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Theorem resqrexlemp1rp 10077
Description: Lemma for resqrex 10097. Applying the recursion rule yields a positive real (expressed in a way that will help apply iseqfcl 9571 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.)
Hypotheses
Ref Expression
resqrexlemex.seq  |-  F  =  seq 1 ( ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y ) )  /  2 ) ) ,  ( NN  X.  { ( 1  +  A ) } ) ,  RR+ )
resqrexlemex.a  |-  ( ph  ->  A  e.  RR )
resqrexlemex.agt0  |-  ( ph  ->  0  <_  A )
Assertion
Ref Expression
resqrexlemp1rp  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  ( B ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
) )  /  2
) ) C )  e.  RR+ )
Distinct variable groups:    y, A, z    ph, y, z    y, B, z    y, C, z
Allowed substitution hints:    F( y, z)

Proof of Theorem resqrexlemp1rp
StepHypRef Expression
1 eqidd 2084 . . 3  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  (
y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y ) )  /  2 ) )  =  ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
) )  /  2
) ) )
2 id 19 . . . . . 6  |-  ( y  =  B  ->  y  =  B )
3 oveq2 5572 . . . . . 6  |-  ( y  =  B  ->  ( A  /  y )  =  ( A  /  B
) )
42, 3oveq12d 5582 . . . . 5  |-  ( y  =  B  ->  (
y  +  ( A  /  y ) )  =  ( B  +  ( A  /  B
) ) )
54oveq1d 5579 . . . 4  |-  ( y  =  B  ->  (
( y  +  ( A  /  y ) )  /  2 )  =  ( ( B  +  ( A  /  B ) )  / 
2 ) )
65ad2antrl 474 . . 3  |-  ( ( ( ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  /\  ( y  =  B  /\  z  =  C ) )  ->  (
( y  +  ( A  /  y ) )  /  2 )  =  ( ( B  +  ( A  /  B ) )  / 
2 ) )
7 simprl 498 . . 3  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  B  e.  RR+ )
8 simprr 499 . . 3  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  C  e.  RR+ )
97rpred 8890 . . . . 5  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  B  e.  RR )
10 resqrexlemex.a . . . . . . 7  |-  ( ph  ->  A  e.  RR )
1110adantr 270 . . . . . 6  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  A  e.  RR )
1211, 7rerpdivcld 8922 . . . . 5  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  ( A  /  B )  e.  RR )
139, 12readdcld 7246 . . . 4  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  ( B  +  ( A  /  B ) )  e.  RR )
1413rehalfcld 8380 . . 3  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  (
( B  +  ( A  /  B ) )  /  2 )  e.  RR )
151, 6, 7, 8, 14ovmpt2d 5680 . 2  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  ( B ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
) )  /  2
) ) C )  =  ( ( B  +  ( A  /  B ) )  / 
2 ) )
167rpgt0d 8893 . . . . 5  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  0  <  B )
17 resqrexlemex.agt0 . . . . . . 7  |-  ( ph  ->  0  <_  A )
1817adantr 270 . . . . . 6  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  0  <_  A )
1911, 7, 18divge0d 8931 . . . . 5  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  0  <_  ( A  /  B
) )
20 addgtge0 7657 . . . . 5  |-  ( ( ( B  e.  RR  /\  ( A  /  B
)  e.  RR )  /\  ( 0  < 
B  /\  0  <_  ( A  /  B ) ) )  ->  0  <  ( B  +  ( A  /  B ) ) )
219, 12, 16, 19, 20syl22anc 1171 . . . 4  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  0  <  ( B  +  ( A  /  B ) ) )
2213, 21elrpd 8888 . . 3  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  ( B  +  ( A  /  B ) )  e.  RR+ )
2322rphalfcld 8903 . 2  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  (
( B  +  ( A  /  B ) )  /  2 )  e.  RR+ )
2415, 23eqeltrd 2159 1  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  ( B ( y  e.  RR+ ,  z  e.  RR+  |->  ( ( y  +  ( A  /  y
) )  /  2
) ) C )  e.  RR+ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   {csn 3417   class class class wbr 3806    X. cxp 4390  (class class class)co 5564    |-> cmpt2 5566   RRcr 7078   0cc0 7079   1c1 7080    + caddc 7082    < clt 7251    <_ cle 7252    / cdiv 7863   NNcn 8142   2c2 8192   RR+crp 8851    seqcseq 9557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-cnex 7165  ax-resscn 7166  ax-1cn 7167  ax-1re 7168  ax-icn 7169  ax-addcl 7170  ax-addrcl 7171  ax-mulcl 7172  ax-mulrcl 7173  ax-addcom 7174  ax-mulcom 7175  ax-addass 7176  ax-mulass 7177  ax-distr 7178  ax-i2m1 7179  ax-0lt1 7180  ax-1rid 7181  ax-0id 7182  ax-rnegex 7183  ax-precex 7184  ax-cnre 7185  ax-pre-ltirr 7186  ax-pre-ltwlin 7187  ax-pre-lttrn 7188  ax-pre-apti 7189  ax-pre-ltadd 7190  ax-pre-mulgt0 7191  ax-pre-mulext 7192
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2826  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-opab 3861  df-id 4077  df-po 4080  df-iso 4081  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-iota 4918  df-fun 4955  df-fv 4961  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-pnf 7253  df-mnf 7254  df-xr 7255  df-ltxr 7256  df-le 7257  df-sub 7384  df-neg 7385  df-reap 7778  df-ap 7785  df-div 7864  df-2 8201  df-rp 8852
This theorem is referenced by:  resqrexlemf  10078  resqrexlemf1  10079  resqrexlemfp1  10080
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