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Theorem 3oalem2 23118
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 735 . . 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 736 . . . 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 23117 . . . . . 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 22493 . . . . . . . . . 10  |-  ( ( y  e.  ~H  /\  w  e.  ~H  /\  w  e.  ~H )  ->  (
y  +h  ( w  -h  w ) )  =  ( w  +h  ( y  -h  w
) ) )
983anidm23 1243 . . . . . . . . 9  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( y  +h  (
w  -h  w ) )  =  ( w  +h  ( y  -h  w ) ) )
10 hvsubid 22481 . . . . . . . . . . 11  |-  ( w  e.  ~H  ->  (
w  -h  w )  =  0h )
1110oveq2d 6056 . . . . . . . . . 10  |-  ( w  e.  ~H  ->  (
y  +h  ( w  -h  w ) )  =  ( y  +h 
0h ) )
12 ax-hvaddid 22460 . . . . . . . . . 10  |-  ( y  e.  ~H  ->  (
y  +h  0h )  =  y )
1311, 12sylan9eqr 2458 . . . . . . . . 9  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( y  +h  (
w  -h  w ) )  =  y )
149, 13eqtr3d 2438 . . . . . . . 8  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( w  +h  (
y  -h  w ) )  =  y )
1514ad2ant2l 727 . . . . . . 7  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( w  +h  ( y  -h  w
) )  =  y )
1615adantlr 696 . . . . . 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 740 . . . . . 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 2422 . . . . . . . . . . 11  |-  ( ( v  =  ( x  +h  y )  /\  v  =  ( z  +h  w ) )  -> 
( x  +h  y
)  =  ( z  +h  w ) )
2019oveq1d 6055 . . . . . . . . . 10  |-  ( ( v  =  ( x  +h  y )  /\  v  =  ( z  +h  w ) )  -> 
( ( x  +h  y )  -h  (
x  +h  w ) )  =  ( ( z  +h  w )  -h  ( x  +h  w ) ) )
2120ad2ant2l 727 . . . . . . . . 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 444 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  x  e.  ~H )
2322anim1i 552 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( x  e. 
~H  /\  w  e.  ~H ) )
24 hvsub4 22492 . . . . . . . . . . . . 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 457 . . . . . . . . . . . 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 22481 . . . . . . . . . . . . . 14  |-  ( x  e.  ~H  ->  (
x  -h  x )  =  0h )
2726ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( x  -h  x )  =  0h )
2827oveq1d 6055 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( ( x  -h  x )  +h  ( y  -h  w
) )  =  ( 0h  +h  ( y  -h  w ) ) )
29 hvsubcl 22473 . . . . . . . . . . . . . 14  |-  ( ( y  e.  ~H  /\  w  e.  ~H )  ->  ( y  -h  w
)  e.  ~H )
30 hvaddid2 22478 . . . . . . . . . . . . . 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 695 . . . . . . . . . . . 12  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( 0h  +h  ( y  -h  w
) )  =  ( y  -h  w ) )
3325, 28, 323eqtrd 2440 . . . . . . . . . . 11  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  w  e.  ~H )  ->  ( ( x  +h  y )  -h  ( x  +h  w
) )  =  ( y  -h  w ) )
3433ad2ant2rl 730 . . . . . . . . . 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 448 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( z  e.  ~H  /\  w  e. 
~H ) )
37 simpr 448 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  ~H  /\  w  e.  ~H )  ->  w  e.  ~H )
3837anim2i 553 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( x  e.  ~H  /\  w  e. 
~H ) )
39 hvsub4 22492 . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . 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 710 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( w  -h  w )  =  0h )
4241oveq2d 6056 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  -h  x )  +h  ( w  -h  w ) )  =  ( ( z  -h  x )  +h  0h ) )
43 hvsubcl 22473 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  ~H  /\  x  e.  ~H )  ->  ( z  -h  x
)  e.  ~H )
44 ax-hvaddid 22460 . . . . . . . . . . . . . . . 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 440 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ~H  /\  z  e.  ~H )  ->  ( ( z  -h  x )  +h  0h )  =  ( z  -h  x ) )
4746adantrr 698 . . . . . . . . . . . . 13  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  -h  x )  +h  0h )  =  ( z  -h  x
) )
4840, 42, 473eqtrd 2440 . . . . . . . . . . . 12  |-  ( ( x  e.  ~H  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( x  +h  w ) )  =  ( z  -h  x
) )
4948adantlr 696 . . . . . . . . . . 11  |-  ( ( ( x  e.  ~H  /\  y  e.  ~H )  /\  ( z  e.  ~H  /\  w  e.  ~H )
)  ->  ( (
z  +h  w )  -h  ( x  +h  w ) )  =  ( z  -h  x
) )
5049adantlr 696 . . . . . . . . . 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 2444 . . . . . . . 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 731 . . . . . . . . 9  |-  ( ( ( x  e.  B  /\  y  e.  R
)  /\  v  =  ( x  +h  y
) )  ->  x  e.  B )
54 simpll 731 . . . . . . . . 9  |-  ( ( ( z  e.  C  /\  w  e.  S
)  /\  v  =  ( z  +h  w
) )  ->  z  e.  C )
554chshii 22683 . . . . . . . . . . . 12  |-  C  e.  SH
563chshii 22683 . . . . . . . . . . . 12  |-  B  e.  SH
5755, 56shsvsi 22822 . . . . . . . . . . 11  |-  ( ( z  e.  C  /\  x  e.  B )  ->  ( z  -h  x
)  e.  ( C  +H  B ) )
5857ancoms 440 . . . . . . . . . 10  |-  ( ( x  e.  B  /\  z  e.  C )  ->  ( z  -h  x
)  e.  ( C  +H  B ) )
5956, 55shscomi 22818 . . . . . . . . . 10  |-  ( B  +H  C )  =  ( C  +H  B
)
6058, 59syl6eleqr 2495 . . . . . . . . 9  |-  ( ( x  e.  B  /\  z  e.  C )  ->  ( z  -h  x
)  e.  ( B  +H  C ) )
6153, 54, 60syl2an 464 . . . . . . . 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 2478 . . . . . . 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 732 . . . . . . . 8  |-  ( ( ( x  e.  B  /\  y  e.  R
)  /\  v  =  ( x  +h  y
) )  ->  y  e.  R )
64 simplr 732 . . . . . . . 8  |-  ( ( ( z  e.  C  /\  w  e.  S
)  /\  v  =  ( z  +h  w
) )  ->  w  e.  S )
655chshii 22683 . . . . . . . . 9  |-  R  e.  SH
666chshii 22683 . . . . . . . . 9  |-  S  e.  SH
6765, 66shsvsi 22822 . . . . . . . 8  |-  ( ( y  e.  R  /\  w  e.  S )  ->  ( y  -h  w
)  e.  ( R  +H  S ) )
6863, 64, 67syl2an 464 . . . . . . 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 3490 . . . . . . 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 646 . . . . . 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 22772 . . . . . . . 8  |-  ( B  +H  C )  e.  SH
7265, 66shscli 22772 . . . . . . . 8  |-  ( R  +H  S )  e.  SH
7371, 72shincli 22817 . . . . . . 7  |-  ( ( B  +H  C )  i^i  ( R  +H  S ) )  e.  SH
7466, 73shsvai 22819 . . . . . 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 643 . . . . 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 2479 . . . 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 3490 . . . 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 646 . . 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 22772 . . . . 5  |-  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S
) ) )  e.  SH
8065, 79shincli 22817 . . . 4  |-  ( R  i^i  ( S  +H  ( ( B  +H  C )  i^i  ( R  +H  S ) ) ) )  e.  SH
8156, 80shsvai 22819 . . 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 643 . 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 2464 . . 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 708 . 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 224 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    i^i cin 3279  (class class class)co 6040   ~Hchil 22375    +h cva 22376   0hc0v 22380    -h cmv 22381   CHcch 22385    +H cph 22387
This theorem is referenced by:  3oalem3  23119
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-hilex 22455  ax-hfvadd 22456  ax-hvcom 22457  ax-hvass 22458  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  ax-hvdistr1 22464  ax-hvdistr2 22465  ax-hvmul0 22466
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-ltxr 9081  df-sub 9249  df-neg 9250  df-nn 9957  df-grpo 21732  df-ablo 21823  df-hvsub 22427  df-hlim 22428  df-sh 22662  df-ch 22677  df-shs 22763
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