Theorem resqrexlemp1rp 11691

Description: Lemma for resqrex 11711. Applying the recursion rule yields a positive real (expressed in a way that will help apply seqf 10826 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 2233	. . 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 6058	. . . . . 6 |- ( y = B -> ( A / y ) = ( A / B ) )
4	2, 3	oveq12d 6068	. . . . 5 |- ( y = B -> ( y + ( A / y ) ) = ( B + ( A / B ) ) )
5	4	oveq1d 6065	. . . 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 531	. . 3 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> B e. RR+ )
8		simprr 533	. . 3 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> C e. RR+ )
9	7	rpred 10029	. . . . 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 10061	. . . . 5 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( A / B ) e. RR )
13	9, 12	readdcld 8303	. . . 4 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( B + ( A / B ) ) e. RR )
14	13	rehalfcld 9485	. . 3 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( ( B + ( A / B ) ) / 2 ) e. RR )
15	1, 6, 7, 8, 14	ovmpod 6181	. 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 10032	. . . . 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 10070	. . . . 5 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> 0 <_ ( A / B ) )
20		addgtge0 8724	. . . . 5 |- ( ( ( B e. RR /\ ( A / B ) e. RR ) /\ ( 0 < B /\ 0 <_ ( A / B ) ) ) -> 0 < ( B + ( A / B ) ) )
21	9, 12, 16, 19, 20	syl22anc 1275	. . . 4 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> 0 < ( B + ( A / B ) ) )
22	13, 21	elrpd 10026	. . 3 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( B + ( A / B ) ) e. RR+ )
23	22	rphalfcld 10042	. 2 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( ( B + ( A / B ) ) / 2 ) e. RR+ )
24	15, 23	eqeltrd 2309	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 1398   e. wcel 2203   class class class wbr 4109  (class class class)co 6050   e. cmpo 6052   RRcr 8126   0cc0 8127   + caddc 8130   < clt 8308   <_ cle 8309   / cdiv 8946   2c2 9288   RR+crp 9986

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-2 9296  df-rp 9987

This theorem is referenced by:  resqrexlemf  11692  resqrexlemf1  11693  resqrexlemfp1  11694