44 ax-hvaddid 24409 . . . . . . . . . . . . . . . 16  |-  ( ( z  -h  x )  e.  ~H  ->  (
( z  -h  x
)  +h  0h )  =  ( z  -h  x ) )
4543, 44syl 16 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  ~H  /\  x  e.  ~H )  ->  ( ( z  -h  x )  +h  0h )  =  ( z  -h  x ) )
4645ancoms 453 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( ( z  -h  x )  +h  0h )  =  ( z  -h  x ) )
4746adantrr 716 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  -h  x )  +h  0h )  =  ( z  -h  x
) )
4840, 42, 473eqtrd 2479 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( x  +h  w ) )  =  ( z  -h  x
) )
4948adantlr 714 . . . . . . . . . . 11  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( x  +h  w ) )  =  ( z  -h  x
) )
5049adantlr 714 . . . . . . . . . 10  |-  ( ( ( ( x  e. 
~H  /\  y  e.  ~H )  /\  v  e.  ~H )  /\  (
z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( x  +h  w ) )  =  ( z  -h  x
) )
517, 50syl 16 . . . . . . . . 9  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( ( z  +h  w )  -h  ( x  +h  w
) )  =  ( z  -h  x ) )
5221, 35, 513eqtr3d 2483 . . . . . . . 8  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( y  -h  w )  =  ( z  -h  x ) )
53 simpll 753 . . . . . . . . 9  |-  ( ( ( x  e.  B  /\  y  e.  R
)  /\  v  =  ( x  +h  y
) )  ->  x  e.  B )
54 simpll 753 . . . . . . . . 9  |-  ( ( ( z  e.  C  /\  w  e.  S
)  /\  v  =  ( z  +h  w
) )  ->  z  e.  C )
554chshii 24633 . . . . . . . . . . . 12  |-  C  e.  SH
563chshii 24633 . . . . . . . . . . . 12  |-  B  e.  SH
5755, 56shsvsi 24773 . . . . . . . . . . 11  |-  ( ( z  e.  C  /\  x  e.  B )  ->  ( z  -h  x
)  e.  ( C  +H  B ) )
5857ancoms 453 . . . . . . . . . 10  |-  ( ( x  e.  B  /\  z  e.  C )  ->  ( z  -h  x
)  e.  ( C  +H  B ) )
5956, 55shscomi 24769 . . . . . . . . . 10  |-  ( B  +H  C )  =  ( C  +H  B
)
6058, 59syl6eleqr 2534 . . . . . . . . 9  |-  ( ( x  e.  B  /\  z  e.  C )  ->  ( z  -h  x
)  e.  ( B  +H  C ) )
6153, 54, 60syl2an 477 . . . . . . . 8  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( z  -h  x )  e.  ( B  +H  C ) )
6252, 61eqeltrd 2517 . . . . . . 7  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( y  -h  w )  e.  ( B  +H  C ) )
63 simplr 754 . . . . . . . 8  |-  ( ( ( x  e.  B  /\  y  e.  R
)  /\  v  =  ( x  +h  y
) )  ->  y  e.  R )
64 simplr 754 . . . . . . . 8  |-  ( ( ( z  e.  C  /\  w  e.  S
)  /\  v  =  ( z  +h  w
) )  ->  w  e.  S )
655chshii 24633 . . . . . . . . 9  |-  R  e.  SH
666chshii 24633 . . . . . . . . 9  |-  S  e.  SH
6765, 66shsvsi 24773 . . . . . . . 8  |-  ( ( y  e.  R  /\  w  e.  S )  ->  ( y  -h  w
)  e.  ( R  +H  S ) )
6863, 64, 67syl2an 477 . . . . . . 7  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( y  -h  w )  e.  ( R  +H  S ) )
6962, 68elind 3543 . . . . . 6  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( y  -h  w )  e.  ( ( B  +H  C
)  i^i  ( R  +H  S ) ) )
7056, 55shscli 24723 . . . . . . . 8  |-  ( B  +H  C )  e.  SH
7165, 66shscli 24723 . . . . . . . 8  |-  ( R  +H  S )  e.  SH
7270, 71shincli 24768 . . . . . . 7  |-  ( ( B  +H  C )  i^i  ( R  +H  S ) )  e.  SH
7366, 72shsvai 24770 . . . . . 6  |-  ( ( w  e.  S  /\  ( y  -h  w
)  e.  ( ( B  +H  C )  i^i  ( R  +H  S ) ) )  ->  ( w  +h  ( y  -h  w
) )  e.  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) )
7418, 69, 73syl2anc 661 . . . . 5  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( w  +h  ( y  -h  w
) )  e.  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) )
7517, 74eqeltrrd 2518 . . . 4  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  y  e.  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) )
762, 75elind 3543 . . 3  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  y  e.  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S
) ) ) ) )
7766, 72shscli 24723 . . . . 5  |-  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S
) ) )  e.  SH
7865, 77shincli 24768 . . . 4  |-  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) )  e.  SH
7956, 78shsvai 24770 . . 3  |-  ( ( x  e.  B  /\  y  e.  ( R  i^i  ( S  +H  (
( B  +H  C
)  i^i  ( R  +H  S ) ) ) ) )  ->  (
x  +h  y )  e.  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S
) ) ) ) ) )
801, 76, 79syl2anc 661 . 2  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( x  +h  y )  e.  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) ) )
81 eleq1 2503 . . 3  |-  ( v  =  ( x  +h  y )  ->  (
v  e.  ( B  +H  ( R  i^i  ( S  +H  (
( B  +H  C
)  i^i  ( R  +H  S ) ) ) ) )  <->  ( x  +h  y )  e.  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) ) ) )
8281ad2antlr 726 . 2  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  ( v  e.  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) )  <->  ( x  +h  y )  e.  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) ) ) )
8380, 82mpbird 232 1  |-  ( ( ( ( x  e.  B  /\  y  e.  R )  /\  v  =  ( x  +h  y ) )  /\  ( ( z  e.  C  /\  w  e.  S )  /\  v  =  ( z  +h  w ) ) )  ->  v  e.  ( B  +H  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    i^i cin 3330  (class class class)co 6094   ~Hchil 24324    +h cva 24325   0hc0v 24329    -h cmv 24330   CHcch 24334    +H cph 24336
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 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-hilex 24404  ax-hfvadd 24405  ax-hvcom 24406  ax-hvass 24407  ax-hv0cl 24408  ax-hvaddid 24409  ax-hfvmul 24410  ax-hvmulid 24411  ax-hvdistr1 24413  ax-hvdistr2 24414  ax-hvmul0 24415
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-recs 6835  df-rdg 6869  df-er 7104  df-map 7219  df-en 7314  df-dom 7315  df-sdom 7316  df-pnf 9423  df-mnf 9424  df-ltxr 9426  df-sub 9600  df-neg 9601  df-nn 10326  df-grpo 23681  df-ablo 23772  df-hvsub 24376  df-hlim 24377  df-sh 24612  df-ch 24627  df-shs 24714
This theorem is referenced by:  3oalem3  25070
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