Theorem resqrexlemp1rp 9892

Description: Lemma for resqrex 9912. Applying the recursion rule yields a positive real (expressed in a way that will help apply iseqf 9444 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

Step	Hyp	Ref	Expression
1		eqidd 2082	. . 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 5540	. . . . . 6 |- ( y = B -> ( A / y ) = ( A / B ) )
4	2, 3	oveq12d 5550	. . . . 5 |- ( y = B -> ( y + ( A / y ) ) = ( B + ( A / B ) ) )
5	4	oveq1d 5547	. . . 4 |- ( y = B -> ( ( y + ( A / y ) ) / 2 ) = ( ( B + ( A / B ) ) / 2 ) )
6	5	ad2antrl 473	. . 3 |- ( ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) /\ ( y = B /\ z = C ) ) -> ( ( y + ( A / y ) ) / 2 ) = ( ( B + ( A / B ) ) / 2 ) )
7		simprl 497	. . 3 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> B e. RR+ )
8		simprr 498	. . 3 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> C e. RR+ )
9	7	rpred 8773	. . . . 5 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> B e. RR )
10		resqrexlemex.a	. . . . . . 7 |- ( ph -> A e. RR )
11	10	adantr 270	. . . . . 6 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> A e. RR )
12	11, 7	rerpdivcld 8805	. . . . 5 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( A / B ) e. RR )
13	9, 12	readdcld 7148	. . . 4 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( B + ( A / B ) ) e. RR )
14	13	rehalfcld 8277	. . 3 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( ( B + ( A / B ) ) / 2 ) e. RR )
15	1, 6, 7, 8, 14	ovmpt2d 5648	. 2 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( B ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) C ) = ( ( B + ( A / B ) ) / 2 ) )
16	7	rpgt0d 8776	. . . . 5 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> 0 < B )
17		resqrexlemex.agt0	. . . . . . 7 |- ( ph -> 0 <_ A )
18	17	adantr 270	. . . . . 6 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> 0 <_ A )
19	11, 7, 18	divge0d 8814	. . . . 5 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> 0 <_ ( A / B ) )
20		addgtge0 7554	. . . . 5 |- ( ( ( B e. RR /\ ( A / B ) e. RR ) /\ ( 0 < B /\ 0 <_ ( A / B ) ) ) -> 0 < ( B + ( A / B ) ) )
21	9, 12, 16, 19, 20	syl22anc 1170	. . . 4 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> 0 < ( B + ( A / B ) ) )
22	13, 21	elrpd 8771	. . 3 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( B + ( A / B ) ) e. RR+ )
23	22	rphalfcld 8786	. 2 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( ( B + ( A / B ) ) / 2 ) e. RR+ )
24	15, 23	eqeltrd 2155	1 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( B ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) C ) e. RR+ )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   {csn 3398   class class class wbr 3785   X. cxp 4361  (class class class)co 5532   |-> cmpt2 5534   RRcr 6980   0cc0 6981   1c1 6982   + caddc 6984   < clt 7153   <_ cle 7154   / cdiv 7760   NNcn 8039   2c2 8089   RR+crp 8734   seqcseq 9431

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-po 4051  df-iso 4052  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-2 8098  df-rp 8735

This theorem is referenced by:  resqrexlemf  9893  resqrexlemf1  9894  resqrexlemfp1  9895