trilpolemlt1 - Mathbox for Jim Kingdon

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Theorem trilpolemlt1 14073

Description: Lemma for trilpo 14075. 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 7116 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 7935	. . . 4
2		trilpolemgt1.f	. . . . 5
3		trilpolemgt1.a	. . . . 5
4	2, 3	trilpolemcl 14069	. . . 4
5	1, 4	resubcld 8300	. . 3
6		trilpolemlt1.a	. . . 4
7	4, 1	posdifd 8451	. . . 4
8	6, 7	mpbid 146	. . 3
9		nnrecl 9133	. . 3 $/\$
10	5, 8, 9	syl2anc 409	. 2
11		elfznn 10010	. . . . . . 7
12	11	ad2antrl 487	. . . . . 6 $/\$ $/\$ $/\$ $/\$
13		simprl 526	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
14	13	fvresd 5521	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
15		simprr 527	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
16	14, 15	eqtr3d 2205	. . . . . 6 $/\$ $/\$ $/\$ $/\$
17	12, 16	jca 304	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
18	17	ex 114	. . . 4 $/\$ $/\$ $/\$ $/\$
19	18	reximdv2 2569	. . 3 $/\$ $/\$
20		2rp 9615	. . . . . . . . . 10
21	20	a1i 9	. . . . . . . . 9 $/\$ $/\$
22		simprl 526	. . . . . . . . . 10 $/\$ $/\$
23	22	nnzd 9333	. . . . . . . . 9 $/\$ $/\$
24	21, 23	rpexpcld 10633	. . . . . . . 8 $/\$ $/\$
25	24	rprecred 9665	. . . . . . 7 $/\$ $/\$
26	22	nnrecred 8925	. . . . . . 7 $/\$ $/\$
27	5	adantr 274	. . . . . . 7 $/\$ $/\$
28		2z 9240	. . . . . . . . . 10
29		uzid 9501	. . . . . . . . . 10
30	28, 29	mp1i 10	. . . . . . . . 9 $/\$ $/\$
31	22	nnnn0d 9188	. . . . . . . . 9 $/\$ $/\$
32		bernneq3 10598	. . . . . . . . 9 $/\$
33	30, 31, 32	syl2anc 409	. . . . . . . 8 $/\$ $/\$
34	22	nnrpd 9651	. . . . . . . . 9 $/\$ $/\$
35	34, 24	ltrecd 9672	. . . . . . . 8 $/\$ $/\$
36	33, 35	mpbid 146	. . . . . . 7 $/\$ $/\$
37		simprr 527	. . . . . . 7 $/\$ $/\$
38	25, 26, 27, 36, 37	lttrd 8045	. . . . . 6 $/\$ $/\$
39	38	adantr 274	. . . . 5 $/\$ $/\$ $/\$
40	27	adantr 274	. . . . . 6 $/\$ $/\$ $/\$
41	25	adantr 274	. . . . . 6 $/\$ $/\$ $/\$
42		1red 7935	. . . . . . 7 $/\$ $/\$ $/\$
43	4	ad2antrr 485	. . . . . . 7 $/\$ $/\$ $/\$
44		0red 7921	. . . . . . . . . 10 $/\$ $/\$ $/\$
45		eqid 2170	. . . . . . . . . . 11
46	22	adantr 274	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
47	46	peano2nnd 8893	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
48	47	nnzd 9333	. . . . . . . . . . 11 $/\$ $/\$ $/\$
49		eluznn 9559	. . . . . . . . . . . . 13 $/\$
50	47, 49	sylan 281	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
51		eqid 2170	. . . . . . . . . . . . 13
52		oveq2 5861	. . . . . . . . . . . . . . 15
53	52	oveq2d 5869	. . . . . . . . . . . . . 14
54		fveq2 5496	. . . . . . . . . . . . . 14
55	53, 54	oveq12d 5871	. . . . . . . . . . . . 13
56		simpr 109	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
57	20	a1i 9	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
58	56	nnzd 9333	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
59	57, 58	rpexpcld 10633	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
60	59	rprecred 9665	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
61		0re 7920	. . . . . . . . . . . . . . . 16
62		1re 7919	. . . . . . . . . . . . . . . 16
63		prssi 3738	. . . . . . . . . . . . . . . 16 $/\$
64	61, 62, 63	mp2an 424	. . . . . . . . . . . . . . 15
65	2	adantr 274	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
66	65	ad2antrr 485	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
67	66, 56	ffvelrnd 5632	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
68	64, 67	sselid 3145	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
69	60, 68	remulcld 7950	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
70	51, 55, 56, 69	fvmptd3 5589	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
71	50, 70	syldan 280	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
72	50, 69	syldan 280	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
73	65	adantr 274	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
74	73, 51	trilpolemclim 14068	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
75		nnuz 9522	. . . . . . . . . . . . 13
76	69	recnd 7948	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
77	70, 76	eqeltrd 2247	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
78	75, 47, 77	iserex 11302	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
79	74, 78	mpbid 146	. . . . . . . . . . 11 $/\$ $/\$ $/\$
80	45, 48, 71, 72, 79	isumrecl 11392	. . . . . . . . . 10 $/\$ $/\$ $/\$
81		1zzd 9239	. . . . . . . . . . . . 13 $/\$ $/\$
82	81, 23	fzfigd 10387	. . . . . . . . . . . 12 $/\$ $/\$
83	82	adantr 274	. . . . . . . . . . 11 $/\$ $/\$ $/\$
84	20	a1i 9	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
85		elfzelz 9981	. . . . . . . . . . . . . 14
86	85	adantl 275	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
87	84, 86	rpexpcld 10633	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
88	87	rprecred 9665	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	83, 88	fsumrecl 11364	. . . . . . . . . 10 $/\$ $/\$ $/\$
90	50, 60	syldan 280	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
91	50, 68	syldan 280	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
92	59	rpreccld 9664	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
93	50, 92	syldan 280	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
94	93	rpge0d 9657	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
95		0le0 8967	. . . . . . . . . . . . . 14
96		simpr 109	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
97	95, 96	breqtrrid 4027	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
98		0le1 8400	. . . . . . . . . . . . . 14
99		simpr 109	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
100	98, 99	breqtrrid 4027	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
101	73	adantr 274	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
102	101, 50	ffvelrnd 5632	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
103		elpri 3606	. . . . . . . . . . . . . 14 $\/$
104	102, 103	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $\/$
105	97, 100, 104	mpjaodan 793	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
106	90, 91, 94, 105	mulge0d 8540	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
107	45, 48, 71, 72, 79, 106	isumge0 11393	. . . . . . . . . 10 $/\$ $/\$ $/\$
108	44, 80, 89, 107	leadd2dd 8479	. . . . . . . . 9 $/\$ $/\$ $/\$
109	89	recnd 7948	. . . . . . . . . . 11 $/\$ $/\$ $/\$
110	109	addid1d 8068	. . . . . . . . . 10 $/\$ $/\$ $/\$
111	110	eqcomd 2176	. . . . . . . . 9 $/\$ $/\$ $/\$
112	75, 45, 47, 70, 76, 74	isumsplit 11454	. . . . . . . . . . 11 $/\$ $/\$ $/\$
113	3, 112	eqtrid 2215	. . . . . . . . . 10 $/\$ $/\$ $/\$
114	46	nncnd 8892	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
115		1cnd 7936	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
116	114, 115	pncand 8231	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
117	116	oveq2d 5869	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
118		simpr 109	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
119	118	fvresd 5521	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
120		fveqeq2 5505	. . . . . . . . . . . . . . . 16
121		simplr 525	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
122	120, 121, 118	rspcdva 2839	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
123	119, 122	eqtr3d 2205	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
124	123	oveq2d 5869	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
125	87	rpreccld 9664	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
126	125	rpcnd 9655	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
127	126	mulid1d 7937	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
128	124, 127	eqtrd 2203	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
129	117, 128	sumeq12rdv 11336	. . . . . . . . . . 11 $/\$ $/\$ $/\$
130	129	oveq1d 5868	. . . . . . . . . 10 $/\$ $/\$ $/\$
131	113, 130	eqtrd 2203	. . . . . . . . 9 $/\$ $/\$ $/\$
132	108, 111, 131	3brtr4d 4021	. . . . . . . 8 $/\$ $/\$ $/\$
133		geo2sum 11477	. . . . . . . . . 10 $/\$
134	133	breq1d 3999	. . . . . . . . 9 $/\$
135	46, 115, 134	syl2anc 409	. . . . . . . 8 $/\$ $/\$ $/\$
136	132, 135	mpbid 146	. . . . . . 7 $/\$ $/\$ $/\$
137	42, 41, 43, 136	subled 8467	. . . . . 6 $/\$ $/\$ $/\$
138	40, 41, 137	lensymd 8041	. . . . 5 $/\$ $/\$ $/\$
139	39, 138	pm2.21dd 615	. . . 4 $/\$ $/\$ $/\$
140	139	ex 114	. . 3 $/\$ $/\$
141		fveq1 5495	. . . . . . 7
142	141	eqeq1d 2179	. . . . . 6
143	142	rexbidv 2471	. . . . 5
144	141	eqeq1d 2179	. . . . . 6
145	144	ralbidv 2470	. . . . 5
146	143, 145	orbi12d 788	. . . 4 $\/$ $\/$
147		finomni 7116	. . . . . 6 Omni
148	82, 147	syl 14	. . . . 5 $/\$ $/\$ Omni
149		isomninn 14063	. . . . . 6 Omni Omni $\/$
150	148, 149	syl 14	. . . . 5 $/\$ $/\$ Omni $\/$
151	148, 150	mpbid 146	. . . 4 $/\$ $/\$ $\/$
152		fz1ssnn 10012	. . . . . . 7
153	152	a1i 9	. . . . . 6 $/\$ $/\$
154	65, 153	fssresd 5374	. . . . 5 $/\$ $/\$
155		0red 7921	. . . . . . 7 $/\$ $/\$
156		1red 7935	. . . . . . 7 $/\$ $/\$
157		prexg 4196	. . . . . . 7 $/\$
158	155, 156, 157	syl2anc 409	. . . . . 6 $/\$ $/\$
159	158, 82	elmapd 6640	. . . . 5 $/\$ $/\$
160	154, 159	mpbird 166	. . . 4 $/\$ $/\$
161	146, 151, 160	rspcdva 2839	. . 3 $/\$ $/\$ $\/$
162	19, 140, 161	mpjaod 713	. 2 $/\$ $/\$
163	10, 162	rexlimddv 2592	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $\/$ wo 703

wceq 1348

wcel 2141

wral 2448

wrex 2449

cvv 2730

wss 3121

cpr 3584 class class class wbr 3989

cmpt 4050

cdm 4611

cres 4613

wf 5194

cfv 5198 (class class class)co 5853

cmap 6626

cfn 6718 Omnicomni 7110

cc 7772

cr 7773

cc0 7774

c1 7775

caddc 7777

cmul 7779

clt 7954

cle 7955

cmin 8090

cdiv 8589

cn 8878

c2 8929

cn0 9135

cz 9212

cuz 9487

crp 9610

cfz 9965

cseq 10401

cexp 10475

cli 11241

csu 11316

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-2o 6396 df-oadd 6399 df-er 6513 df-map 6628 df-en 6719 df-dom 6720 df-fin 6721 df-omni 7111 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-ico 9851 df-fz 9966 df-fzo 10099 df-seqfrec 10402 df-exp 10476 df-ihash 10710 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-clim 11242 df-sumdc 11317

This theorem is referenced by: trilpolemres 14074 neapmkvlem 14098

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