trilpolemlt1 - Mathbox for Jim Kingdon

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Theorem trilpolemlt1 16951

Description: Lemma for trilpo 16953. 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 7444 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 8305	. . . 4
2		trilpolemgt1.f	. . . . 5
3		trilpolemgt1.a	. . . . 5
4	2, 3	trilpolemcl 16947	. . . 4
5	1, 4	resubcld 8671	. . 3
6		trilpolemlt1.a	. . . 4
7	4, 1	posdifd 8823	. . . 4
8	6, 7	mpbid 147	. . 3
9		nnrecl 9511	. . 3 $/\$
10	5, 8, 9	syl2anc 411	. 2
11		elfznn 10409	. . . . . . 7
12	11	ad2antrl 490	. . . . . 6 $/\$ $/\$ $/\$ $/\$
13		simprl 531	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
14	13	fvresd 5700	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
15		simprr 533	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
16	14, 15	eqtr3d 2269	. . . . . 6 $/\$ $/\$ $/\$ $/\$
17	12, 16	jca 306	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
18	17	ex 115	. . . 4 $/\$ $/\$ $/\$ $/\$
19	18	reximdv2 2643	. . 3 $/\$ $/\$
20		2rp 10009	. . . . . . . . . 10
21	20	a1i 9	. . . . . . . . 9 $/\$ $/\$
22		simprl 531	. . . . . . . . . 10 $/\$ $/\$
23	22	nnzd 9717	. . . . . . . . 9 $/\$ $/\$
24	21, 23	rpexpcld 11084	. . . . . . . 8 $/\$ $/\$
25	24	rprecred 10059	. . . . . . 7 $/\$ $/\$
26	22	nnrecred 9301	. . . . . . 7 $/\$ $/\$
27	5	adantr 276	. . . . . . 7 $/\$ $/\$
28		2z 9622	. . . . . . . . . 10
29		uzid 9886	. . . . . . . . . 10
30	28, 29	mp1i 10	. . . . . . . . 9 $/\$ $/\$
31	22	nnnn0d 9570	. . . . . . . . 9 $/\$ $/\$
32		bernneq3 11049	. . . . . . . . 9 $/\$
33	30, 31, 32	syl2anc 411	. . . . . . . 8 $/\$ $/\$
34	22	nnrpd 10045	. . . . . . . . 9 $/\$ $/\$
35	34, 24	ltrecd 10066	. . . . . . . 8 $/\$ $/\$
36	33, 35	mpbid 147	. . . . . . 7 $/\$ $/\$
37		simprr 533	. . . . . . 7 $/\$ $/\$
38	25, 26, 27, 36, 37	lttrd 8415	. . . . . 6 $/\$ $/\$
39	38	adantr 276	. . . . 5 $/\$ $/\$ $/\$
40	27	adantr 276	. . . . . 6 $/\$ $/\$ $/\$
41	25	adantr 276	. . . . . 6 $/\$ $/\$ $/\$
42		1red 8305	. . . . . . 7 $/\$ $/\$ $/\$
43	4	ad2antrr 488	. . . . . . 7 $/\$ $/\$ $/\$
44		0red 8291	. . . . . . . . . 10 $/\$ $/\$ $/\$
45		eqid 2234	. . . . . . . . . . 11
46	22	adantr 276	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
47	46	peano2nnd 9269	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
48	47	nnzd 9717	. . . . . . . . . . 11 $/\$ $/\$ $/\$
49		eluznn 9950	. . . . . . . . . . . . 13 $/\$
50	47, 49	sylan 283	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
51		eqid 2234	. . . . . . . . . . . . 13
52		oveq2 6066	. . . . . . . . . . . . . . 15
53	52	oveq2d 6074	. . . . . . . . . . . . . 14
54		fveq2 5675	. . . . . . . . . . . . . 14
55	53, 54	oveq12d 6076	. . . . . . . . . . . . 13
56		simpr 110	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
57	20	a1i 9	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
58	56	nnzd 9717	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
59	57, 58	rpexpcld 11084	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
60	59	rprecred 10059	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
61		0re 8290	. . . . . . . . . . . . . . . 16
62		1re 8289	. . . . . . . . . . . . . . . 16
63		prssi 3857	. . . . . . . . . . . . . . . 16 $/\$
64	61, 62, 63	mp2an 426	. . . . . . . . . . . . . . 15
65	2	adantr 276	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
66	65	ad2antrr 488	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
67	66, 56	ffvelcdmd 5818	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
68	64, 67	sselid 3240	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
69	60, 68	remulcld 8320	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
70	51, 55, 56, 69	fvmptd3 5776	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
71	50, 70	syldan 282	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
72	50, 69	syldan 282	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
73	65	adantr 276	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
74	73, 51	trilpolemclim 16946	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
75		nnuz 9908	. . . . . . . . . . . . 13
76	69	recnd 8318	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
77	70, 76	eqeltrd 2311	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
78	75, 47, 77	iserex 12049	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
79	74, 78	mpbid 147	. . . . . . . . . . 11 $/\$ $/\$ $/\$
80	45, 48, 71, 72, 79	isumrecl 12140	. . . . . . . . . 10 $/\$ $/\$ $/\$
81		1zzd 9621	. . . . . . . . . . . . 13 $/\$ $/\$
82	81, 23	fzfigd 10817	. . . . . . . . . . . 12 $/\$ $/\$
83	82	adantr 276	. . . . . . . . . . 11 $/\$ $/\$ $/\$
84	20	a1i 9	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
85		elfzelz 10378	. . . . . . . . . . . . . 14
86	85	adantl 277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
87	84, 86	rpexpcld 11084	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
88	87	rprecred 10059	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	83, 88	fsumrecl 12112	. . . . . . . . . 10 $/\$ $/\$ $/\$
90	50, 60	syldan 282	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
91	50, 68	syldan 282	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
92	59	rpreccld 10058	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
93	50, 92	syldan 282	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
94	93	rpge0d 10051	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
95		0le0 9343	. . . . . . . . . . . . . 14
96		simpr 110	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
97	95, 96	breqtrrid 4152	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
98		0le1 8772	. . . . . . . . . . . . . 14
99		simpr 110	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
100	98, 99	breqtrrid 4152	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
101	73	adantr 276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
102	101, 50	ffvelcdmd 5818	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
103		elpri 3717	. . . . . . . . . . . . . 14 $\/$
104	102, 103	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $\/$
105	97, 100, 104	mpjaodan 806	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
106	90, 91, 94, 105	mulge0d 8912	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
107	45, 48, 71, 72, 79, 106	isumge0 12141	. . . . . . . . . 10 $/\$ $/\$ $/\$
108	44, 80, 89, 107	leadd2dd 8851	. . . . . . . . 9 $/\$ $/\$ $/\$
109	89	recnd 8318	. . . . . . . . . . 11 $/\$ $/\$ $/\$
110	109	addridd 8438	. . . . . . . . . 10 $/\$ $/\$ $/\$
111	110	eqcomd 2240	. . . . . . . . 9 $/\$ $/\$ $/\$
112	75, 45, 47, 70, 76, 74	isumsplit 12202	. . . . . . . . . . 11 $/\$ $/\$ $/\$
113	3, 112	eqtrid 2279	. . . . . . . . . 10 $/\$ $/\$ $/\$
114	46	nncnd 9268	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
115		1cnd 8306	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
116	114, 115	pncand 8601	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
117	116	oveq2d 6074	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
118		simpr 110	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
119	118	fvresd 5700	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
120		fveqeq2 5684	. . . . . . . . . . . . . . . 16
121		simplr 529	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
122	120, 121, 118	rspcdva 2928	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
123	119, 122	eqtr3d 2269	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
124	123	oveq2d 6074	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
125	87	rpreccld 10058	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
126	125	rpcnd 10049	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
127	126	mulridd 8307	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
128	124, 127	eqtrd 2267	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
129	117, 128	sumeq12rdv 12083	. . . . . . . . . . 11 $/\$ $/\$ $/\$
130	129	oveq1d 6073	. . . . . . . . . 10 $/\$ $/\$ $/\$
131	113, 130	eqtrd 2267	. . . . . . . . 9 $/\$ $/\$ $/\$
132	108, 111, 131	3brtr4d 4146	. . . . . . . 8 $/\$ $/\$ $/\$
133		geo2sum 12225	. . . . . . . . . 10 $/\$
134	133	breq1d 4124	. . . . . . . . 9 $/\$
135	46, 115, 134	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
136	132, 135	mpbid 147	. . . . . . 7 $/\$ $/\$ $/\$
137	42, 41, 43, 136	subled 8839	. . . . . 6 $/\$ $/\$ $/\$
138	40, 41, 137	lensymd 8411	. . . . 5 $/\$ $/\$ $/\$
139	39, 138	pm2.21dd 625	. . . 4 $/\$ $/\$ $/\$
140	139	ex 115	. . 3 $/\$ $/\$
141		fveq1 5674	. . . . . . 7
142	141	eqeq1d 2243	. . . . . 6
143	142	rexbidv 2545	. . . . 5
144	141	eqeq1d 2243	. . . . . 6
145	144	ralbidv 2544	. . . . 5
146	143, 145	orbi12d 801	. . . 4 $\/$ $\/$
147		finomni 7444	. . . . . 6 Omni
148	82, 147	syl 14	. . . . 5 $/\$ $/\$ Omni
149		isomninn 16941	. . . . . 6 Omni Omni $\/$
150	148, 149	syl 14	. . . . 5 $/\$ $/\$ Omni $\/$
151	148, 150	mpbid 147	. . . 4 $/\$ $/\$ $\/$
152		fz1ssnn 10411	. . . . . . 7
153	152	a1i 9	. . . . . 6 $/\$ $/\$
154	65, 153	fssresd 5546	. . . . 5 $/\$ $/\$
155		0red 8291	. . . . . . 7 $/\$ $/\$
156		1red 8305	. . . . . . 7 $/\$ $/\$
157		prexg 4330	. . . . . . 7 $/\$
158	155, 156, 157	syl2anc 411	. . . . . 6 $/\$ $/\$
159	158, 82	elmapd 6909	. . . . 5 $/\$ $/\$
160	154, 159	mpbird 167	. . . 4 $/\$ $/\$
161	146, 151, 160	rspcdva 2928	. . 3 $/\$ $/\$ $\/$
162	19, 140, 161	mpjaod 726	. 2 $/\$ $/\$
163	10, 162	rexlimddv 2667	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 716

wceq 1398

wcel 2205

wral 2522

wrex 2523

cvv 2815

wss 3214

cpr 3695 class class class wbr 4114

cmpt 4176

cdm 4754

cres 4756

wf 5353

cfv 5357 (class class class)co 6058

cmap 6895

cfn 6988 Omnicomni 7438

cc 8141

cr 8142

cc0 8143

c1 8144

caddc 8146

cmul 8148

clt 8324

cle 8325

cmin 8460

cdiv 8963

cn 9254

c2 9305

cn0 9513

cz 9594

cuz 9871

crp 10004

cfz 10361

cseq 10833

cexp 10924

cli 11988

csu 12063

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-isom 5366 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-irdg 6614 df-frec 6635 df-1o 6660 df-2o 6661 df-oadd 6664 df-er 6780 df-map 6897 df-en 6989 df-dom 6990 df-fin 6991 df-omni 7439 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-n0 9514 df-z 9595 df-uz 9872 df-q 9970 df-rp 10005 df-ico 10246 df-fz 10362 df-fzo 10499 df-seqfrec 10834 df-exp 10925 df-ihash 11164 df-cj 11552 df-re 11553 df-im 11554 df-rsqrt 11708 df-abs 11709 df-clim 11989 df-sumdc 12064

This theorem is referenced by: trilpolemres 16952 neapmkvlem 16979

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