Theorem knoppndvlem19 32521

Description: Lemma for knoppndv 32525. (Contributed by Asger C. Ipsen, 17-Aug-2021.)

Hypotheses

Ref	Expression
knoppndvlem19.a	|- A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. m )
knoppndvlem19.b	|- B = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( m + 1 ) )
knoppndvlem19.j	|- ( ph -> J e. NN0 )
knoppndvlem19.h	|- ( ph -> H e. RR )
knoppndvlem19.n	|- ( ph -> N e. NN )

Assertion

Ref	Expression
knoppndvlem19	|- ( ph -> E. m e. ZZ ( A <_ H /\ H <_ B ) )

Distinct variable groups:   ph, m   m, J   m, H   m, N

Allowed substitution hints:   A( m)   B( m)

Proof of Theorem knoppndvlem19

Step	Hyp	Ref	Expression
1		knoppndvlem19.h	. . . 4 |- ( ph -> H e. RR )
2		2re 11090	. . . . . . . 8 |- 2 e. RR
3	2	a1i 11	. . . . . . 7 |- ( ph -> 2 e. RR )
4		knoppndvlem19.n	. . . . . . . 8 |- ( ph -> N e. NN )
5	4	nnred 11035	. . . . . . 7 |- ( ph -> N e. RR )
6	3, 5	remulcld 10070	. . . . . 6 |- ( ph -> ( 2 x. N ) e. RR )
7		2pos 11112	. . . . . . . . 9 |- 0 < 2
8	7	a1i 11	. . . . . . . 8 |- ( ph -> 0 < 2 )
9	4	nngt0d 11064	. . . . . . . 8 |- ( ph -> 0 < N )
10	3, 5, 8, 9	mulgt0d 10192	. . . . . . 7 |- ( ph -> 0 < ( 2 x. N ) )
11	10	gt0ne0d 10592	. . . . . 6 |- ( ph -> ( 2 x. N ) =/= 0 )
12		knoppndvlem19.j	. . . . . . . 8 |- ( ph -> J e. NN0 )
13	12	nn0zd 11480	. . . . . . 7 |- ( ph -> J e. ZZ )
14	13	znegcld 11484	. . . . . 6 |- ( ph -> -u J e. ZZ )
15	6, 11, 14	reexpclzd 13034	. . . . 5 |- ( ph -> ( ( 2 x. N ) ^ -u J ) e. RR )
16	3	recnd 10068	. . . . . 6 |- ( ph -> 2 e. CC )
17	5	recnd 10068	. . . . . 6 |- ( ph -> N e. CC )
18	16, 17, 11	mulne0bad 10682	. . . . 5 |- ( ph -> 2 =/= 0 )
19	15, 3, 18	redivcld 10853	. . . 4 |- ( ph -> ( ( ( 2 x. N ) ^ -u J ) / 2 ) e. RR )
20	6, 14, 10	3jca 1242	. . . . . . 7 |- ( ph -> ( ( 2 x. N ) e. RR /\ -u J e. ZZ /\ 0 < ( 2 x. N ) ) )
21		expgt0 12893	. . . . . . 7 |- ( ( ( 2 x. N ) e. RR /\ -u J e. ZZ /\ 0 < ( 2 x. N ) ) -> 0 < ( ( 2 x. N ) ^ -u J ) )
22	20, 21	syl 17	. . . . . 6 |- ( ph -> 0 < ( ( 2 x. N ) ^ -u J ) )
23	15, 3, 22, 8	divgt0d 10959	. . . . 5 |- ( ph -> 0 < ( ( ( 2 x. N ) ^ -u J ) / 2 ) )
24	23	gt0ne0d 10592	. . . 4 |- ( ph -> ( ( ( 2 x. N ) ^ -u J ) / 2 ) =/= 0 )
25	1, 19, 24	redivcld 10853	. . 3 |- ( ph -> ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) e. RR )
26	25	flcld 12599	. 2 |- ( ph -> ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) e. ZZ )
27		knoppndvlem19.a	. . . . . . 7 |- A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. m )
28	27	a1i 11	. . . . . 6 |- ( m = ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) -> A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. m ) )
29		id 22	. . . . . . 7 |- ( m = ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) -> m = ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) )
30	29	oveq2d 6666	. . . . . 6 |- ( m = ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) -> ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. m ) = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) ) )
31	28, 30	eqtrd 2656	. . . . 5 |- ( m = ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) -> A = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) ) )
32	31	breq1d 4663	. . . 4 |- ( m = ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) -> ( A <_ H <-> ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) ) <_ H ) )
33		knoppndvlem19.b	. . . . . . 7 |- B = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( m + 1 ) )
34	33	a1i 11	. . . . . 6 |- ( m = ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) -> B = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( m + 1 ) ) )
35	29	oveq1d 6665	. . . . . . 7 |- ( m = ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) -> ( m + 1 ) = ( ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) + 1 ) )
36	35	oveq2d 6666	. . . . . 6 |- ( m = ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) -> ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( m + 1 ) ) = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) + 1 ) ) )
37	34, 36	eqtrd 2656	. . . . 5 |- ( m = ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) -> B = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) + 1 ) ) )
38	37	breq2d 4665	. . . 4 |- ( m = ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) -> ( H <_ B <-> H <_ ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) + 1 ) ) ) )
39	32, 38	anbi12d 747	. . 3 |- ( m = ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) -> ( ( A <_ H /\ H <_ B ) <-> ( ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) ) <_ H /\ H <_ ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) + 1 ) ) ) ) )
40	39	adantl 482	. 2 |- ( ( ph /\ m = ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) ) -> ( ( A <_ H /\ H <_ B ) <-> ( ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) ) <_ H /\ H <_ ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) + 1 ) ) ) ) )
41	26	zred 11482	. . . . 5 |- ( ph -> ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) e. RR )
42		0red 10041	. . . . . 6 |- ( ph -> 0 e. RR )
43	42, 19, 23	ltled 10185	. . . . 5 |- ( ph -> 0 <_ ( ( ( 2 x. N ) ^ -u J ) / 2 ) )
44		flle 12600	. . . . . 6 |- ( ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) e. RR -> ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) <_ ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) )
45	25, 44	syl 17	. . . . 5 |- ( ph -> ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) <_ ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) )
46	41, 25, 19, 43, 45	lemul2ad 10964	. . . 4 |- ( ph -> ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) ) <_ ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) )
47	1	recnd 10068	. . . . 5 |- ( ph -> H e. CC )
48	19	recnd 10068	. . . . 5 |- ( ph -> ( ( ( 2 x. N ) ^ -u J ) / 2 ) e. CC )
49	47, 48, 24	divcan2d 10803	. . . 4 |- ( ph -> ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) = H )
50	46, 49	breqtrd 4679	. . 3 |- ( ph -> ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) ) <_ H )
51	49	eqcomd 2628	. . . 4 |- ( ph -> H = ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) )
52		peano2re 10209	. . . . . 6 |- ( ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) e. RR -> ( ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) + 1 ) e. RR )
53	41, 52	syl 17	. . . . 5 |- ( ph -> ( ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) + 1 ) e. RR )
54		fllep1 12602	. . . . . 6 |- ( ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) e. RR -> ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) <_ ( ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) + 1 ) )
55	25, 54	syl 17	. . . . 5 |- ( ph -> ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) <_ ( ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) + 1 ) )
56	25, 53, 19, 43, 55	lemul2ad 10964	. . . 4 |- ( ph -> ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) <_ ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) + 1 ) ) )
57	51, 56	eqbrtrd 4675	. . 3 |- ( ph -> H <_ ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) + 1 ) ) )
58	50, 57	jca 554	. 2 |- ( ph -> ( ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) ) <_ H /\ H <_ ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. ( ( |_ ` ( H / ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) ) + 1 ) ) ) )
59	26, 40, 58	rspcedvd 3317	1 |- ( ph -> E. m e. ZZ ( A <_ H /\ H <_ B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   |_cfl 12591   ^cexp 12860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593  df-seq 12802  df-exp 12861

This theorem is referenced by:  knoppndvlem21  32523