trilpolemlt1 - Mathbox for Jim Kingdon

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Theorem trilpolemlt1 15531

Description: Lemma for trilpo 15533. 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 7199 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 8034	. . . 4
2		trilpolemgt1.f	. . . . 5
3		trilpolemgt1.a	. . . . 5
4	2, 3	trilpolemcl 15527	. . . 4
5	1, 4	resubcld 8400	. . 3
6		trilpolemlt1.a	. . . 4
7	4, 1	posdifd 8551	. . . 4
8	6, 7	mpbid 147	. . 3
9		nnrecl 9238	. . 3 $/\$
10	5, 8, 9	syl2anc 411	. 2
11		elfznn 10120	. . . . . . 7
12	11	ad2antrl 490	. . . . . 6 $/\$ $/\$ $/\$ $/\$
13		simprl 529	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
14	13	fvresd 5579	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
15		simprr 531	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
16	14, 15	eqtr3d 2228	. . . . . 6 $/\$ $/\$ $/\$ $/\$
17	12, 16	jca 306	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
18	17	ex 115	. . . 4 $/\$ $/\$ $/\$ $/\$
19	18	reximdv2 2593	. . 3 $/\$ $/\$
20		2rp 9724	. . . . . . . . . 10
21	20	a1i 9	. . . . . . . . 9 $/\$ $/\$
22		simprl 529	. . . . . . . . . 10 $/\$ $/\$
23	22	nnzd 9438	. . . . . . . . 9 $/\$ $/\$
24	21, 23	rpexpcld 10768	. . . . . . . 8 $/\$ $/\$
25	24	rprecred 9774	. . . . . . 7 $/\$ $/\$
26	22	nnrecred 9029	. . . . . . 7 $/\$ $/\$
27	5	adantr 276	. . . . . . 7 $/\$ $/\$
28		2z 9345	. . . . . . . . . 10
29		uzid 9606	. . . . . . . . . 10
30	28, 29	mp1i 10	. . . . . . . . 9 $/\$ $/\$
31	22	nnnn0d 9293	. . . . . . . . 9 $/\$ $/\$
32		bernneq3 10733	. . . . . . . . 9 $/\$
33	30, 31, 32	syl2anc 411	. . . . . . . 8 $/\$ $/\$
34	22	nnrpd 9760	. . . . . . . . 9 $/\$ $/\$
35	34, 24	ltrecd 9781	. . . . . . . 8 $/\$ $/\$
36	33, 35	mpbid 147	. . . . . . 7 $/\$ $/\$
37		simprr 531	. . . . . . 7 $/\$ $/\$
38	25, 26, 27, 36, 37	lttrd 8145	. . . . . 6 $/\$ $/\$
39	38	adantr 276	. . . . 5 $/\$ $/\$ $/\$
40	27	adantr 276	. . . . . 6 $/\$ $/\$ $/\$
41	25	adantr 276	. . . . . 6 $/\$ $/\$ $/\$
42		1red 8034	. . . . . . 7 $/\$ $/\$ $/\$
43	4	ad2antrr 488	. . . . . . 7 $/\$ $/\$ $/\$
44		0red 8020	. . . . . . . . . 10 $/\$ $/\$ $/\$
45		eqid 2193	. . . . . . . . . . 11
46	22	adantr 276	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
47	46	peano2nnd 8997	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
48	47	nnzd 9438	. . . . . . . . . . 11 $/\$ $/\$ $/\$
49		eluznn 9665	. . . . . . . . . . . . 13 $/\$
50	47, 49	sylan 283	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
51		eqid 2193	. . . . . . . . . . . . 13
52		oveq2 5926	. . . . . . . . . . . . . . 15
53	52	oveq2d 5934	. . . . . . . . . . . . . 14
54		fveq2 5554	. . . . . . . . . . . . . 14
55	53, 54	oveq12d 5936	. . . . . . . . . . . . 13
56		simpr 110	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
57	20	a1i 9	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
58	56	nnzd 9438	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
59	57, 58	rpexpcld 10768	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
60	59	rprecred 9774	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
61		0re 8019	. . . . . . . . . . . . . . . 16
62		1re 8018	. . . . . . . . . . . . . . . 16
63		prssi 3776	. . . . . . . . . . . . . . . 16 $/\$
64	61, 62, 63	mp2an 426	. . . . . . . . . . . . . . 15
65	2	adantr 276	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
66	65	ad2antrr 488	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
67	66, 56	ffvelcdmd 5694	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
68	64, 67	sselid 3177	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
69	60, 68	remulcld 8050	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
70	51, 55, 56, 69	fvmptd3 5651	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
71	50, 70	syldan 282	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
72	50, 69	syldan 282	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
73	65	adantr 276	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
74	73, 51	trilpolemclim 15526	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
75		nnuz 9628	. . . . . . . . . . . . 13
76	69	recnd 8048	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
77	70, 76	eqeltrd 2270	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
78	75, 47, 77	iserex 11482	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
79	74, 78	mpbid 147	. . . . . . . . . . 11 $/\$ $/\$ $/\$
80	45, 48, 71, 72, 79	isumrecl 11572	. . . . . . . . . 10 $/\$ $/\$ $/\$
81		1zzd 9344	. . . . . . . . . . . . 13 $/\$ $/\$
82	81, 23	fzfigd 10502	. . . . . . . . . . . 12 $/\$ $/\$
83	82	adantr 276	. . . . . . . . . . 11 $/\$ $/\$ $/\$
84	20	a1i 9	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
85		elfzelz 10091	. . . . . . . . . . . . . 14
86	85	adantl 277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
87	84, 86	rpexpcld 10768	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
88	87	rprecred 9774	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	83, 88	fsumrecl 11544	. . . . . . . . . 10 $/\$ $/\$ $/\$
90	50, 60	syldan 282	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
91	50, 68	syldan 282	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
92	59	rpreccld 9773	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
93	50, 92	syldan 282	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
94	93	rpge0d 9766	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
95		0le0 9071	. . . . . . . . . . . . . 14
96		simpr 110	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
97	95, 96	breqtrrid 4067	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
98		0le1 8500	. . . . . . . . . . . . . 14
99		simpr 110	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
100	98, 99	breqtrrid 4067	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
101	73	adantr 276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
102	101, 50	ffvelcdmd 5694	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
103		elpri 3641	. . . . . . . . . . . . . 14 $\/$
104	102, 103	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $\/$
105	97, 100, 104	mpjaodan 799	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
106	90, 91, 94, 105	mulge0d 8640	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
107	45, 48, 71, 72, 79, 106	isumge0 11573	. . . . . . . . . 10 $/\$ $/\$ $/\$
108	44, 80, 89, 107	leadd2dd 8579	. . . . . . . . 9 $/\$ $/\$ $/\$
109	89	recnd 8048	. . . . . . . . . . 11 $/\$ $/\$ $/\$
110	109	addridd 8168	. . . . . . . . . 10 $/\$ $/\$ $/\$
111	110	eqcomd 2199	. . . . . . . . 9 $/\$ $/\$ $/\$
112	75, 45, 47, 70, 76, 74	isumsplit 11634	. . . . . . . . . . 11 $/\$ $/\$ $/\$
113	3, 112	eqtrid 2238	. . . . . . . . . 10 $/\$ $/\$ $/\$
114	46	nncnd 8996	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
115		1cnd 8035	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
116	114, 115	pncand 8331	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
117	116	oveq2d 5934	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
118		simpr 110	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
119	118	fvresd 5579	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
120		fveqeq2 5563	. . . . . . . . . . . . . . . 16
121		simplr 528	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
122	120, 121, 118	rspcdva 2869	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
123	119, 122	eqtr3d 2228	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
124	123	oveq2d 5934	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
125	87	rpreccld 9773	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
126	125	rpcnd 9764	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
127	126	mulridd 8036	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
128	124, 127	eqtrd 2226	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
129	117, 128	sumeq12rdv 11516	. . . . . . . . . . 11 $/\$ $/\$ $/\$
130	129	oveq1d 5933	. . . . . . . . . 10 $/\$ $/\$ $/\$
131	113, 130	eqtrd 2226	. . . . . . . . 9 $/\$ $/\$ $/\$
132	108, 111, 131	3brtr4d 4061	. . . . . . . 8 $/\$ $/\$ $/\$
133		geo2sum 11657	. . . . . . . . . 10 $/\$
134	133	breq1d 4039	. . . . . . . . 9 $/\$
135	46, 115, 134	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$
136	132, 135	mpbid 147	. . . . . . 7 $/\$ $/\$ $/\$
137	42, 41, 43, 136	subled 8567	. . . . . 6 $/\$ $/\$ $/\$
138	40, 41, 137	lensymd 8141	. . . . 5 $/\$ $/\$ $/\$
139	39, 138	pm2.21dd 621	. . . 4 $/\$ $/\$ $/\$
140	139	ex 115	. . 3 $/\$ $/\$
141		fveq1 5553	. . . . . . 7
142	141	eqeq1d 2202	. . . . . 6
143	142	rexbidv 2495	. . . . 5
144	141	eqeq1d 2202	. . . . . 6
145	144	ralbidv 2494	. . . . 5
146	143, 145	orbi12d 794	. . . 4 $\/$ $\/$
147		finomni 7199	. . . . . 6 Omni
148	82, 147	syl 14	. . . . 5 $/\$ $/\$ Omni
149		isomninn 15521	. . . . . 6 Omni Omni $\/$
150	148, 149	syl 14	. . . . 5 $/\$ $/\$ Omni $\/$
151	148, 150	mpbid 147	. . . 4 $/\$ $/\$ $\/$
152		fz1ssnn 10122	. . . . . . 7
153	152	a1i 9	. . . . . 6 $/\$ $/\$
154	65, 153	fssresd 5430	. . . . 5 $/\$ $/\$
155		0red 8020	. . . . . . 7 $/\$ $/\$
156		1red 8034	. . . . . . 7 $/\$ $/\$
157		prexg 4240	. . . . . . 7 $/\$
158	155, 156, 157	syl2anc 411	. . . . . 6 $/\$ $/\$
159	158, 82	elmapd 6716	. . . . 5 $/\$ $/\$
160	154, 159	mpbird 167	. . . 4 $/\$ $/\$
161	146, 151, 160	rspcdva 2869	. . 3 $/\$ $/\$ $\/$
162	19, 140, 161	mpjaod 719	. 2 $/\$ $/\$
163	10, 162	rexlimddv 2616	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 709

wceq 1364

wcel 2164

wral 2472

wrex 2473

cvv 2760

wss 3153

cpr 3619 class class class wbr 4029

cmpt 4090

cdm 4659

cres 4661

wf 5250

cfv 5254 (class class class)co 5918

cmap 6702

cfn 6794 Omnicomni 7193

cc 7870

cr 7871

cc0 7872

c1 7873

caddc 7875

cmul 7877

clt 8054

cle 8055

cmin 8190

cdiv 8691

cn 8982

c2 9033

cn0 9240

cz 9317

cuz 9592

crp 9719

cfz 10074

cseq 10518

cexp 10609

cli 11421

csu 11496

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-isom 5263 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-frec 6444 df-1o 6469 df-2o 6470 df-oadd 6473 df-er 6587 df-map 6704 df-en 6795 df-dom 6796 df-fin 6797 df-omni 7194 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-ico 9960 df-fz 10075 df-fzo 10209 df-seqfrec 10519 df-exp 10610 df-ihash 10847 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-clim 11422 df-sumdc 11497

This theorem is referenced by: trilpolemres 15532 neapmkvlem 15557

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