Theorem resqrexlemgt0 11764

Description: Lemma for resqrex 11770. A limit is nonnegative. (Contributed by Jim Kingdon, 7-Aug-2021.)

Hypotheses

Ref	Expression
resqrexlemex.seq	|- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )
resqrexlemex.a	|- ( ph -> A e. RR )
resqrexlemex.agt0	|- ( ph -> 0 <_ A )
resqrexlemgt0.rr	|- ( ph -> L e. RR )
resqrexlemgt0.lim	|- ( ph -> A. e e. RR+ E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( L + e ) /\ L < ( ( F ` i ) + e ) ) )

Assertion

Ref	Expression
resqrexlemgt0	|- ( ph -> 0 <_ L )

Distinct variable groups:   y, A, z   e, F   e, L, i, j   ph, i, j   z, j, ph   ph, y

Allowed substitution hints:   ph( e)   A( e, i, j)   F( y, z, i, j)   L( y, z)

Proof of Theorem resqrexlemgt0

Step	Hyp	Ref	Expression
1		oveq2 6083	. . . . . . . . 9 |- ( e = -u L -> ( L + e ) = ( L + -u L ) )
2	1	breq2d 4137	. . . . . . . 8 |- ( e = -u L -> ( ( F ` i ) < ( L + e ) <-> ( F ` i ) < ( L + -u L ) ) )
3		oveq2 6083	. . . . . . . . 9 |- ( e = -u L -> ( ( F ` i ) + e ) = ( ( F ` i ) + -u L ) )
4	3	breq2d 4137	. . . . . . . 8 |- ( e = -u L -> ( L < ( ( F ` i ) + e ) <-> L < ( ( F ` i ) + -u L ) ) )
5	2, 4	anbi12d 477	. . . . . . 7 |- ( e = -u L -> ( ( ( F ` i ) < ( L + e ) /\ L < ( ( F ` i ) + e ) ) <-> ( ( F ` i ) < ( L + -u L ) /\ L < ( ( F ` i ) + -u L ) ) ) )
6	5	rexralbidv 2576	. . . . . 6 |- ( e = -u L -> ( E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( L + e ) /\ L < ( ( F ` i ) + e ) ) <-> E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( L + -u L ) /\ L < ( ( F ` i ) + -u L ) ) ) )
7		resqrexlemgt0.lim	. . . . . . 7 |- ( ph -> A. e e. RR+ E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( L + e ) /\ L < ( ( F ` i ) + e ) ) )
8	7	adantr 276	. . . . . 6 |- ( ( ph /\ L < 0 ) -> A. e e. RR+ E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( L + e ) /\ L < ( ( F ` i ) + e ) ) )
9		resqrexlemgt0.rr	. . . . . . . . 9 |- ( ph -> L e. RR )
10	9	adantr 276	. . . . . . . 8 |- ( ( ph /\ L < 0 ) -> L e. RR )
11	10	renegcld 8697	. . . . . . 7 |- ( ( ph /\ L < 0 ) -> -u L e. RR )
12	9	lt0neg1d 8833	. . . . . . . 8 |- ( ph -> ( L < 0 <-> 0 < -u L ) )
13	12	biimpa 296	. . . . . . 7 |- ( ( ph /\ L < 0 ) -> 0 < -u L )
14	11, 13	elrpd 10073	. . . . . 6 |- ( ( ph /\ L < 0 ) -> -u L e. RR+ )
15	6, 8, 14	rspcdva 2934	. . . . 5 |- ( ( ph /\ L < 0 ) -> E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( L + -u L ) /\ L < ( ( F ` i ) + -u L ) ) )
16		simpl 109	. . . . . . . 8 |- ( ( ( F ` i ) < ( L + -u L ) /\ L < ( ( F ` i ) + -u L ) ) -> ( F ` i ) < ( L + -u L ) )
17	10	recnd 8344	. . . . . . . . . 10 |- ( ( ph /\ L < 0 ) -> L e. CC )
18	17	negidd 8617	. . . . . . . . 9 |- ( ( ph /\ L < 0 ) -> ( L + -u L ) = 0 )
19	18	breq2d 4137	. . . . . . . 8 |- ( ( ph /\ L < 0 ) -> ( ( F ` i ) < ( L + -u L ) <-> ( F ` i ) < 0 ) )
20	16, 19	imbitrid 154	. . . . . . 7 |- ( ( ph /\ L < 0 ) -> ( ( ( F ` i ) < ( L + -u L ) /\ L < ( ( F ` i ) + -u L ) ) -> ( F ` i ) < 0 ) )
21	20	ralimdv 2618	. . . . . 6 |- ( ( ph /\ L < 0 ) -> ( A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( L + -u L ) /\ L < ( ( F ` i ) + -u L ) ) -> A. i e. ( ZZ>= ` j ) ( F ` i ) < 0 ) )
22	21	reximdv 2651	. . . . 5 |- ( ( ph /\ L < 0 ) -> ( E. j e. NN A. i e. ( ZZ>= ` j ) ( ( F ` i ) < ( L + -u L ) /\ L < ( ( F ` i ) + -u L ) ) -> E. j e. NN A. i e. ( ZZ>= ` j ) ( F ` i ) < 0 ) )
23	15, 22	mpd 13	. . . 4 |- ( ( ph /\ L < 0 ) -> E. j e. NN A. i e. ( ZZ>= ` j ) ( F ` i ) < 0 )
24		0red 8317	. . . . . . . . . . 11 |- ( ( ph /\ ( j e. NN /\ i e. ( ZZ>= ` j ) ) ) -> 0 e. RR )
25		eluznn 9979	. . . . . . . . . . . . 13 |- ( ( j e. NN /\ i e. ( ZZ>= ` j ) ) -> i e. NN )
26		resqrexlemex.seq	. . . . . . . . . . . . . . 15 |- F = seq 1 ( ( y e. RR+ , z e. RR+ |-> ( ( y + ( A / y ) ) / 2 ) ) , ( NN X. { ( 1 + A ) } ) )
27		resqrexlemex.a	. . . . . . . . . . . . . . 15 |- ( ph -> A e. RR )
28		resqrexlemex.agt0	. . . . . . . . . . . . . . 15 |- ( ph -> 0 <_ A )
29	26, 27, 28	resqrexlemf 11751	. . . . . . . . . . . . . 14 |- ( ph -> F : NN --> RR+ )
30	29	ffvelcdmda 5834	. . . . . . . . . . . . 13 |- ( ( ph /\ i e. NN ) -> ( F ` i ) e. RR+ )
31	25, 30	sylan2 286	. . . . . . . . . . . 12 |- ( ( ph /\ ( j e. NN /\ i e. ( ZZ>= ` j ) ) ) -> ( F ` i ) e. RR+ )
32	31	rpred 10076	. . . . . . . . . . 11 |- ( ( ph /\ ( j e. NN /\ i e. ( ZZ>= ` j ) ) ) -> ( F ` i ) e. RR )
33	31	rpgt0d 10079	. . . . . . . . . . 11 |- ( ( ph /\ ( j e. NN /\ i e. ( ZZ>= ` j ) ) ) -> 0 < ( F ` i ) )
34	24, 32, 33	ltnsymd 8436	. . . . . . . . . 10 |- ( ( ph /\ ( j e. NN /\ i e. ( ZZ>= ` j ) ) ) -> -. ( F ` i ) < 0 )
35	34	pm2.21d 628	. . . . . . . . 9 |- ( ( ph /\ ( j e. NN /\ i e. ( ZZ>= ` j ) ) ) -> ( ( F ` i ) < 0 -> F. ) )
36	35	anassrs 404	. . . . . . . 8 |- ( ( ( ph /\ j e. NN ) /\ i e. ( ZZ>= ` j ) ) -> ( ( F ` i ) < 0 -> F. ) )
37	36	ralimdva 2617	. . . . . . 7 |- ( ( ph /\ j e. NN ) -> ( A. i e. ( ZZ>= ` j ) ( F ` i ) < 0 -> A. i e. ( ZZ>= ` j ) F. ) )
38		nnz 9642	. . . . . . . . 9 |- ( j e. NN -> j e. ZZ )
39		uzid 9915	. . . . . . . . . 10 |- ( j e. ZZ -> j e. ( ZZ>= ` j ) )
40		elex2 2838	. . . . . . . . . 10 |- ( j e. ( ZZ>= ` j ) -> E. z z e. ( ZZ>= ` j ) )
41		r19.3rmv 3615	. . . . . . . . . 10 |- ( E. z z e. ( ZZ>= ` j ) -> ( F. <-> A. i e. ( ZZ>= ` j ) F. ) )
42	39, 40, 41	3syl 17	. . . . . . . . 9 |- ( j e. ZZ -> ( F. <-> A. i e. ( ZZ>= ` j ) F. ) )
43	38, 42	syl 14	. . . . . . . 8 |- ( j e. NN -> ( F. <-> A. i e. ( ZZ>= ` j ) F. ) )
44	43	adantl 277	. . . . . . 7 |- ( ( ph /\ j e. NN ) -> ( F. <-> A. i e. ( ZZ>= ` j ) F. ) )
45	37, 44	sylibrd 169	. . . . . 6 |- ( ( ph /\ j e. NN ) -> ( A. i e. ( ZZ>= ` j ) ( F ` i ) < 0 -> F. ) )
46	45	rexlimdva 2668	. . . . 5 |- ( ph -> ( E. j e. NN A. i e. ( ZZ>= ` j ) ( F ` i ) < 0 -> F. ) )
47	46	adantr 276	. . . 4 |- ( ( ph /\ L < 0 ) -> ( E. j e. NN A. i e. ( ZZ>= ` j ) ( F ` i ) < 0 -> F. ) )
48	23, 47	mpd 13	. . 3 |- ( ( ph /\ L < 0 ) -> F. )
49	48	inegd 1421	. 2 |- ( ph -> -. L < 0 )
50		0re 8316	. . 3 |- 0 e. RR
51		lenlt 8391	. . 3 |- ( ( 0 e. RR /\ L e. RR ) -> ( 0 <_ L <-> -. L < 0 ) )
52	50, 9, 51	sylancr 418	. 2 |- ( ph -> ( 0 <_ L <-> -. L < 0 ) )
53	49, 52	mpbird 167	1 |- ( ph -> 0 <_ L )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   F. wfal 1407   E.wex 1545   e. wcel 2209   A.wral 2528   E.wrex 2529   {csn 3705   class class class wbr 4125   X. cxp 4767   ` cfv 5372  (class class class)co 6075   e. cmpo 6077   RRcr 8168   0cc0 8169   1c1 8170   + caddc 8172   < clt 8350   <_ cle 8351   -ucneg 8488   / cdiv 8992   NNcn 9283   2c2 9334   ZZcz 9623   ZZ>=cuz 9900   RR+crp 10033   seqcseq 10862

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-rp 10034  df-seqfrec 10863

This theorem is referenced by:  resqrexlemglsq  11766  resqrexlemex  11769