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Theorem recvguniqlem 11020
Description: Lemma for recvguniq 11021. 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, 3ffvelcdmd 5667 . . . 4  |-  ( ph  ->  ( F `  K
)  e.  RR )
5 recvguniqlem.b . . . . . 6  |-  ( ph  ->  B  e.  RR )
61, 5resubcld 8355 . . . . 5  |-  ( ph  ->  ( A  -  B
)  e.  RR )
76rehalfcld 9182 . . . 4  |-  ( ph  ->  ( ( A  -  B )  /  2
)  e.  RR )
84, 7readdcld 8004 . . 3  |-  ( ph  ->  ( ( F `  K )  +  ( ( A  -  B
)  /  2 ) )  e.  RR )
9 recvguniqlem.lt1 . . 3  |-  ( ph  ->  A  <  ( ( F `  K )  +  ( ( A  -  B )  / 
2 ) ) )
105, 7readdcld 8004 . . . . 5  |-  ( ph  ->  ( B  +  ( ( A  -  B
)  /  2 ) )  e.  RR )
11 recvguniqlem.lt2 . . . . 5  |-  ( ph  ->  ( F `  K
)  <  ( B  +  ( ( A  -  B )  / 
2 ) ) )
124, 10, 7, 11ltadd1dd 8530 . . . 4  |-  ( ph  ->  ( ( F `  K )  +  ( ( A  -  B
)  /  2 ) )  <  ( ( B  +  ( ( A  -  B )  /  2 ) )  +  ( ( A  -  B )  / 
2 ) ) )
135recnd 8003 . . . . . 6  |-  ( ph  ->  B  e.  CC )
147recnd 8003 . . . . . 6  |-  ( ph  ->  ( ( A  -  B )  /  2
)  e.  CC )
1513, 14, 14addassd 7997 . . . . 5  |-  ( ph  ->  ( ( B  +  ( ( A  -  B )  /  2
) )  +  ( ( A  -  B
)  /  2 ) )  =  ( B  +  ( ( ( A  -  B )  /  2 )  +  ( ( A  -  B )  /  2
) ) ) )
166recnd 8003 . . . . . . 7  |-  ( ph  ->  ( A  -  B
)  e.  CC )
17162halvesd 9181 . . . . . 6  |-  ( ph  ->  ( ( ( A  -  B )  / 
2 )  +  ( ( A  -  B
)  /  2 ) )  =  ( A  -  B ) )
1817oveq2d 5906 . . . . 5  |-  ( ph  ->  ( B  +  ( ( ( A  -  B )  /  2
)  +  ( ( A  -  B )  /  2 ) ) )  =  ( B  +  ( A  -  B ) ) )
191recnd 8003 . . . . . 6  |-  ( ph  ->  A  e.  CC )
2013, 19pncan3d 8288 . . . . 5  |-  ( ph  ->  ( B  +  ( A  -  B ) )  =  A )
2115, 18, 203eqtrd 2225 . . . 4  |-  ( ph  ->  ( ( B  +  ( ( A  -  B )  /  2
) )  +  ( ( A  -  B
)  /  2 ) )  =  A )
2212, 21breqtrd 4043 . . 3  |-  ( ph  ->  ( ( F `  K )  +  ( ( A  -  B
)  /  2 ) )  <  A )
231, 8, 1, 9, 22lttrd 8100 . 2  |-  ( ph  ->  A  <  A )
241ltnrd 8086 . 2  |-  ( ph  ->  -.  A  <  A
)
2523, 24pm2.21fal 1383 1  |-  ( ph  -> F.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4   F. wfal 1368    e. wcel 2159   class class class wbr 4017   -->wf 5226   ` cfv 5230  (class class class)co 5890   RRcr 7827    + caddc 7831    < clt 8009    - cmin 8145    / cdiv 8646   NNcn 8936   2c2 8987
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-mulrcl 7927  ax-addcom 7928  ax-mulcom 7929  ax-addass 7930  ax-mulass 7931  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-1rid 7935  ax-0id 7936  ax-rnegex 7937  ax-precex 7938  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-apti 7943  ax-pre-ltadd 7944  ax-pre-mulgt0 7945  ax-pre-mulext 7946
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-br 4018  df-opab 4079  df-id 4307  df-po 4310  df-iso 4311  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-fv 5238  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-reap 8549  df-ap 8556  df-div 8647  df-2 8995
This theorem is referenced by:  recvguniq  11021
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