Users' Mathboxes Mathbox for Jim Kingdon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  apdifflemf Unicode version

Theorem apdifflemf 14335
Description: Lemma for apdiff 14337. Being apart from the point halfway between  Q and  R suffices for  A to be a different distance from  Q and from  R. (Contributed by Jim Kingdon, 18-May-2024.)
Hypotheses
Ref Expression
apdifflemf.a  |-  ( ph  ->  A  e.  RR )
apdifflemf.q  |-  ( ph  ->  Q  e.  QQ )
apdifflemf.r  |-  ( ph  ->  R  e.  QQ )
apdifflemf.qr  |-  ( ph  ->  Q  <  R )
apdifflemf.ap  |-  ( ph  ->  ( ( Q  +  R )  /  2
) #  A )
Assertion
Ref Expression
apdifflemf  |-  ( ph  ->  ( abs `  ( A  -  Q )
) #  ( abs `  ( A  -  R )
) )

Proof of Theorem apdifflemf
StepHypRef Expression
1 apdifflemf.a . . . . . . 7  |-  ( ph  ->  A  e.  RR )
21recnd 7960 . . . . . 6  |-  ( ph  ->  A  e.  CC )
3 apdifflemf.r . . . . . . 7  |-  ( ph  ->  R  e.  QQ )
4 qcn 9605 . . . . . . 7  |-  ( R  e.  QQ  ->  R  e.  CC )
53, 4syl 14 . . . . . 6  |-  ( ph  ->  R  e.  CC )
62, 5subcld 8242 . . . . 5  |-  ( ph  ->  ( A  -  R
)  e.  CC )
76adantr 276 . . . 4  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( A  -  R )  e.  CC )
87abscld 11156 . . 3  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( abs `  ( A  -  R ) )  e.  RR )
9 apdifflemf.q . . . . . . 7  |-  ( ph  ->  Q  e.  QQ )
10 qcn 9605 . . . . . . 7  |-  ( Q  e.  QQ  ->  Q  e.  CC )
119, 10syl 14 . . . . . 6  |-  ( ph  ->  Q  e.  CC )
122, 11subcld 8242 . . . . 5  |-  ( ph  ->  ( A  -  Q
)  e.  CC )
1312abscld 11156 . . . 4  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  e.  RR )
1413adantr 276 . . 3  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( abs `  ( A  -  Q ) )  e.  RR )
15 qre 9596 . . . . . . . . . 10  |-  ( Q  e.  QQ  ->  Q  e.  RR )
169, 15syl 14 . . . . . . . . 9  |-  ( ph  ->  Q  e.  RR )
1716adantr 276 . . . . . . . 8  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  Q  e.  RR )
181adantr 276 . . . . . . . 8  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  A  e.  RR )
19 qaddcl 9606 . . . . . . . . . . . . . 14  |-  ( ( Q  e.  QQ  /\  R  e.  QQ )  ->  ( Q  +  R
)  e.  QQ )
209, 3, 19syl2anc 411 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Q  +  R
)  e.  QQ )
21 qre 9596 . . . . . . . . . . . . 13  |-  ( ( Q  +  R )  e.  QQ  ->  ( Q  +  R )  e.  RR )
2220, 21syl 14 . . . . . . . . . . . 12  |-  ( ph  ->  ( Q  +  R
)  e.  RR )
2322rehalfcld 9136 . . . . . . . . . . 11  |-  ( ph  ->  ( ( Q  +  R )  /  2
)  e.  RR )
2423adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  (
( Q  +  R
)  /  2 )  e.  RR )
25 apdifflemf.qr . . . . . . . . . . . 12  |-  ( ph  ->  Q  <  R )
26 qre 9596 . . . . . . . . . . . . . 14  |-  ( R  e.  QQ  ->  R  e.  RR )
273, 26syl 14 . . . . . . . . . . . . 13  |-  ( ph  ->  R  e.  RR )
28 avglt1 9128 . . . . . . . . . . . . 13  |-  ( ( Q  e.  RR  /\  R  e.  RR )  ->  ( Q  <  R  <->  Q  <  ( ( Q  +  R )  / 
2 ) ) )
2916, 27, 28syl2anc 411 . . . . . . . . . . . 12  |-  ( ph  ->  ( Q  <  R  <->  Q  <  ( ( Q  +  R )  / 
2 ) ) )
3025, 29mpbid 147 . . . . . . . . . . 11  |-  ( ph  ->  Q  <  ( ( Q  +  R )  /  2 ) )
3130adantr 276 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  Q  <  ( ( Q  +  R )  /  2
) )
32 simpr 110 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  (
( Q  +  R
)  /  2 )  <  A )
3317, 24, 18, 31, 32lttrd 8057 . . . . . . . . 9  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  Q  <  A )
3417, 18, 33ltled 8050 . . . . . . . 8  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  Q  <_  A )
3517, 18, 34abssubge0d 11151 . . . . . . 7  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( abs `  ( A  -  Q ) )  =  ( A  -  Q
) )
3635oveq2d 5881 . . . . . 6  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  -  ( abs `  ( A  -  Q
) ) )  =  ( R  -  ( A  -  Q )
) )
375adantr 276 . . . . . . 7  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  R  e.  CC )
382adantr 276 . . . . . . 7  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  A  e.  CC )
3911adantr 276 . . . . . . 7  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  Q  e.  CC )
4037, 38, 39subsub3d 8272 . . . . . 6  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  -  ( A  -  Q ) )  =  ( ( R  +  Q )  -  A
) )
4137, 39addcomd 8082 . . . . . . 7  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  +  Q )  =  ( Q  +  R ) )
4241oveq1d 5880 . . . . . 6  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  (
( R  +  Q
)  -  A )  =  ( ( Q  +  R )  -  A ) )
4336, 40, 423eqtrd 2212 . . . . 5  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  -  ( abs `  ( A  -  Q
) ) )  =  ( ( Q  +  R )  -  A
) )
4422adantr 276 . . . . . . . . 9  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( Q  +  R )  e.  RR )
45 2rp 9627 . . . . . . . . . 10  |-  2  e.  RR+
4645a1i 9 . . . . . . . . 9  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  2  e.  RR+ )
4744, 18, 46ltdivmuld 9717 . . . . . . . 8  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  (
( ( Q  +  R )  /  2
)  <  A  <->  ( Q  +  R )  <  (
2  x.  A ) ) )
4832, 47mpbid 147 . . . . . . 7  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( Q  +  R )  <  ( 2  x.  A
) )
49382timesd 9132 . . . . . . 7  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  (
2  x.  A )  =  ( A  +  A ) )
5048, 49breqtrd 4024 . . . . . 6  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( Q  +  R )  <  ( A  +  A
) )
5144, 18, 18ltsubaddd 8472 . . . . . 6  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  (
( ( Q  +  R )  -  A
)  <  A  <->  ( Q  +  R )  <  ( A  +  A )
) )
5250, 51mpbird 167 . . . . 5  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  (
( Q  +  R
)  -  A )  <  A )
5343, 52eqbrtrd 4020 . . . 4  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  -  ( abs `  ( A  -  Q
) ) )  < 
A )
5425adantr 276 . . . . . . 7  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  Q  <  R )
5527adantr 276 . . . . . . . 8  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  R  e.  RR )
56 difrp 9661 . . . . . . . 8  |-  ( ( Q  e.  RR  /\  R  e.  RR )  ->  ( Q  <  R  <->  ( R  -  Q )  e.  RR+ ) )
5717, 55, 56syl2anc 411 . . . . . . 7  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( Q  <  R  <->  ( R  -  Q )  e.  RR+ ) )
5854, 57mpbid 147 . . . . . 6  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  -  Q )  e.  RR+ )
5918, 58ltaddrpd 9699 . . . . 5  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  A  <  ( A  +  ( R  -  Q ) ) )
6035oveq2d 5881 . . . . . 6  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  +  ( abs `  ( A  -  Q
) ) )  =  ( R  +  ( A  -  Q ) ) )
6137, 38, 39addsub12d 8265 . . . . . 6  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  +  ( A  -  Q ) )  =  ( A  +  ( R  -  Q ) ) )
6260, 61eqtrd 2208 . . . . 5  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  +  ( abs `  ( A  -  Q
) ) )  =  ( A  +  ( R  -  Q ) ) )
6359, 62breqtrrd 4026 . . . 4  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  A  <  ( R  +  ( abs `  ( A  -  Q ) ) ) )
6418, 55, 14absdifltd 11153 . . . 4  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  (
( abs `  ( A  -  R )
)  <  ( abs `  ( A  -  Q
) )  <->  ( ( R  -  ( abs `  ( A  -  Q
) ) )  < 
A  /\  A  <  ( R  +  ( abs `  ( A  -  Q
) ) ) ) ) )
6553, 63, 64mpbir2and 944 . . 3  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( abs `  ( A  -  R ) )  < 
( abs `  ( A  -  Q )
) )
668, 14, 65gtapd 8568 . 2  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( abs `  ( A  -  Q ) ) #  ( abs `  ( A  -  R ) ) )
6713adantr 276 . . 3  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( abs `  ( A  -  Q
) )  e.  RR )
686adantr 276 . . . 4  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( A  -  R )  e.  CC )
6968abscld 11156 . . 3  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( abs `  ( A  -  R
) )  e.  RR )
7011, 5, 2subsubd 8270 . . . . . . 7  |-  ( ph  ->  ( Q  -  ( R  -  A )
)  =  ( ( Q  -  R )  +  A ) )
7116, 27sublt0d 8501 . . . . . . . . 9  |-  ( ph  ->  ( ( Q  -  R )  <  0  <->  Q  <  R ) )
7225, 71mpbird 167 . . . . . . . 8  |-  ( ph  ->  ( Q  -  R
)  <  0 )
7316, 27resubcld 8312 . . . . . . . . 9  |-  ( ph  ->  ( Q  -  R
)  e.  RR )
74 ltaddnegr 8356 . . . . . . . . 9  |-  ( ( ( Q  -  R
)  e.  RR  /\  A  e.  RR )  ->  ( ( Q  -  R )  <  0  <->  ( ( Q  -  R
)  +  A )  <  A ) )
7573, 1, 74syl2anc 411 . . . . . . . 8  |-  ( ph  ->  ( ( Q  -  R )  <  0  <->  ( ( Q  -  R
)  +  A )  <  A ) )
7672, 75mpbid 147 . . . . . . 7  |-  ( ph  ->  ( ( Q  -  R )  +  A
)  <  A )
7770, 76eqbrtrd 4020 . . . . . 6  |-  ( ph  ->  ( Q  -  ( R  -  A )
)  <  A )
7877adantr 276 . . . . 5  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( Q  -  ( R  -  A ) )  < 
A )
791adantr 276 . . . . . . . 8  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  A  e.  RR )
8022adantr 276 . . . . . . . 8  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( Q  +  R )  e.  RR )
81 simpr 110 . . . . . . . 8  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  A  <  ( ( Q  +  R
)  /  2 ) )
8279, 79, 80, 81, 81lt2halvesd 9137 . . . . . . 7  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( A  +  A )  <  ( Q  +  R )
)
8379, 79, 80ltaddsub2d 8477 . . . . . . 7  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( ( A  +  A )  <  ( Q  +  R
)  <->  A  <  ( ( Q  +  R )  -  A ) ) )
8482, 83mpbid 147 . . . . . 6  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  A  <  ( ( Q  +  R
)  -  A ) )
8511adantr 276 . . . . . . 7  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  Q  e.  CC )
865adantr 276 . . . . . . 7  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  R  e.  CC )
872adantr 276 . . . . . . 7  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  A  e.  CC )
8885, 86, 87addsubassd 8262 . . . . . 6  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( ( Q  +  R )  -  A )  =  ( Q  +  ( R  -  A ) ) )
8984, 88breqtrd 4024 . . . . 5  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  A  <  ( Q  +  ( R  -  A ) ) )
9016adantr 276 . . . . . 6  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  Q  e.  RR )
9127adantr 276 . . . . . . 7  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  R  e.  RR )
9291, 79resubcld 8312 . . . . . 6  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( R  -  A )  e.  RR )
9379, 90, 92absdifltd 11153 . . . . 5  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( ( abs `  ( A  -  Q ) )  < 
( R  -  A
)  <->  ( ( Q  -  ( R  -  A ) )  < 
A  /\  A  <  ( Q  +  ( R  -  A ) ) ) ) )
9478, 89, 93mpbir2and 944 . . . 4  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( abs `  ( A  -  Q
) )  <  ( R  -  A )
)
9523adantr 276 . . . . . . 7  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( ( Q  +  R )  /  2 )  e.  RR )
96 avglt2 9129 . . . . . . . . . 10  |-  ( ( Q  e.  RR  /\  R  e.  RR )  ->  ( Q  <  R  <->  ( ( Q  +  R
)  /  2 )  <  R ) )
9716, 27, 96syl2anc 411 . . . . . . . . 9  |-  ( ph  ->  ( Q  <  R  <->  ( ( Q  +  R
)  /  2 )  <  R ) )
9825, 97mpbid 147 . . . . . . . 8  |-  ( ph  ->  ( ( Q  +  R )  /  2
)  <  R )
9998adantr 276 . . . . . . 7  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( ( Q  +  R )  /  2 )  < 
R )
10079, 95, 91, 81, 99lttrd 8057 . . . . . 6  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  A  <  R )
10179, 91, 100ltled 8050 . . . . 5  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  A  <_  R )
10279, 91, 101abssuble0d 11152 . . . 4  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( abs `  ( A  -  R
) )  =  ( R  -  A ) )
10394, 102breqtrrd 4026 . . 3  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( abs `  ( A  -  Q
) )  <  ( abs `  ( A  -  R ) ) )
10467, 69, 103ltapd 8569 . 2  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( abs `  ( A  -  Q
) ) #  ( abs `  ( A  -  R
) ) )
105 apdifflemf.ap . . 3  |-  ( ph  ->  ( ( Q  +  R )  /  2
) #  A )
106 reaplt 8519 . . . 4  |-  ( ( ( ( Q  +  R )  /  2
)  e.  RR  /\  A  e.  RR )  ->  ( ( ( Q  +  R )  / 
2 ) #  A  <->  ( (
( Q  +  R
)  /  2 )  <  A  \/  A  <  ( ( Q  +  R )  /  2
) ) ) )
10723, 1, 106syl2anc 411 . . 3  |-  ( ph  ->  ( ( ( Q  +  R )  / 
2 ) #  A  <->  ( (
( Q  +  R
)  /  2 )  <  A  \/  A  <  ( ( Q  +  R )  /  2
) ) ) )
108105, 107mpbid 147 . 2  |-  ( ph  ->  ( ( ( Q  +  R )  / 
2 )  <  A  \/  A  <  ( ( Q  +  R )  /  2 ) ) )
10966, 104, 108mpjaodan 798 1  |-  ( ph  ->  ( abs `  ( A  -  Q )
) #  ( abs `  ( A  -  R )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708    e. wcel 2146   class class class wbr 3998   ` cfv 5208  (class class class)co 5865   CCcc 7784   RRcr 7785   0cc0 7786    + caddc 7789    x. cmul 7791    < clt 7966    - cmin 8102   # cap 8512    / cdiv 8601   2c2 8941   QQcq 9590   RR+crp 9622   abscabs 10972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904  ax-arch 7905  ax-caucvg 7906
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-n0 9148  df-z 9225  df-uz 9500  df-q 9591  df-rp 9623  df-seqfrec 10414  df-exp 10488  df-cj 10817  df-re 10818  df-im 10819  df-rsqrt 10973  df-abs 10974
This theorem is referenced by:  apdiff  14337
  Copyright terms: Public domain W3C validator