trilpolemlt1 - Mathbox for Jim Kingdon

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Theorem trilpolemlt1 15980

Description: Lemma for trilpo 15982. The

case. We can use the distance between

and one (that is,

) to find a position in the sequence

where terms after that point will not add up to as much as

. By finomni 7242 we know the terms up to

either contain a zero or are all one. But if they are all one that contradicts the way we constructed

, so we know that the sequence contains a zero. (Contributed by Jim Kingdon, 23-Aug-2023.)

**Hypotheses**
Ref	Expression
trilpolemgt1.f
trilpolemgt1.a
trilpolemlt1.a

**Assertion**
Ref	Expression
trilpolemlt1

Distinct variable groups:

Proof of Theorem trilpolemlt1 Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1red 8087	. . . 4
2		trilpolemgt1.f	. . . . 5
3		trilpolemgt1.a	. . . . 5
4	2, 3	trilpolemcl 15976	. . . 4
5	1, 4	resubcld 8453	. . 3
6		trilpolemlt1.a	. . . 4
7	4, 1	posdifd 8605	. . . 4
8	6, 7	mpbid 147	. . 3
9		nnrecl 9293	. . 3 $/\$
10	5, 8, 9	syl2anc 411	. 2
11		elfznn 10176	. . . . . . 7
12	11	ad2antrl 490	. . . . . 6 $/\$ $/\$ $/\$ $/\$
13		simprl 529	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
14	13	fvresd 5601	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
15		simprr 531	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
16	14, 15	eqtr3d 2240	. . . . . 6 $/\$ $/\$ $/\$ $/\$
17	12, 16	jca 306	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
18	17	ex 115	. . . 4 $/\$ $/\$ $/\$ $/\$
19	18	reximdv2 2605	. . 3 $/\$ $/\$
20		2rp 9780	. . . . . . . . . 10
21	20	a1i 9	. . . . . . . . 9 $/\$ $/\$
22		simprl 529	. . . . . . . . . 10 $/\$ $/\$
23	22	nnzd 9494	. . . . . . . . 9 $/\$ $/\$
24	21, 23	rpexpcld 10842	. . . . . . . 8 $/\$ $/\$
25	24	rprecred 9830	. . . . . . 7 $/\$ $/\$
26	22	nnrecred 9083	. . . . . . 7 $/\$ $/\$
27	5	adantr 276	. . . . . . 7 $/\$ $/\$
28		2z 9400	. . . . . . . . . 10
29		uzid 9662	. . . . . . . . . 10
30	28, 29	mp1i 10	. . . . . . . . 9 $/\$ $/\$
31	22	nnnn0d 9348	. . . . . . . . 9 $/\$ $/\$
32		bernneq3 10807	. . . . . . . . 9 $/\$
33	30, 31, 32	syl2anc 411	. . . . . . . 8 $/\$ $/\$
34	22	nnrpd 9816	. . . . . . . . 9 $/\$ $/\$
35	34, 24	ltrecd 9837	. . . . . . . 8 $/\$ $/\$
36	33, 35	mpbid 147	. . . . . . 7 $/\$ $/\$
37		simprr 531	. . . . . . 7 $/\$ $/\$
38	25, 26, 27, 36, 37	lttrd 8198	. . . . . 6 $/\$ $/\$
39	38	adantr 276	. . . . 5 $/\$ $/\$ $/\$
40	27	adantr 276	. . . . . 6 $/\$ $/\$ $/\$
41	25	adantr 276	. . . . . 6 $/\$ $/\$ $/\$
42		1red 8087	. . . . . . 7 $/\$ $/\$ $/\$
43	4	ad2antrr 488	. . . . . . 7 $/\$ $/\$ $/\$
44		0red 8073	. . . . . . . . . 10 $/\$ $/\$ $/\$
45		eqid 2205	. . . . . . . . . . 11
46	22	adantr 276	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
47	46	peano2nnd 9051	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
48	47	nnzd 9494	. . . . . . . . . . 11 $/\$ $/\$ $/\$
49		eluznn 9721	. . . . . . . . . . . . 13 $/\$
50	47, 49	sylan 283	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
51		eqid 2205	. . . . . . . . . . . . 13
52		oveq2 5952	. . . . . . . . . . . . . . 15
53	52	oveq2d 5960	. . . . . . . . . . . . . 14
54		fveq2 5576	. . . . . . . . . . . . . 14
55	53, 54	oveq12d 5962	. . . . . . . . . . . . 13
56		simpr 110	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
57	20	a1i 9	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
58	56	nnzd 9494	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
59	57, 58	rpexpcld 10842	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
60	59	rprecred 9830	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
61		0re 8072	. . . . . . . . . . . . . . . 16
62		1re 8071	. . . . . . . . . . . . . . . 16
63		prssi 3791	. . . . . . . . . . . . . . . 16 $/\$
64	61, 62, 63	mp2an 426	. . . . . . . . . . . . . . 15
65	2	adantr 276	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
66	65	ad2antrr 488	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
67	66, 56	ffvelcdmd 5716	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
68	64, 67	sselid 3191	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
69	60, 68	remulcld 8103	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
70	51, 55, 56, 69	fvmptd3 5673	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
71	50, 70	syldan 282	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
72	50, 69	syldan 282	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
73	65	adantr 276	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
74	73, 51	trilpolemclim 15975	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
75		nnuz 9684	. . . . . . . . . . . . 13
76	69	recnd 8101	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
77	70, 76	eqeltrd 2282	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
78	75, 47, 77	iserex 11650	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
79	74, 78	mpbid 147	. . . . . . . . . . 11 $/\$ $/\$ $/\$
80	45, 48, 71, 72, 79	isumrecl 11740	. . . . . . . . . 10 $/\$ $/\$ $/\$
81		1zzd 9399	. . . . . . . . . . . . 13 $/\$ $/\$
82	81, 23	fzfigd 10576	. . . . . . . . . . . 12 $/\$ $/\$
83	82	adantr 276	. . . . . . . . . . 11 $/\$ $/\$ $/\$
84	20	a1i 9	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
85		elfzelz 10147	. . . . . . . . . . . . . 14
86	85	adantl 277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
87	84, 86	rpexpcld 10842	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
88	87	rprecred 9830	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	83, 88	fsumrecl 11712	. . . . . . . . . 10 $/\$ $/\$ $/\$
90	50, 60	syldan 282	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
91	50, 68	syldan 282	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
92	59	rpreccld 9829	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
93	50, 92	syldan 282	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
94	93	rpge0d 9822	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
95		0le0 9125	. . . . . . . . . . . . . 14
96		simpr 110	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
97	95, 96	breqtrrid 4082	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
98		0le1 8554	. . . . . . . . . . . . . 14
99		simpr 110	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
100	98, 99	breqtrrid 4082	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
101	73	adantr 276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
102	101, 50	ffvelcdmd 5716	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
103		elpri 3656	. . . . . . . . . . . . . 14 $\/$
104	102, 103	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $\/$
105	97, 100, 104	mpjaodan 800	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
106	90, 91, 94, 105	mulge0d 8694	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
107	45, 48, 71, 72, 79, 106	isumge0 11741	. . . . . . . . . 10 $/\$ $/\$ $/\$
108	44, 80, 89, 107	leadd2dd 8633	. . . . . . . . 9 $/\$ $/\$ $/\$
109	89	recnd 8101	. . . . . . . . . . 11 $/\$ $/\$ $/\$
110	109	addridd 8221	. . . . . . . . . 10 $/\$ $/\$ $/\$
111	110	eqcomd 2211	. . . . . . . . 9 $/\$ $/\$ $/\$
112	75, 45, 47, 70, 76, 74	isumsplit 11802	. . . . . . . . . . 11 $/\$ $/\$ $/\$
113	3, 112	eqtrid 2250	. . . . . . . . . 10 $/\$ $/\$ $/\$
114	46	nncnd 9050	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
115		1cnd 8088	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
116	114, 115	pncand 8384	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
117	116	oveq2d 5960	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
118		simpr 110	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
119	118	fvresd 5601	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
120		fveqeq2 5585	. . . . . . . . . . . . . . . 16
121		simplr 528	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
122	120, 121, 118	rspcdva 2882	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
123	119, 122	eqtr3d 2240	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
124	123	oveq2d 5960	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
125	87	rpreccld 9829	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
126	125	rpcnd 9820	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
127	126	mulridd 8089	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
128	124, 127	eqtrd 2238	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
129	117, 128	sumeq12rdv 11684	. . . . . . . . . . 11 $/\$ $/\$ $/\$
130	129	oveq1d 5959	. . . . . . . . . 10 $/\$ $/\$ $/\$
131	113, 130	eqtrd 2238	. . . . . . . . 9 $/\$ $/\$ $/\$
132	108, 111, 131	3brtr4d 4076	. . . . . . . 8 $/\$ $/\$ $/\$
133		geo2sum 11825	. . . . . . . . . 10 $/\$
134	133	breq1d 4054	. . . . . . . . 9 $/\$
135	46, 115, 134	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
136	132, 135	mpbid 147	. . . . . . 7 $/\$ $/\$ $/\$
137	42, 41, 43, 136	subled 8621	. . . . . 6 $/\$ $/\$ $/\$
138	40, 41, 137	lensymd 8194	. . . . 5 $/\$ $/\$ $/\$
139	39, 138	pm2.21dd 621	. . . 4 $/\$ $/\$ $/\$
140	139	ex 115	. . 3 $/\$ $/\$
141		fveq1 5575	. . . . . . 7
142	141	eqeq1d 2214	. . . . . 6
143	142	rexbidv 2507	. . . . 5
144	141	eqeq1d 2214	. . . . . 6
145	144	ralbidv 2506	. . . . 5
146	143, 145	orbi12d 795	. . . 4 $\/$ $\/$
147		finomni 7242	. . . . . 6 Omni
148	82, 147	syl 14	. . . . 5 $/\$ $/\$ Omni
149		isomninn 15970	. . . . . 6 Omni Omni $\/$
150	148, 149	syl 14	. . . . 5 $/\$ $/\$ Omni $\/$
151	148, 150	mpbid 147	. . . 4 $/\$ $/\$ $\/$
152		fz1ssnn 10178	. . . . . . 7
153	152	a1i 9	. . . . . 6 $/\$ $/\$
154	65, 153	fssresd 5452	. . . . 5 $/\$ $/\$
155		0red 8073	. . . . . . 7 $/\$ $/\$
156		1red 8087	. . . . . . 7 $/\$ $/\$
157		prexg 4255	. . . . . . 7 $/\$
158	155, 156, 157	syl2anc 411	. . . . . 6 $/\$ $/\$
159	158, 82	elmapd 6749	. . . . 5 $/\$ $/\$
160	154, 159	mpbird 167	. . . 4 $/\$ $/\$
161	146, 151, 160	rspcdva 2882	. . 3 $/\$ $/\$ $\/$
162	19, 140, 161	mpjaod 720	. 2 $/\$ $/\$
163	10, 162	rexlimddv 2628	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 710

wceq 1373

wcel 2176

wral 2484

wrex 2485

cvv 2772

wss 3166

cpr 3634 class class class wbr 4044

cmpt 4105

cdm 4675

cres 4677

wf 5267

cfv 5271 (class class class)co 5944

cmap 6735

cfn 6827 Omnicomni 7236

cc 7923

cr 7924

cc0 7925

c1 7926

caddc 7928

cmul 7930

clt 8107

cle 8108

cmin 8243

cdiv 8745

cn 9036

c2 9087

cn0 9295

cz 9372

cuz 9648

crp 9775

cfz 10130

cseq 10592

cexp 10683

cli 11589

csu 11664

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 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-iinf 4636 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 ax-pre-mulgt0 8042 ax-pre-mulext 8043 ax-arch 8044 ax-caucvg 8045

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-tr 4143 df-id 4340 df-po 4343 df-iso 4344 df-iord 4413 df-on 4415 df-ilim 4416 df-suc 4418 df-iom 4639 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-isom 5280 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-recs 6391 df-irdg 6456 df-frec 6477 df-1o 6502 df-2o 6503 df-oadd 6506 df-er 6620 df-map 6737 df-en 6828 df-dom 6829 df-fin 6830 df-omni 7237 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-reap 8648 df-ap 8655 df-div 8746 df-inn 9037 df-2 9095 df-3 9096 df-4 9097 df-n0 9296 df-z 9373 df-uz 9649 df-q 9741 df-rp 9776 df-ico 10016 df-fz 10131 df-fzo 10265 df-seqfrec 10593 df-exp 10684 df-ihash 10921 df-cj 11153 df-re 11154 df-im 11155 df-rsqrt 11309 df-abs 11310 df-clim 11590 df-sumdc 11665

This theorem is referenced by: trilpolemres 15981 neapmkvlem 16006

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