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Theorem resqrexlemp1rp 10500
Description: Lemma for resqrex 10520. Applying the recursion rule yields a positive real (expressed in a way that will help apply seqf 9941 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.)
Hypotheses
Ref Expression
resqrexlem1arp.a  |-  ( ph  ->  A  e.  RR )
resqrexlem1arp.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

Proof of Theorem resqrexlemp1rp
StepHypRef Expression
1 eqidd 2090 . . 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 5674 . . . . . 6  |-  ( y  =  B  ->  ( A  /  y )  =  ( A  /  B
) )
42, 3oveq12d 5684 . . . . 5  |-  ( y  =  B  ->  (
y  +  ( A  /  y ) )  =  ( B  +  ( A  /  B
) ) )
54oveq1d 5681 . . . 4  |-  ( y  =  B  ->  (
( y  +  ( A  /  y ) )  /  2 )  =  ( ( B  +  ( A  /  B ) )  / 
2 ) )
65ad2antrl 475 . . 3  |-  ( ( ( ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  /\  ( y  =  B  /\  z  =  C ) )  ->  (
( y  +  ( A  /  y ) )  /  2 )  =  ( ( B  +  ( A  /  B ) )  / 
2 ) )
7 simprl 499 . . 3  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  B  e.  RR+ )
8 simprr 500 . . 3  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  C  e.  RR+ )
97rpred 9234 . . . . 5  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  B  e.  RR )
10 resqrexlem1arp.a . . . . . . 7  |-  ( ph  ->  A  e.  RR )
1110adantr 271 . . . . . 6  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  A  e.  RR )
1211, 7rerpdivcld 9266 . . . . 5  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  ( A  /  B )  e.  RR )
139, 12readdcld 7578 . . . 4  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  ( B  +  ( A  /  B ) )  e.  RR )
1413rehalfcld 8723 . . 3  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  (
( B  +  ( A  /  B ) )  /  2 )  e.  RR )
151, 6, 7, 8, 14ovmpt2d 5786 . 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 9237 . . . . 5  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  0  <  B )
17 resqrexlem1arp.agt0 . . . . . . 7  |-  ( ph  ->  0  <_  A )
1817adantr 271 . . . . . 6  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  0  <_  A )
1911, 7, 18divge0d 9275 . . . . 5  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  0  <_  ( A  /  B
) )
20 addgtge0 7989 . . . . 5  |-  ( ( ( B  e.  RR  /\  ( A  /  B
)  e.  RR )  /\  ( 0  < 
B  /\  0  <_  ( A  /  B ) ) )  ->  0  <  ( B  +  ( A  /  B ) ) )
219, 12, 16, 19, 20syl22anc 1176 . . . 4  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  0  <  ( B  +  ( A  /  B ) ) )
2213, 21elrpd 9232 . . 3  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  ( B  +  ( A  /  B ) )  e.  RR+ )
2322rphalfcld 9247 . 2  |-  ( (
ph  /\  ( B  e.  RR+  /\  C  e.  RR+ ) )  ->  (
( B  +  ( A  /  B ) )  /  2 )  e.  RR+ )
2415, 23eqeltrd 2165 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 103    = wceq 1290    e. wcel 1439   class class class wbr 3851  (class class class)co 5666    |-> cmpt2 5668   RRcr 7410   0cc0 7411    + caddc 7414    < clt 7583    <_ cle 7584    / cdiv 8200   2c2 8534   RR+crp 9195
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 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-mulrcl 7505  ax-addcom 7506  ax-mulcom 7507  ax-addass 7508  ax-mulass 7509  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-1rid 7513  ax-0id 7514  ax-rnegex 7515  ax-precex 7516  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522  ax-pre-mulgt0 7523  ax-pre-mulext 7524
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-po 4132  df-iso 4133  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-iota 4993  df-fun 5030  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-reap 8113  df-ap 8120  df-div 8201  df-2 8542  df-rp 9196
This theorem is referenced by:  resqrexlemf  10501  resqrexlemf1  10502  resqrexlemfp1  10503
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