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Theorem recvguniqlem 10606
Description: Lemma for recvguniq 10607. Some of the rearrangements of the expressions. (Contributed by Jim Kingdon, 8-Aug-2021.)
Hypotheses
Ref Expression
recvguniqlem.f  |-  ( ph  ->  F : NN --> RR )
recvguniqlem.a  |-  ( ph  ->  A  e.  RR )
recvguniqlem.b  |-  ( ph  ->  B  e.  RR )
recvguniqlem.k  |-  ( ph  ->  K  e.  NN )
recvguniqlem.lt1  |-  ( ph  ->  A  <  ( ( F `  K )  +  ( ( A  -  B )  / 
2 ) ) )
recvguniqlem.lt2  |-  ( ph  ->  ( F `  K
)  <  ( B  +  ( ( A  -  B )  / 
2 ) ) )
Assertion
Ref Expression
recvguniqlem  |-  ( ph  -> F.  )

Proof of Theorem recvguniqlem
StepHypRef Expression
1 recvguniqlem.a . . 3  |-  ( ph  ->  A  e.  RR )
2 recvguniqlem.f . . . . 5  |-  ( ph  ->  F : NN --> RR )
3 recvguniqlem.k . . . . 5  |-  ( ph  ->  K  e.  NN )
42, 3ffvelrnd 5488 . . . 4  |-  ( ph  ->  ( F `  K
)  e.  RR )
5 recvguniqlem.b . . . . . 6  |-  ( ph  ->  B  e.  RR )
61, 5resubcld 8010 . . . . 5  |-  ( ph  ->  ( A  -  B
)  e.  RR )
76rehalfcld 8818 . . . 4  |-  ( ph  ->  ( ( A  -  B )  /  2
)  e.  RR )
84, 7readdcld 7667 . . 3  |-  ( ph  ->  ( ( F `  K )  +  ( ( A  -  B
)  /  2 ) )  e.  RR )
9 recvguniqlem.lt1 . . 3  |-  ( ph  ->  A  <  ( ( F `  K )  +  ( ( A  -  B )  / 
2 ) ) )
105, 7readdcld 7667 . . . . 5  |-  ( ph  ->  ( B  +  ( ( A  -  B
)  /  2 ) )  e.  RR )
11 recvguniqlem.lt2 . . . . 5  |-  ( ph  ->  ( F `  K
)  <  ( B  +  ( ( A  -  B )  / 
2 ) ) )
124, 10, 7, 11ltadd1dd 8184 . . . 4  |-  ( ph  ->  ( ( F `  K )  +  ( ( A  -  B
)  /  2 ) )  <  ( ( B  +  ( ( A  -  B )  /  2 ) )  +  ( ( A  -  B )  / 
2 ) ) )
135recnd 7666 . . . . . 6  |-  ( ph  ->  B  e.  CC )
147recnd 7666 . . . . . 6  |-  ( ph  ->  ( ( A  -  B )  /  2
)  e.  CC )
1513, 14, 14addassd 7660 . . . . 5  |-  ( ph  ->  ( ( B  +  ( ( A  -  B )  /  2
) )  +  ( ( A  -  B
)  /  2 ) )  =  ( B  +  ( ( ( A  -  B )  /  2 )  +  ( ( A  -  B )  /  2
) ) ) )
166recnd 7666 . . . . . . 7  |-  ( ph  ->  ( A  -  B
)  e.  CC )
17162halvesd 8817 . . . . . 6  |-  ( ph  ->  ( ( ( A  -  B )  / 
2 )  +  ( ( A  -  B
)  /  2 ) )  =  ( A  -  B ) )
1817oveq2d 5722 . . . . 5  |-  ( ph  ->  ( B  +  ( ( ( A  -  B )  /  2
)  +  ( ( A  -  B )  /  2 ) ) )  =  ( B  +  ( A  -  B ) ) )
191recnd 7666 . . . . . 6  |-  ( ph  ->  A  e.  CC )
2013, 19pncan3d 7947 . . . . 5  |-  ( ph  ->  ( B  +  ( A  -  B ) )  =  A )
2115, 18, 203eqtrd 2136 . . . 4  |-  ( ph  ->  ( ( B  +  ( ( A  -  B )  /  2
) )  +  ( ( A  -  B
)  /  2 ) )  =  A )
2212, 21breqtrd 3899 . . 3  |-  ( ph  ->  ( ( F `  K )  +  ( ( A  -  B
)  /  2 ) )  <  A )
231, 8, 1, 9, 22lttrd 7759 . 2  |-  ( ph  ->  A  <  A )
241ltnrd 7746 . 2  |-  ( ph  ->  -.  A  <  A
)
2523, 24pm2.21fal 1319 1  |-  ( ph  -> F.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F. wfal 1304    e. wcel 1448   class class class wbr 3875   -->wf 5055   ` cfv 5059  (class class class)co 5706   RRcr 7499    + caddc 7503    < clt 7672    - cmin 7804    / cdiv 8293   NNcn 8578   2c2 8629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-po 4156  df-iso 4157  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-2 8637
This theorem is referenced by:  recvguniq  10607
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