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Theorem 3oalem2 25001
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 752 . . 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 753 . . . 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 25000 . . . . . 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 24375 . . . . . . . . . 10  |-  ( ( y  e.  ~H  /\  w  e.  ~H  /\  w  e.  ~H )  ->  (
y  +h  ( w  -h  w ) )  =  ( w  +h  ( y  -h  w
) ) )
983anidm23 1272 . . . . . . . . 9  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( y  +h  (
w  -h  w ) )  =  ( w  +h  ( y  -h  w ) ) )
10 hvsubid 24363 . . . . . . . . . . 11  |-  ( w  e.  ~H  ->  (
w  -h  w )  =  0h )
1110oveq2d 6106 . . . . . . . . . 10  |-  ( w  e.  ~H  ->  (
y  +h  ( w  -h  w ) )  =  ( y  +h 
0h ) )
12 ax-hvaddid 24341 . . . . . . . . . 10  |-  ( y  e.  ~H  ->  (
y  +h  0h )  =  y )
1311, 12sylan9eqr 2495 . . . . . . . . 9  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( y  +h  (
w  -h  w ) )  =  y )
149, 13eqtr3d 2475 . . . . . . . 8  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( w  +h  (
y  -h  w ) )  =  y )
1514ad2ant2l 740 . . . . . . 7  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( w  +h  ( y  -h  w
) )  =  y )
1615adantlr 709 . . . . . 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 757 . . . . . 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 2459 . . . . . . . . . . 11  |-  ( ( v  =  ( x  +h  y )  /\  v  =  ( z  +h  w ) )  -> 
( x  +h  y
)  =  ( z  +h  w ) )
2019oveq1d 6105 . . . . . . . . . 10  |-  ( ( v  =  ( x  +h  y )  /\  v  =  ( z  +h  w ) )  -> 
( ( x  +h  y )  -h  (
x  +h  w ) )  =  ( ( z  +h  w )  -h  ( x  +h  w ) ) )
2120ad2ant2l 740 . . . . . . . . 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 454 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  x  e.  ~H )
2322anim1i 565 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( x  e. 
~H  /\  w  e.  ~H ) )
24 hvsub4 24374 . . . . . . . . . . . . 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 467 . . . . . . . . . . . 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 24363 . . . . . . . . . . . . . 14  |-  ( x  e.  ~H  ->  (
x  -h  x )  =  0h )
2726ad2antrr 720 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( x  -h  x )  =  0h )
2827oveq1d 6105 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( ( x  -h  x )  +h  ( y  -h  w
) )  =  ( 0h  +h  ( y  -h  w ) ) )
29 hvsubcl 24354 . . . . . . . . . . . . . 14  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( y  -h  w
)  e.  ~H )
30 hvaddid2 24360 . . . . . . . . . . . . . 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 708 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( 0h  +h  ( y  -h  w
) )  =  ( y  -h  w ) )
3325, 28, 323eqtrd 2477 . . . . . . . . . . 11  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( ( x  +h  y )  -h  ( x  +h  w
) )  =  ( y  -h  w ) )
3433ad2ant2rl 743 . . . . . . . . . 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 458 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( z  e.  ~H  /\  w  e. 
~H ) )
37 simpr 458 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  ~H  /\  w  e.  ~H )  ->  w  e.  ~H )
3837anim2i 566 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( x  e.  ~H  /\  w  e. 
~H ) )
39 hvsub4 24374 . . . . . . . . . . . . . 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 656 . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( w  -h  w )  =  0h )
4241oveq2d 6106 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  -h  x )  +h  ( w  -h  w ) )  =  ( ( z  -h  x )  +h  0h ) )
43 hvsubcl 24354 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  ~H  /\  x  e.  ~H )  ->  ( z  -h  x
)  e.  ~H )
44 ax-hvaddid 24341 . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( ( z  -h  x )  +h  0h )  =  ( z  -h  x ) )
4746adantrr 711 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  -h  x )  +h  0h )  =  ( z  -h  x
) )
4840, 42, 473eqtrd 2477 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( x  +h  w ) )  =  ( z  -h  x
) )
4948adantlr 709 . . . . . . . . . . 11  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( x  +h  w ) )  =  ( z  -h  x
) )
5049adantlr 709 . . . . . . . . . 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 2481 . . . . . . . 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 748 . . . . . . . . 9  |-  ( ( ( x  e.  B  /\  y  e.  R
)  /\  v  =  ( x  +h  y
) )  ->  x  e.  B )
54 simpll 748 . . . . . . . . 9  |-  ( ( ( z  e.  C  /\  w  e.  S
)  /\  v  =  ( z  +h  w
) )  ->  z  e.  C )
554chshii 24565 . . . . . . . . . . . 12  |-  C  e.  SH
563chshii 24565 . . . . . . . . . . . 12  |-  B  e.  SH
5755, 56shsvsi 24705 . . . . . . . . . . 11  |-  ( ( z  e.  C  /\  x  e.  B )  ->  ( z  -h  x
)  e.  ( C  +H  B ) )
5857ancoms 450 . . . . . . . . . 10  |-  ( ( x  e.  B  /\  z  e.  C )  ->  ( z  -h  x
)  e.  ( C  +H  B ) )
5956, 55shscomi 24701 . . . . . . . . . 10  |-  ( B  +H  C )  =  ( C  +H  B
)
6058, 59syl6eleqr 2532 . . . . . . . . 9  |-  ( ( x  e.  B  /\  z  e.  C )  ->  ( z  -h  x
)  e.  ( B  +H  C ) )
6153, 54, 60syl2an 474 . . . . . . . 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 2515 . . . . . . 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 749 . . . . . . . 8  |-  ( ( ( x  e.  B  /\  y  e.  R
)  /\  v  =  ( x  +h  y
) )  ->  y  e.  R )
64 simplr 749 . . . . . . . 8  |-  ( ( ( z  e.  C  /\  w  e.  S
)  /\  v  =  ( z  +h  w
) )  ->  w  e.  S )
655chshii 24565 . . . . . . . . 9  |-  R  e.  SH
666chshii 24565 . . . . . . . . 9  |-  S  e.  SH
6765, 66shsvsi 24705 . . . . . . . 8  |-  ( ( y  e.  R  /\  w  e.  S )  ->  ( y  -h  w
)  e.  ( R  +H  S ) )
6863, 64, 67syl2an 474 . . . . . . 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 3537 . . . . . 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 24655 . . . . . . . 8  |-  ( B  +H  C )  e.  SH
7165, 66shscli 24655 . . . . . . . 8  |-  ( R  +H  S )  e.  SH
7270, 71shincli 24700 . . . . . . 7  |-  ( ( B  +H  C )  i^i  ( R  +H  S ) )  e.  SH
7366, 72shsvai 24702 . . . . . 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 656 . . . . 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 2516 . . . 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 3537 . . 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 24655 . . . . 5  |-  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S
) ) )  e.  SH
7865, 77shincli 24700 . . . 4  |-  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) )  e.  SH
7956, 78shsvai 24702 . . 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 656 . 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 2501 . . 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 721 . 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 1364    e. wcel 1761    i^i cin 3324  (class class class)co 6090   ~Hchil 24256    +h cva 24257   0hc0v 24261    -h cmv 24262   CHcch 24266    +H cph 24268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-hilex 24336  ax-hfvadd 24337  ax-hvcom 24338  ax-hvass 24339  ax-hv0cl 24340  ax-hvaddid 24341  ax-hfvmul 24342  ax-hvmulid 24343  ax-hvdistr1 24345  ax-hvdistr2 24346  ax-hvmul0 24347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-ltxr 9419  df-sub 9593  df-neg 9594  df-nn 10319  df-grpo 23613  df-ablo 23704  df-hvsub 24308  df-hlim 24309  df-sh 24544  df-ch 24559  df-shs 24646
This theorem is referenced by:  3oalem3  25002
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