Theorem knoppndvlem2 32504

Description: Lemma for knoppndv 32525. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)

Hypotheses

Ref	Expression
knoppndvlem2.n	|- ( ph -> N e. NN )
knoppndvlem2.i	|- ( ph -> I e. ZZ )
knoppndvlem2.j	|- ( ph -> J e. ZZ )
knoppndvlem2.m	|- ( ph -> M e. ZZ )
knoppndvlem2.1	|- ( ph -> J < I )

Assertion

Ref	Expression
knoppndvlem2	|- ( ph -> ( ( ( 2 x. N ) ^ I ) x. ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) ) e. ZZ )

Proof of Theorem knoppndvlem2

Step	Hyp	Ref	Expression
1		2cnd 11093	. . . . . . 7 |- ( ph -> 2 e. CC )
2		knoppndvlem2.n	. . . . . . . . 9 |- ( ph -> N e. NN )
3		nnz 11399	. . . . . . . . 9 |- ( N e. NN -> N e. ZZ )
4	2, 3	syl 17	. . . . . . . 8 |- ( ph -> N e. ZZ )
5	4	zcnd 11483	. . . . . . 7 |- ( ph -> N e. CC )
6	1, 5	mulcld 10060	. . . . . 6 |- ( ph -> ( 2 x. N ) e. CC )
7		2ne0 11113	. . . . . . . 8 |- 2 =/= 0
8	7	a1i 11	. . . . . . 7 |- ( ph -> 2 =/= 0 )
9		0red 10041	. . . . . . . . 9 |- ( ph -> 0 e. RR )
10		1red 10055	. . . . . . . . . 10 |- ( ph -> 1 e. RR )
11	4	zred 11482	. . . . . . . . . 10 |- ( ph -> N e. RR )
12		0lt1 10550	. . . . . . . . . . 11 |- 0 < 1
13	12	a1i 11	. . . . . . . . . 10 |- ( ph -> 0 < 1 )
14		nnge1 11046	. . . . . . . . . . 11 |- ( N e. NN -> 1 <_ N )
15	2, 14	syl 17	. . . . . . . . . 10 |- ( ph -> 1 <_ N )
16	9, 10, 11, 13, 15	ltletrd 10197	. . . . . . . . 9 |- ( ph -> 0 < N )
17	9, 16	ltned 10173	. . . . . . . 8 |- ( ph -> 0 =/= N )
18	17	necomd 2849	. . . . . . 7 |- ( ph -> N =/= 0 )
19	1, 5, 8, 18	mulne0d 10679	. . . . . 6 |- ( ph -> ( 2 x. N ) =/= 0 )
20		knoppndvlem2.i	. . . . . 6 |- ( ph -> I e. ZZ )
21	6, 19, 20	expclzd 13013	. . . . 5 |- ( ph -> ( ( 2 x. N ) ^ I ) e. CC )
22		knoppndvlem2.j	. . . . . . . 8 |- ( ph -> J e. ZZ )
23	22	znegcld 11484	. . . . . . 7 |- ( ph -> -u J e. ZZ )
24	6, 19, 23	expclzd 13013	. . . . . 6 |- ( ph -> ( ( 2 x. N ) ^ -u J ) e. CC )
25	24, 1, 8	divcld 10801	. . . . 5 |- ( ph -> ( ( ( 2 x. N ) ^ -u J ) / 2 ) e. CC )
26		knoppndvlem2.m	. . . . . 6 |- ( ph -> M e. ZZ )
27	26	zcnd 11483	. . . . 5 |- ( ph -> M e. CC )
28	21, 25, 27	mulassd 10063	. . . 4 |- ( ph -> ( ( ( ( 2 x. N ) ^ I ) x. ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) x. M ) = ( ( ( 2 x. N ) ^ I ) x. ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) ) )
29	28	eqcomd 2628	. . 3 |- ( ph -> ( ( ( 2 x. N ) ^ I ) x. ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) ) = ( ( ( ( 2 x. N ) ^ I ) x. ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) x. M ) )
30	21, 24, 1, 8	divassd 10836	. . . . . 6 |- ( ph -> ( ( ( ( 2 x. N ) ^ I ) x. ( ( 2 x. N ) ^ -u J ) ) / 2 ) = ( ( ( 2 x. N ) ^ I ) x. ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) )
31	30	eqcomd 2628	. . . . 5 |- ( ph -> ( ( ( 2 x. N ) ^ I ) x. ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) = ( ( ( ( 2 x. N ) ^ I ) x. ( ( 2 x. N ) ^ -u J ) ) / 2 ) )
32	6, 19	jca 554	. . . . . . . . . 10 |- ( ph -> ( ( 2 x. N ) e. CC /\ ( 2 x. N ) =/= 0 ) )
33	20, 23	jca 554	. . . . . . . . . 10 |- ( ph -> ( I e. ZZ /\ -u J e. ZZ ) )
34	32, 33	jca 554	. . . . . . . . 9 |- ( ph -> ( ( ( 2 x. N ) e. CC /\ ( 2 x. N ) =/= 0 ) /\ ( I e. ZZ /\ -u J e. ZZ ) ) )
35		expaddz 12904	. . . . . . . . 9 |- ( ( ( ( 2 x. N ) e. CC /\ ( 2 x. N ) =/= 0 ) /\ ( I e. ZZ /\ -u J e. ZZ ) ) -> ( ( 2 x. N ) ^ ( I + -u J ) ) = ( ( ( 2 x. N ) ^ I ) x. ( ( 2 x. N ) ^ -u J ) ) )
36	34, 35	syl 17	. . . . . . . 8 |- ( ph -> ( ( 2 x. N ) ^ ( I + -u J ) ) = ( ( ( 2 x. N ) ^ I ) x. ( ( 2 x. N ) ^ -u J ) ) )
37	36	eqcomd 2628	. . . . . . 7 |- ( ph -> ( ( ( 2 x. N ) ^ I ) x. ( ( 2 x. N ) ^ -u J ) ) = ( ( 2 x. N ) ^ ( I + -u J ) ) )
38	20	zcnd 11483	. . . . . . . . 9 |- ( ph -> I e. CC )
39	22	zcnd 11483	. . . . . . . . 9 |- ( ph -> J e. CC )
40	38, 39	negsubd 10398	. . . . . . . 8 |- ( ph -> ( I + -u J ) = ( I - J ) )
41	40	oveq2d 6666	. . . . . . 7 |- ( ph -> ( ( 2 x. N ) ^ ( I + -u J ) ) = ( ( 2 x. N ) ^ ( I - J ) ) )
42		knoppndvlem2.1	. . . . . . . . . 10 |- ( ph -> J < I )
43	22, 20	jca 554	. . . . . . . . . . 11 |- ( ph -> ( J e. ZZ /\ I e. ZZ ) )
44		znnsub 11423	. . . . . . . . . . 11 |- ( ( J e. ZZ /\ I e. ZZ ) -> ( J < I <-> ( I - J ) e. NN ) )
45	43, 44	syl 17	. . . . . . . . . 10 |- ( ph -> ( J < I <-> ( I - J ) e. NN ) )
46	42, 45	mpbid 222	. . . . . . . . 9 |- ( ph -> ( I - J ) e. NN )
47	6, 46	jca 554	. . . . . . . 8 |- ( ph -> ( ( 2 x. N ) e. CC /\ ( I - J ) e. NN ) )
48		expm1t 12888	. . . . . . . 8 |- ( ( ( 2 x. N ) e. CC /\ ( I - J ) e. NN ) -> ( ( 2 x. N ) ^ ( I - J ) ) = ( ( ( 2 x. N ) ^ ( ( I - J ) - 1 ) ) x. ( 2 x. N ) ) )
49	47, 48	syl 17	. . . . . . 7 |- ( ph -> ( ( 2 x. N ) ^ ( I - J ) ) = ( ( ( 2 x. N ) ^ ( ( I - J ) - 1 ) ) x. ( 2 x. N ) ) )
50	37, 41, 49	3eqtrd 2660	. . . . . 6 |- ( ph -> ( ( ( 2 x. N ) ^ I ) x. ( ( 2 x. N ) ^ -u J ) ) = ( ( ( 2 x. N ) ^ ( ( I - J ) - 1 ) ) x. ( 2 x. N ) ) )
51	50	oveq1d 6665	. . . . 5 |- ( ph -> ( ( ( ( 2 x. N ) ^ I ) x. ( ( 2 x. N ) ^ -u J ) ) / 2 ) = ( ( ( ( 2 x. N ) ^ ( ( I - J ) - 1 ) ) x. ( 2 x. N ) ) / 2 ) )
52	20, 22	jca 554	. . . . . . . . . . . 12 |- ( ph -> ( I e. ZZ /\ J e. ZZ ) )
53		zsubcl 11419	. . . . . . . . . . . 12 |- ( ( I e. ZZ /\ J e. ZZ ) -> ( I - J ) e. ZZ )
54	52, 53	syl 17	. . . . . . . . . . 11 |- ( ph -> ( I - J ) e. ZZ )
55		peano2zm 11420	. . . . . . . . . . 11 |- ( ( I - J ) e. ZZ -> ( ( I - J ) - 1 ) e. ZZ )
56	54, 55	syl 17	. . . . . . . . . 10 |- ( ph -> ( ( I - J ) - 1 ) e. ZZ )
57	22	zred 11482	. . . . . . . . . . . . 13 |- ( ph -> J e. RR )
58	20	zred 11482	. . . . . . . . . . . . 13 |- ( ph -> I e. RR )
59	57, 58	posdifd 10614	. . . . . . . . . . . 12 |- ( ph -> ( J < I <-> 0 < ( I - J ) ) )
60	42, 59	mpbid 222	. . . . . . . . . . 11 |- ( ph -> 0 < ( I - J ) )
61		0zd 11389	. . . . . . . . . . . . 13 |- ( ph -> 0 e. ZZ )
62	61, 54	jca 554	. . . . . . . . . . . 12 |- ( ph -> ( 0 e. ZZ /\ ( I - J ) e. ZZ ) )
63		zltlem1 11430	. . . . . . . . . . . 12 |- ( ( 0 e. ZZ /\ ( I - J ) e. ZZ ) -> ( 0 < ( I - J ) <-> 0 <_ ( ( I - J ) - 1 ) ) )
64	62, 63	syl 17	. . . . . . . . . . 11 |- ( ph -> ( 0 < ( I - J ) <-> 0 <_ ( ( I - J ) - 1 ) ) )
65	60, 64	mpbid 222	. . . . . . . . . 10 |- ( ph -> 0 <_ ( ( I - J ) - 1 ) )
66	56, 65	jca 554	. . . . . . . . 9 |- ( ph -> ( ( ( I - J ) - 1 ) e. ZZ /\ 0 <_ ( ( I - J ) - 1 ) ) )
67		elnn0z 11390	. . . . . . . . 9 |- ( ( ( I - J ) - 1 ) e. NN0 <-> ( ( ( I - J ) - 1 ) e. ZZ /\ 0 <_ ( ( I - J ) - 1 ) ) )
68	66, 67	sylibr 224	. . . . . . . 8 |- ( ph -> ( ( I - J ) - 1 ) e. NN0 )
69	6, 68	expcld 13008	. . . . . . 7 |- ( ph -> ( ( 2 x. N ) ^ ( ( I - J ) - 1 ) ) e. CC )
70	69, 6, 1, 8	divassd 10836	. . . . . 6 |- ( ph -> ( ( ( ( 2 x. N ) ^ ( ( I - J ) - 1 ) ) x. ( 2 x. N ) ) / 2 ) = ( ( ( 2 x. N ) ^ ( ( I - J ) - 1 ) ) x. ( ( 2 x. N ) / 2 ) ) )
71	5, 1, 8	divcan3d 10806	. . . . . . 7 |- ( ph -> ( ( 2 x. N ) / 2 ) = N )
72	71	oveq2d 6666	. . . . . 6 |- ( ph -> ( ( ( 2 x. N ) ^ ( ( I - J ) - 1 ) ) x. ( ( 2 x. N ) / 2 ) ) = ( ( ( 2 x. N ) ^ ( ( I - J ) - 1 ) ) x. N ) )
73	70, 72	eqtrd 2656	. . . . 5 |- ( ph -> ( ( ( ( 2 x. N ) ^ ( ( I - J ) - 1 ) ) x. ( 2 x. N ) ) / 2 ) = ( ( ( 2 x. N ) ^ ( ( I - J ) - 1 ) ) x. N ) )
74	31, 51, 73	3eqtrd 2660	. . . 4 |- ( ph -> ( ( ( 2 x. N ) ^ I ) x. ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) = ( ( ( 2 x. N ) ^ ( ( I - J ) - 1 ) ) x. N ) )
75	74	oveq1d 6665	. . 3 |- ( ph -> ( ( ( ( 2 x. N ) ^ I ) x. ( ( ( 2 x. N ) ^ -u J ) / 2 ) ) x. M ) = ( ( ( ( 2 x. N ) ^ ( ( I - J ) - 1 ) ) x. N ) x. M ) )
76	29, 75	eqtrd 2656	. 2 |- ( ph -> ( ( ( 2 x. N ) ^ I ) x. ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) ) = ( ( ( ( 2 x. N ) ^ ( ( I - J ) - 1 ) ) x. N ) x. M ) )
77		2z 11409	. . . . . . . . 9 |- 2 e. ZZ
78	77	a1i 11	. . . . . . . 8 |- ( ph -> 2 e. ZZ )
79	78, 4	jca 554	. . . . . . 7 |- ( ph -> ( 2 e. ZZ /\ N e. ZZ ) )
80		zmulcl 11426	. . . . . . 7 |- ( ( 2 e. ZZ /\ N e. ZZ ) -> ( 2 x. N ) e. ZZ )
81	79, 80	syl 17	. . . . . 6 |- ( ph -> ( 2 x. N ) e. ZZ )
82	81, 68	jca 554	. . . . 5 |- ( ph -> ( ( 2 x. N ) e. ZZ /\ ( ( I - J ) - 1 ) e. NN0 ) )
83		zexpcl 12875	. . . . 5 |- ( ( ( 2 x. N ) e. ZZ /\ ( ( I - J ) - 1 ) e. NN0 ) -> ( ( 2 x. N ) ^ ( ( I - J ) - 1 ) ) e. ZZ )
84	82, 83	syl 17	. . . 4 |- ( ph -> ( ( 2 x. N ) ^ ( ( I - J ) - 1 ) ) e. ZZ )
85	84, 4	zmulcld 11488	. . 3 |- ( ph -> ( ( ( 2 x. N ) ^ ( ( I - J ) - 1 ) ) x. N ) e. ZZ )
86	85, 26	zmulcld 11488	. 2 |- ( ph -> ( ( ( ( 2 x. N ) ^ ( ( I - J ) - 1 ) ) x. N ) x. M ) e. ZZ )
87	76, 86	eqeltrd 2701	1 |- ( ph -> ( ( ( 2 x. N ) ^ I ) x. ( ( ( ( 2 x. N ) ^ -u J ) / 2 ) x. M ) ) e. ZZ )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ^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

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-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-seq 12802  df-exp 12861

This theorem is referenced by:  knoppndvlem6  32508