Theorem resqrexlemp1rp 11018

Description: Lemma for resqrex 11038. Applying the recursion rule yields a positive real (expressed in a way that will help apply seqf 10464 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 2178	. . 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 5886	. . . . . 6 |- ( y = B -> ( A / y ) = ( A / B ) )
4	2, 3	oveq12d 5896	. . . . 5 |- ( y = B -> ( y + ( A / y ) ) = ( B + ( A / B ) ) )
5	4	oveq1d 5893	. . . 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 9699	. . . . 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 9731	. . . . 5 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( A / B ) e. RR )
13	9, 12	readdcld 7990	. . . 4 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( B + ( A / B ) ) e. RR )
14	13	rehalfcld 9168	. . 3 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( ( B + ( A / B ) ) / 2 ) e. RR )
15	1, 6, 7, 8, 14	ovmpod 6005	. 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 9702	. . . . 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 9740	. . . . 5 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> 0 <_ ( A / B ) )
20		addgtge0 8410	. . . . 5 |- ( ( ( B e. RR /\ ( A / B ) e. RR ) /\ ( 0 < B /\ 0 <_ ( A / B ) ) ) -> 0 < ( B + ( A / B ) ) )
21	9, 12, 16, 19, 20	syl22anc 1239	. . . 4 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> 0 < ( B + ( A / B ) ) )
22	13, 21	elrpd 9696	. . 3 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( B + ( A / B ) ) e. RR+ )
23	22	rphalfcld 9712	. 2 |- ( ( ph /\ ( B e. RR+ /\ C e. RR+ ) ) -> ( ( B + ( A / B ) ) / 2 ) e. RR+ )
24	15, 23	eqeltrd 2254	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 1353   e. wcel 2148   class class class wbr 4005  (class class class)co 5878   e. cmpo 5880   RRcr 7813   0cc0 7814   + caddc 7817   < clt 7995   <_ cle 7996   / cdiv 8632   2c2 8973   RR+crp 9656

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-precex 7924  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930  ax-pre-mulgt0 7931  ax-pre-mulext 7932

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-po 4298  df-iso 4299  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-reap 8535  df-ap 8542  df-div 8633  df-2 8981  df-rp 9657

This theorem is referenced by:  resqrexlemf  11019  resqrexlemf1  11020  resqrexlemfp1  11021