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Theorem apdifflemf 14450
Description: Lemma for apdiff 14452. 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 7976 . . . . . 6  |-  ( ph  ->  A  e.  CC )
3 apdifflemf.r . . . . . . 7  |-  ( ph  ->  R  e.  QQ )
4 qcn 9623 . . . . . . 7  |-  ( R  e.  QQ  ->  R  e.  CC )
53, 4syl 14 . . . . . 6  |-  ( ph  ->  R  e.  CC )
62, 5subcld 8258 . . . . 5  |-  ( ph  ->  ( A  -  R
)  e.  CC )
76adantr 276 . . . 4  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( A  -  R )  e.  CC )
87abscld 11174 . . 3  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( abs `  ( A  -  R ) )  e.  RR )
9 apdifflemf.q . . . . . . 7  |-  ( ph  ->  Q  e.  QQ )
10 qcn 9623 . . . . . . 7  |-  ( Q  e.  QQ  ->  Q  e.  CC )
119, 10syl 14 . . . . . 6  |-  ( ph  ->  Q  e.  CC )
122, 11subcld 8258 . . . . 5  |-  ( ph  ->  ( A  -  Q
)  e.  CC )
1312abscld 11174 . . . 4  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  e.  RR )
1413adantr 276 . . 3  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( abs `  ( A  -  Q ) )  e.  RR )
15 qre 9614 . . . . . . . . . 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 9624 . . . . . . . . . . . . . 14  |-  ( ( Q  e.  QQ  /\  R  e.  QQ )  ->  ( Q  +  R
)  e.  QQ )
209, 3, 19syl2anc 411 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Q  +  R
)  e.  QQ )
21 qre 9614 . . . . . . . . . . . . 13  |-  ( ( Q  +  R )  e.  QQ  ->  ( Q  +  R )  e.  RR )
2220, 21syl 14 . . . . . . . . . . . 12  |-  ( ph  ->  ( Q  +  R
)  e.  RR )
2322rehalfcld 9154 . . . . . . . . . . 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 9614 . . . . . . . . . . . . . 14  |-  ( R  e.  QQ  ->  R  e.  RR )
273, 26syl 14 . . . . . . . . . . . . 13  |-  ( ph  ->  R  e.  RR )
28 avglt1 9146 . . . . . . . . . . . . 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 8073 . . . . . . . . 9  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  Q  <  A )
3417, 18, 33ltled 8066 . . . . . . . 8  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  Q  <_  A )
3517, 18, 34abssubge0d 11169 . . . . . . 7  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( abs `  ( A  -  Q ) )  =  ( A  -  Q
) )
3635oveq2d 5885 . . . . . 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 8288 . . . . . 6  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  -  ( A  -  Q ) )  =  ( ( R  +  Q )  -  A
) )
4137, 39addcomd 8098 . . . . . . 7  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  +  Q )  =  ( Q  +  R ) )
4241oveq1d 5884 . . . . . 6  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  (
( R  +  Q
)  -  A )  =  ( ( Q  +  R )  -  A ) )
4336, 40, 423eqtrd 2214 . . . . 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 9645 . . . . . . . . . 10  |-  2  e.  RR+
4645a1i 9 . . . . . . . . 9  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  2  e.  RR+ )
4744, 18, 46ltdivmuld 9735 . . . . . . . 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 9150 . . . . . . 7  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  (
2  x.  A )  =  ( A  +  A ) )
5048, 49breqtrd 4026 . . . . . 6  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( Q  +  R )  <  ( A  +  A
) )
5144, 18, 18ltsubaddd 8488 . . . . . 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 4022 . . . 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 9679 . . . . . . . 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 9717 . . . . 5  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  A  <  ( A  +  ( R  -  Q ) ) )
6035oveq2d 5885 . . . . . 6  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  +  ( abs `  ( A  -  Q
) ) )  =  ( R  +  ( A  -  Q ) ) )
6137, 38, 39addsub12d 8281 . . . . . 6  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  +  ( A  -  Q ) )  =  ( A  +  ( R  -  Q ) ) )
6260, 61eqtrd 2210 . . . . 5  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  +  ( abs `  ( A  -  Q
) ) )  =  ( A  +  ( R  -  Q ) ) )
6359, 62breqtrrd 4028 . . . 4  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  A  <  ( R  +  ( abs `  ( A  -  Q ) ) ) )
6418, 55, 14absdifltd 11171 . . . 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 8584 . 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 11174 . . 3  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( abs `  ( A  -  R
) )  e.  RR )
7011, 5, 2subsubd 8286 . . . . . . 7  |-  ( ph  ->  ( Q  -  ( R  -  A )
)  =  ( ( Q  -  R )  +  A ) )
7116, 27sublt0d 8517 . . . . . . . . 9  |-  ( ph  ->  ( ( Q  -  R )  <  0  <->  Q  <  R ) )
7225, 71mpbird 167 . . . . . . . 8  |-  ( ph  ->  ( Q  -  R
)  <  0 )
7316, 27resubcld 8328 . . . . . . . . 9  |-  ( ph  ->  ( Q  -  R
)  e.  RR )
74 ltaddnegr 8372 . . . . . . . . 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 4022 . . . . . 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 9155 . . . . . . 7  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( A  +  A )  <  ( Q  +  R )
)
8379, 79, 80ltaddsub2d 8493 . . . . . . 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 8278 . . . . . 6  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( ( Q  +  R )  -  A )  =  ( Q  +  ( R  -  A ) ) )
8984, 88breqtrd 4026 . . . . 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 8328 . . . . . 6  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( R  -  A )  e.  RR )
9379, 90, 92absdifltd 11171 . . . . 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 9147 . . . . . . . . . 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 8073 . . . . . 6  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  A  <  R )
10179, 91, 100ltled 8066 . . . . 5  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  A  <_  R )
10279, 91, 101abssuble0d 11170 . . . 4  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( abs `  ( A  -  R
) )  =  ( R  -  A ) )
10394, 102breqtrrd 4028 . . 3  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( abs `  ( A  -  Q
) )  <  ( abs `  ( A  -  R ) ) )
10467, 69, 103ltapd 8585 . 2  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( abs `  ( A  -  Q
) ) #  ( abs `  ( A  -  R
) ) )
105 apdifflemf.ap . . 3  |-  ( ph  ->  ( ( Q  +  R )  /  2
) #  A )
106 reaplt 8535 . . . 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 2148   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802    + caddc 7805    x. cmul 7807    < clt 7982    - cmin 8118   # cap 8528    / cdiv 8618   2c2 8959   QQcq 9608   RR+crp 9640   abscabs 10990
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
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 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992
This theorem is referenced by:  apdiff  14452
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