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Theorem 3oalem2 22258
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 734 . . 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 735 . . . 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 22257 . . . . . 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 21633 . . . . . . . . . 10  |-  ( ( y  e.  ~H  /\  w  e.  ~H  /\  w  e.  ~H )  ->  (
y  +h  ( w  -h  w ) )  =  ( w  +h  ( y  -h  w
) ) )
983anidm23 1241 . . . . . . . . 9  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( y  +h  (
w  -h  w ) )  =  ( w  +h  ( y  -h  w ) ) )
10 hvsubid 21621 . . . . . . . . . . 11  |-  ( w  e.  ~H  ->  (
w  -h  w )  =  0h )
1110oveq2d 5890 . . . . . . . . . 10  |-  ( w  e.  ~H  ->  (
y  +h  ( w  -h  w ) )  =  ( y  +h 
0h ) )
12 ax-hvaddid 21600 . . . . . . . . . 10  |-  ( y  e.  ~H  ->  (
y  +h  0h )  =  y )
1311, 12sylan9eqr 2350 . . . . . . . . 9  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( y  +h  (
w  -h  w ) )  =  y )
149, 13eqtr3d 2330 . . . . . . . 8  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( w  +h  (
y  -h  w ) )  =  y )
1514ad2ant2l 726 . . . . . . 7  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( w  +h  ( y  -h  w
) )  =  y )
1615adantlr 695 . . . . . 6  |-  ( ( ( ( x  e. 
~H  /\  y  e.  ~H )  /\  v  e.  ~H )  /\  (
z  e.  ~H  /\  w  e.  ~H )
)  ->  ( w  +h  ( y  -h  w
) )  =  y )
177, 16syl 15 . . . . 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 739 . . . . . 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 2314 . . . . . . . . . . 11  |-  ( ( v  =  ( x  +h  y )  /\  v  =  ( z  +h  w ) )  -> 
( x  +h  y
)  =  ( z  +h  w ) )
2019oveq1d 5889 . . . . . . . . . 10  |-  ( ( v  =  ( x  +h  y )  /\  v  =  ( z  +h  w ) )  -> 
( ( x  +h  y )  -h  (
x  +h  w ) )  =  ( ( z  +h  w )  -h  ( x  +h  w ) ) )
2120ad2ant2l 726 . . . . . . . . 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 443 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  x  e.  ~H )
2322anim1i 551 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( x  e. 
~H  /\  w  e.  ~H ) )
24 hvsub4 21632 . . . . . . . . . . . . 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 456 . . . . . . . . . . . 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 21621 . . . . . . . . . . . . . 14  |-  ( x  e.  ~H  ->  (
x  -h  x )  =  0h )
2726ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( x  -h  x )  =  0h )
2827oveq1d 5889 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( ( x  -h  x )  +h  ( y  -h  w
) )  =  ( 0h  +h  ( y  -h  w ) ) )
29 hvsubcl 21613 . . . . . . . . . . . . . 14  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( y  -h  w
)  e.  ~H )
30 hvaddid2 21618 . . . . . . . . . . . . . 14  |-  ( ( y  -h  w )  e.  ~H  ->  ( 0h  +h  ( y  -h  w ) )  =  ( y  -h  w
) )
3129, 30syl 15 . . . . . . . . . . . . 13  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( 0h  +h  (
y  -h  w ) )  =  ( y  -h  w ) )
3231adantll 694 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( 0h  +h  ( y  -h  w
) )  =  ( y  -h  w ) )
3325, 28, 323eqtrd 2332 . . . . . . . . . . 11  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( ( x  +h  y )  -h  ( x  +h  w
) )  =  ( y  -h  w ) )
3433ad2ant2rl 729 . . . . . . . . . 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 15 . . . . . . . . 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 447 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( z  e.  ~H  /\  w  e. 
~H ) )
37 simpr 447 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  ~H  /\  w  e.  ~H )  ->  w  e.  ~H )
3837anim2i 552 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( x  e.  ~H  /\  w  e. 
~H ) )
39 hvsub4 21632 . . . . . . . . . . . . . 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 642 . . . . . . . . . . . . 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 709 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( w  -h  w )  =  0h )
4241oveq2d 5890 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  -h  x )  +h  ( w  -h  w ) )  =  ( ( z  -h  x )  +h  0h ) )
43 hvsubcl 21613 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  ~H  /\  x  e.  ~H )  ->  ( z  -h  x
)  e.  ~H )
44 ax-hvaddid 21600 . . . . . . . . . . . . . . . 16  |-  ( ( z  -h  x )  e.  ~H  ->  (
( z  -h  x
)  +h  0h )  =  ( z  -h  x ) )
4543, 44syl 15 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  ~H  /\  x  e.  ~H )  ->  ( ( z  -h  x )  +h  0h )  =  ( z  -h  x ) )
4645ancoms 439 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( ( z  -h  x )  +h  0h )  =  ( z  -h  x ) )
4746adantrr 697 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  -h  x )  +h  0h )  =  ( z  -h  x
) )
4840, 42, 473eqtrd 2332 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( x  +h  w ) )  =  ( z  -h  x
) )
4948adantlr 695 . . . . . . . . . . 11  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( x  +h  w ) )  =  ( z  -h  x
) )
5049adantlr 695 . . . . . . . . . 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 15 . . . . . . . . 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 2336 . . . . . . . 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 730 . . . . . . . . 9  |-  ( ( ( x  e.  B  /\  y  e.  R
)  /\  v  =  ( x  +h  y
) )  ->  x  e.  B )
54 simpll 730 . . . . . . . . 9  |-  ( ( ( z  e.  C  /\  w  e.  S
)  /\  v  =  ( z  +h  w
) )  ->  z  e.  C )
554chshii 21823 . . . . . . . . . . . 12  |-  C  e.  SH
563chshii 21823 . . . . . . . . . . . 12  |-  B  e.  SH
5755, 56shsvsi 21962 . . . . . . . . . . 11  |-  ( ( z  e.  C  /\  x  e.  B )  ->  ( z  -h  x
)  e.  ( C  +H  B ) )
5857ancoms 439 . . . . . . . . . 10  |-  ( ( x  e.  B  /\  z  e.  C )  ->  ( z  -h  x
)  e.  ( C  +H  B ) )
5956, 55shscomi 21958 . . . . . . . . . 10  |-  ( B  +H  C )  =  ( C  +H  B
)
6058, 59syl6eleqr 2387 . . . . . . . . 9  |-  ( ( x  e.  B  /\  z  e.  C )  ->  ( z  -h  x
)  e.  ( B  +H  C ) )
6153, 54, 60syl2an 463 . . . . . . . 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 2370 . . . . . . 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 731 . . . . . . . 8  |-  ( ( ( x  e.  B  /\  y  e.  R
)  /\  v  =  ( x  +h  y
) )  ->  y  e.  R )
64 simplr 731 . . . . . . . 8  |-  ( ( ( z  e.  C  /\  w  e.  S
)  /\  v  =  ( z  +h  w
) )  ->  w  e.  S )
655chshii 21823 . . . . . . . . 9  |-  R  e.  SH
666chshii 21823 . . . . . . . . 9  |-  S  e.  SH
6765, 66shsvsi 21962 . . . . . . . 8  |-  ( ( y  e.  R  /\  w  e.  S )  ->  ( y  -h  w
)  e.  ( R  +H  S ) )
6863, 64, 67syl2an 463 . . . . . . 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 ) )
69 elin 3371 . . . . . . 7  |-  ( ( y  -h  w )  e.  ( ( B  +H  C )  i^i  ( R  +H  S
) )  <->  ( (
y  -h  w )  e.  ( B  +H  C )  /\  (
y  -h  w )  e.  ( R  +H  S ) ) )
7062, 68, 69sylanbrc 645 . . . . . 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 ) ) )
7156, 55shscli 21912 . . . . . . . 8  |-  ( B  +H  C )  e.  SH
7265, 66shscli 21912 . . . . . . . 8  |-  ( R  +H  S )  e.  SH
7371, 72shincli 21957 . . . . . . 7  |-  ( ( B  +H  C )  i^i  ( R  +H  S ) )  e.  SH
7466, 73shsvai 21959 . . . . . 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 ) ) ) )
7518, 70, 74syl2anc 642 . . . . 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 ) ) ) )
7617, 75eqeltrrd 2371 . . . 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 ) ) ) )
77 elin 3371 . . . 4  |-  ( y  e.  ( R  i^i  ( S  +H  (
( B  +H  C
)  i^i  ( R  +H  S ) ) ) )  <->  ( y  e.  R  /\  y  e.  ( S  +H  (
( B  +H  C
)  i^i  ( R  +H  S ) ) ) ) )
782, 76, 77sylanbrc 645 . . 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
) ) ) ) )
7966, 73shscli 21912 . . . . 5  |-  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S
) ) )  e.  SH
8065, 79shincli 21957 . . . 4  |-  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) )  e.  SH
8156, 80shsvai 21959 . . 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
) ) ) ) ) )
821, 78, 81syl2anc 642 . 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 ) ) ) ) ) )
83 eleq1 2356 . . 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 ) ) ) ) ) ) )
8483ad2antlr 707 . 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 ) ) ) ) ) ) )
8582, 84mpbird 223 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    i^i cin 3164  (class class class)co 5874   ~Hchil 21515    +h cva 21516   0hc0v 21520    -h cmv 21521   CHcch 21525    +H cph 21527
This theorem is referenced by:  3oalem3  22259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-hilex 21595  ax-hfvadd 21596  ax-hvcom 21597  ax-hvass 21598  ax-hv0cl 21599  ax-hvaddid 21600  ax-hfvmul 21601  ax-hvmulid 21602  ax-hvdistr1 21604  ax-hvdistr2 21605  ax-hvmul0 21606
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-sub 9055  df-neg 9056  df-nn 9763  df-grpo 20874  df-ablo 20965  df-hvsub 21567  df-hlim 21568  df-sh 21802  df-ch 21817  df-shs 21903
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