Theorem resqrexlemp1rp 11150

Description: Lemma for resqrex 11170. Applying the recursion rule yields a positive real (expressed in a way that will help apply seqf 10535 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

Step	Hyp	Ref	Expression
1		eqidd 2194	. . 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 5926	. . . . . 6 |- ( y = B -> ( A / y ) = ( A / B ) )
4	2, 3	oveq12d 5936	. . . . 5 |- ( y = B -> ( y + ( A / y ) ) = ( B + ( A / B ) ) )
5	4	oveq1d 5933	. . . 4 |- ( y = B -> ( ( y + ( A / y ) ) / 2 ) = ( ( B + ( A / B ) ) / 2 ) )
6	5	ad2antrl 490	. . 3 |- ( ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) /\ ( y = B /\ z = C ) ) -> ( ( y + ( A / y ) ) / 2 ) = ( ( B + ( A / B ) ) / 2 ) )
7		simprl 529	. . 3 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> B e. RR+ )
8		simprr 531	. . 3 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> C e. RR+ )
9	7	rpred 9762	. . . . 5 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> B e. RR )
10		resqrexlem1arp.a	. . . . . . 7 |- ( ph -> A e. RR )
11	10	adantr 276	. . . . . 6 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> A e. RR )
12	11, 7	rerpdivcld 9794	. . . . 5 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( A / B ) e. RR )
13	9, 12	readdcld 8049	. . . 4 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( B + ( A / B ) ) e. RR )
14	13	rehalfcld 9229	. . 3 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( ( B + ( A / B ) ) / 2 ) e. RR )
15	1, 6, 7, 8, 14	ovmpod 6046	. 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 9765	. . . . 5 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> 0 < B )
17		resqrexlem1arp.agt0	. . . . . . 7 |- ( ph -> 0 <_ A )
18	17	adantr 276	. . . . . 6 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> 0 <_ A )
19	11, 7, 18	divge0d 9803	. . . . 5 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> 0 <_ ( A / B ) )
20		addgtge0 8469	. . . . 5 |- ( ( ( B e. RR /\ ( A / B ) e. RR ) /\ ( 0 < B /\ 0 <_ ( A / B ) ) ) -> 0 < ( B + ( A / B ) ) )
21	9, 12, 16, 19, 20	syl22anc 1250	. . . 4 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> 0 < ( B + ( A / B ) ) )
22	13, 21	elrpd 9759	. . 3 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( B + ( A / B ) ) e. RR+ )
23	22	rphalfcld 9775	. 2 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( ( B + ( A / B ) ) / 2 ) e. RR+ )
24	15, 23	eqeltrd 2270	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 104   = wceq 1364   e. wcel 2164   class class class wbr 4029  (class class class)co 5918   e. cmpo 5920   RRcr 7871   0cc0 7872   + caddc 7875   < clt 8054   <_ cle 8055   / cdiv 8691   2c2 9033   RR+crp 9719

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-po 4327  df-iso 4328  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-2 9041  df-rp 9720

This theorem is referenced by:  resqrexlemf  11151  resqrexlemf1  11152  resqrexlemfp1  11153