Theorem resqrexlemp1rp 11188

Description: Lemma for resqrex 11208. Applying the recursion rule yields a positive real (expressed in a way that will help apply seqf 10573 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 2197	. . 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 5933	. . . . . 6 |- ( y = B -> ( A / y ) = ( A / B ) )
4	2, 3	oveq12d 5943	. . . . 5 |- ( y = B -> ( y + ( A / y ) ) = ( B + ( A / B ) ) )
5	4	oveq1d 5940	. . . 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 9788	. . . . 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 9820	. . . . 5 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( A / B ) e. RR )
13	9, 12	readdcld 8073	. . . 4 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( B + ( A / B ) ) e. RR )
14	13	rehalfcld 9255	. . 3 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( ( B + ( A / B ) ) / 2 ) e. RR )
15	1, 6, 7, 8, 14	ovmpod 6054	. 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 9791	. . . . 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 9829	. . . . 5 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> 0 <_ ( A / B ) )
20		addgtge0 8494	. . . . 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 9785	. . 3 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( B + ( A / B ) ) e. RR+ )
23	22	rphalfcld 9801	. 2 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( ( B + ( A / B ) ) / 2 ) e. RR+ )
24	15, 23	eqeltrd 2273	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 2167   class class class wbr 4034  (class class class)co 5925   e. cmpo 5927   RRcr 7895   0cc0 7896   + caddc 7899   < clt 8078   <_ cle 8079   / cdiv 8716   2c2 9058   RR+crp 9745

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-2 9066  df-rp 9746

This theorem is referenced by:  resqrexlemf  11189  resqrexlemf1  11190  resqrexlemfp1  11191