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Theorem apdifflemf 13325
Description: Lemma for apdiff 13327. 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 7806 . . . . . 6  |-  ( ph  ->  A  e.  CC )
3 apdifflemf.r . . . . . . 7  |-  ( ph  ->  R  e.  QQ )
4 qcn 9438 . . . . . . 7  |-  ( R  e.  QQ  ->  R  e.  CC )
53, 4syl 14 . . . . . 6  |-  ( ph  ->  R  e.  CC )
62, 5subcld 8085 . . . . 5  |-  ( ph  ->  ( A  -  R
)  e.  CC )
76adantr 274 . . . 4  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( A  -  R )  e.  CC )
87abscld 10965 . . 3  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( abs `  ( A  -  R ) )  e.  RR )
9 apdifflemf.q . . . . . . 7  |-  ( ph  ->  Q  e.  QQ )
10 qcn 9438 . . . . . . 7  |-  ( Q  e.  QQ  ->  Q  e.  CC )
119, 10syl 14 . . . . . 6  |-  ( ph  ->  Q  e.  CC )
122, 11subcld 8085 . . . . 5  |-  ( ph  ->  ( A  -  Q
)  e.  CC )
1312abscld 10965 . . . 4  |-  ( ph  ->  ( abs `  ( A  -  Q )
)  e.  RR )
1413adantr 274 . . 3  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( abs `  ( A  -  Q ) )  e.  RR )
15 qre 9429 . . . . . . . . . 10  |-  ( Q  e.  QQ  ->  Q  e.  RR )
169, 15syl 14 . . . . . . . . 9  |-  ( ph  ->  Q  e.  RR )
1716adantr 274 . . . . . . . 8  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  Q  e.  RR )
181adantr 274 . . . . . . . 8  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  A  e.  RR )
19 qaddcl 9439 . . . . . . . . . . . . . 14  |-  ( ( Q  e.  QQ  /\  R  e.  QQ )  ->  ( Q  +  R
)  e.  QQ )
209, 3, 19syl2anc 408 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Q  +  R
)  e.  QQ )
21 qre 9429 . . . . . . . . . . . . 13  |-  ( ( Q  +  R )  e.  QQ  ->  ( Q  +  R )  e.  RR )
2220, 21syl 14 . . . . . . . . . . . 12  |-  ( ph  ->  ( Q  +  R
)  e.  RR )
2322rehalfcld 8978 . . . . . . . . . . 11  |-  ( ph  ->  ( ( Q  +  R )  /  2
)  e.  RR )
2423adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  (
( Q  +  R
)  /  2 )  e.  RR )
25 apdifflemf.qr . . . . . . . . . . . 12  |-  ( ph  ->  Q  <  R )
26 qre 9429 . . . . . . . . . . . . . 14  |-  ( R  e.  QQ  ->  R  e.  RR )
273, 26syl 14 . . . . . . . . . . . . 13  |-  ( ph  ->  R  e.  RR )
28 avglt1 8970 . . . . . . . . . . . . 13  |-  ( ( Q  e.  RR  /\  R  e.  RR )  ->  ( Q  <  R  <->  Q  <  ( ( Q  +  R )  / 
2 ) ) )
2916, 27, 28syl2anc 408 . . . . . . . . . . . 12  |-  ( ph  ->  ( Q  <  R  <->  Q  <  ( ( Q  +  R )  / 
2 ) ) )
3025, 29mpbid 146 . . . . . . . . . . 11  |-  ( ph  ->  Q  <  ( ( Q  +  R )  /  2 ) )
3130adantr 274 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  Q  <  ( ( Q  +  R )  /  2
) )
32 simpr 109 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  (
( Q  +  R
)  /  2 )  <  A )
3317, 24, 18, 31, 32lttrd 7900 . . . . . . . . 9  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  Q  <  A )
3417, 18, 33ltled 7893 . . . . . . . 8  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  Q  <_  A )
3517, 18, 34abssubge0d 10960 . . . . . . 7  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( abs `  ( A  -  Q ) )  =  ( A  -  Q
) )
3635oveq2d 5790 . . . . . 6  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  -  ( abs `  ( A  -  Q
) ) )  =  ( R  -  ( A  -  Q )
) )
375adantr 274 . . . . . . 7  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  R  e.  CC )
382adantr 274 . . . . . . 7  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  A  e.  CC )
3911adantr 274 . . . . . . 7  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  Q  e.  CC )
4037, 38, 39subsub3d 8115 . . . . . 6  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  -  ( A  -  Q ) )  =  ( ( R  +  Q )  -  A
) )
4137, 39addcomd 7925 . . . . . . 7  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  +  Q )  =  ( Q  +  R ) )
4241oveq1d 5789 . . . . . 6  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  (
( R  +  Q
)  -  A )  =  ( ( Q  +  R )  -  A ) )
4336, 40, 423eqtrd 2176 . . . . 5  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  -  ( abs `  ( A  -  Q
) ) )  =  ( ( Q  +  R )  -  A
) )
4422adantr 274 . . . . . . . . 9  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( Q  +  R )  e.  RR )
45 2rp 9458 . . . . . . . . . 10  |-  2  e.  RR+
4645a1i 9 . . . . . . . . 9  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  2  e.  RR+ )
4744, 18, 46ltdivmuld 9547 . . . . . . . 8  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  (
( ( Q  +  R )  /  2
)  <  A  <->  ( Q  +  R )  <  (
2  x.  A ) ) )
4832, 47mpbid 146 . . . . . . 7  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( Q  +  R )  <  ( 2  x.  A
) )
49382timesd 8974 . . . . . . 7  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  (
2  x.  A )  =  ( A  +  A ) )
5048, 49breqtrd 3954 . . . . . 6  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( Q  +  R )  <  ( A  +  A
) )
5144, 18, 18ltsubaddd 8315 . . . . . 6  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  (
( ( Q  +  R )  -  A
)  <  A  <->  ( Q  +  R )  <  ( A  +  A )
) )
5250, 51mpbird 166 . . . . 5  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  (
( Q  +  R
)  -  A )  <  A )
5343, 52eqbrtrd 3950 . . . 4  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  -  ( abs `  ( A  -  Q
) ) )  < 
A )
5425adantr 274 . . . . . . 7  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  Q  <  R )
5527adantr 274 . . . . . . . 8  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  R  e.  RR )
56 difrp 9492 . . . . . . . 8  |-  ( ( Q  e.  RR  /\  R  e.  RR )  ->  ( Q  <  R  <->  ( R  -  Q )  e.  RR+ ) )
5717, 55, 56syl2anc 408 . . . . . . 7  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( Q  <  R  <->  ( R  -  Q )  e.  RR+ ) )
5854, 57mpbid 146 . . . . . 6  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  -  Q )  e.  RR+ )
5918, 58ltaddrpd 9529 . . . . 5  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  A  <  ( A  +  ( R  -  Q ) ) )
6035oveq2d 5790 . . . . . 6  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  +  ( abs `  ( A  -  Q
) ) )  =  ( R  +  ( A  -  Q ) ) )
6137, 38, 39addsub12d 8108 . . . . . 6  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  +  ( A  -  Q ) )  =  ( A  +  ( R  -  Q ) ) )
6260, 61eqtrd 2172 . . . . 5  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( R  +  ( abs `  ( A  -  Q
) ) )  =  ( A  +  ( R  -  Q ) ) )
6359, 62breqtrrd 3956 . . . 4  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  A  <  ( R  +  ( abs `  ( A  -  Q ) ) ) )
6418, 55, 14absdifltd 10962 . . . 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 928 . . 3  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( abs `  ( A  -  R ) )  < 
( abs `  ( A  -  Q )
) )
668, 14, 65gtapd 8411 . 2  |-  ( (
ph  /\  ( ( Q  +  R )  /  2 )  < 
A )  ->  ( abs `  ( A  -  Q ) ) #  ( abs `  ( A  -  R ) ) )
6713adantr 274 . . 3  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( abs `  ( A  -  Q
) )  e.  RR )
686adantr 274 . . . 4  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( A  -  R )  e.  CC )
6968abscld 10965 . . 3  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( abs `  ( A  -  R
) )  e.  RR )
7011, 5, 2subsubd 8113 . . . . . . 7  |-  ( ph  ->  ( Q  -  ( R  -  A )
)  =  ( ( Q  -  R )  +  A ) )
7116, 27sublt0d 8344 . . . . . . . . 9  |-  ( ph  ->  ( ( Q  -  R )  <  0  <->  Q  <  R ) )
7225, 71mpbird 166 . . . . . . . 8  |-  ( ph  ->  ( Q  -  R
)  <  0 )
7316, 27resubcld 8155 . . . . . . . . 9  |-  ( ph  ->  ( Q  -  R
)  e.  RR )
74 ltaddnegr 8199 . . . . . . . . 9  |-  ( ( ( Q  -  R
)  e.  RR  /\  A  e.  RR )  ->  ( ( Q  -  R )  <  0  <->  ( ( Q  -  R
)  +  A )  <  A ) )
7573, 1, 74syl2anc 408 . . . . . . . 8  |-  ( ph  ->  ( ( Q  -  R )  <  0  <->  ( ( Q  -  R
)  +  A )  <  A ) )
7672, 75mpbid 146 . . . . . . 7  |-  ( ph  ->  ( ( Q  -  R )  +  A
)  <  A )
7770, 76eqbrtrd 3950 . . . . . 6  |-  ( ph  ->  ( Q  -  ( R  -  A )
)  <  A )
7877adantr 274 . . . . 5  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( Q  -  ( R  -  A ) )  < 
A )
791adantr 274 . . . . . . . 8  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  A  e.  RR )
8022adantr 274 . . . . . . . 8  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( Q  +  R )  e.  RR )
81 simpr 109 . . . . . . . 8  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  A  <  ( ( Q  +  R
)  /  2 ) )
8279, 79, 80, 81, 81lt2halvesd 8979 . . . . . . 7  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( A  +  A )  <  ( Q  +  R )
)
8379, 79, 80ltaddsub2d 8320 . . . . . . 7  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( ( A  +  A )  <  ( Q  +  R
)  <->  A  <  ( ( Q  +  R )  -  A ) ) )
8482, 83mpbid 146 . . . . . 6  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  A  <  ( ( Q  +  R
)  -  A ) )
8511adantr 274 . . . . . . 7  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  Q  e.  CC )
865adantr 274 . . . . . . 7  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  R  e.  CC )
872adantr 274 . . . . . . 7  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  A  e.  CC )
8885, 86, 87addsubassd 8105 . . . . . 6  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( ( Q  +  R )  -  A )  =  ( Q  +  ( R  -  A ) ) )
8984, 88breqtrd 3954 . . . . 5  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  A  <  ( Q  +  ( R  -  A ) ) )
9016adantr 274 . . . . . 6  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  Q  e.  RR )
9127adantr 274 . . . . . . 7  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  R  e.  RR )
9291, 79resubcld 8155 . . . . . 6  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( R  -  A )  e.  RR )
9379, 90, 92absdifltd 10962 . . . . 5  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( ( abs `  ( A  -  Q ) )  < 
( R  -  A
)  <->  ( ( Q  -  ( R  -  A ) )  < 
A  /\  A  <  ( Q  +  ( R  -  A ) ) ) ) )
9478, 89, 93mpbir2and 928 . . . 4  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( abs `  ( A  -  Q
) )  <  ( R  -  A )
)
9523adantr 274 . . . . . . 7  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( ( Q  +  R )  /  2 )  e.  RR )
96 avglt2 8971 . . . . . . . . . 10  |-  ( ( Q  e.  RR  /\  R  e.  RR )  ->  ( Q  <  R  <->  ( ( Q  +  R
)  /  2 )  <  R ) )
9716, 27, 96syl2anc 408 . . . . . . . . 9  |-  ( ph  ->  ( Q  <  R  <->  ( ( Q  +  R
)  /  2 )  <  R ) )
9825, 97mpbid 146 . . . . . . . 8  |-  ( ph  ->  ( ( Q  +  R )  /  2
)  <  R )
9998adantr 274 . . . . . . 7  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( ( Q  +  R )  /  2 )  < 
R )
10079, 95, 91, 81, 99lttrd 7900 . . . . . 6  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  A  <  R )
10179, 91, 100ltled 7893 . . . . 5  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  A  <_  R )
10279, 91, 101abssuble0d 10961 . . . 4  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( abs `  ( A  -  R
) )  =  ( R  -  A ) )
10394, 102breqtrrd 3956 . . 3  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( abs `  ( A  -  Q
) )  <  ( abs `  ( A  -  R ) ) )
10467, 69, 103ltapd 8412 . 2  |-  ( (
ph  /\  A  <  ( ( Q  +  R
)  /  2 ) )  ->  ( abs `  ( A  -  Q
) ) #  ( abs `  ( A  -  R
) ) )
105 apdifflemf.ap . . 3  |-  ( ph  ->  ( ( Q  +  R )  /  2
) #  A )
106 reaplt 8362 . . . 4  |-  ( ( ( ( Q  +  R )  /  2
)  e.  RR  /\  A  e.  RR )  ->  ( ( ( Q  +  R )  / 
2 ) #  A  <->  ( (
( Q  +  R
)  /  2 )  <  A  \/  A  <  ( ( Q  +  R )  /  2
) ) ) )
10723, 1, 106syl2anc 408 . . 3  |-  ( ph  ->  ( ( ( Q  +  R )  / 
2 ) #  A  <->  ( (
( Q  +  R
)  /  2 )  <  A  \/  A  <  ( ( Q  +  R )  /  2
) ) ) )
108105, 107mpbid 146 . 2  |-  ( ph  ->  ( ( ( Q  +  R )  / 
2 )  <  A  \/  A  <  ( ( Q  +  R )  /  2 ) ) )
10966, 104, 108mpjaodan 787 1  |-  ( ph  ->  ( abs `  ( A  -  Q )
) #  ( abs `  ( A  -  R )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    e. wcel 1480   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7630   RRcr 7631   0cc0 7632    + caddc 7635    x. cmul 7637    < clt 7812    - cmin 7945   # cap 8355    / cdiv 8444   2c2 8783   QQcq 9423   RR+crp 9453   abscabs 10781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750  ax-arch 7751  ax-caucvg 7752
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-2 8791  df-3 8792  df-4 8793  df-n0 8990  df-z 9067  df-uz 9339  df-q 9424  df-rp 9454  df-seqfrec 10231  df-exp 10305  df-cj 10626  df-re 10627  df-im 10628  df-rsqrt 10782  df-abs 10783
This theorem is referenced by:  apdiff  13327
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