trilpolemlt1 - Mathbox for Jim Kingdon

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Theorem trilpolemlt1 16995

Description: Lemma for trilpo 16997. 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 7470 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 8331	. . . 4
2		trilpolemgt1.f	. . . . 5
3		trilpolemgt1.a	. . . . 5
4	2, 3	trilpolemcl 16991	. . . 4
5	1, 4	resubcld 8698	. . 3
6		trilpolemlt1.a	. . . 4
7	4, 1	posdifd 8850	. . . 4
8	6, 7	mpbid 147	. . 3
9		nnrecl 9540	. . 3 $/\$
10	5, 8, 9	syl2anc 415	. 2
11		elfznn 10438	. . . . . . 7
12	11	ad2antrl 494	. . . . . 6 $/\$ $/\$ $/\$ $/\$
13		simprl 535	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
14	13	fvresd 5715	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
15		simprr 537	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
16	14, 15	eqtr3d 2273	. . . . . 6 $/\$ $/\$ $/\$ $/\$
17	12, 16	jca 306	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
18	17	ex 115	. . . 4 $/\$ $/\$ $/\$ $/\$
19	18	reximdv2 2649	. . 3 $/\$ $/\$
20		2rp 10038	. . . . . . . . . 10
21	20	a1i 9	. . . . . . . . 9 $/\$ $/\$
22		simprl 535	. . . . . . . . . 10 $/\$ $/\$
23	22	nnzd 9746	. . . . . . . . 9 $/\$ $/\$
24	21, 23	rpexpcld 11113	. . . . . . . 8 $/\$ $/\$
25	24	rprecred 10088	. . . . . . 7 $/\$ $/\$
26	22	nnrecred 9330	. . . . . . 7 $/\$ $/\$
27	5	adantr 276	. . . . . . 7 $/\$ $/\$
28		2z 9651	. . . . . . . . . 10
29		uzid 9915	. . . . . . . . . 10
30	28, 29	mp1i 10	. . . . . . . . 9 $/\$ $/\$
31	22	nnnn0d 9599	. . . . . . . . 9 $/\$ $/\$
32		bernneq3 11078	. . . . . . . . 9 $/\$
33	30, 31, 32	syl2anc 415	. . . . . . . 8 $/\$ $/\$
34	22	nnrpd 10074	. . . . . . . . 9 $/\$ $/\$
35	34, 24	ltrecd 10095	. . . . . . . 8 $/\$ $/\$
36	33, 35	mpbid 147	. . . . . . 7 $/\$ $/\$
37		simprr 537	. . . . . . 7 $/\$ $/\$
38	25, 26, 27, 36, 37	lttrd 8442	. . . . . 6 $/\$ $/\$
39	38	adantr 276	. . . . 5 $/\$ $/\$ $/\$
40	27	adantr 276	. . . . . 6 $/\$ $/\$ $/\$
41	25	adantr 276	. . . . . 6 $/\$ $/\$ $/\$
42		1red 8331	. . . . . . 7 $/\$ $/\$ $/\$
43	4	ad2antrr 492	. . . . . . 7 $/\$ $/\$ $/\$
44		0red 8317	. . . . . . . . . 10 $/\$ $/\$ $/\$
45		eqid 2238	. . . . . . . . . . 11
46	22	adantr 276	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
47	46	peano2nnd 9298	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
48	47	nnzd 9746	. . . . . . . . . . 11 $/\$ $/\$ $/\$
49		eluznn 9979	. . . . . . . . . . . . 13 $/\$
50	47, 49	sylan 283	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
51		eqid 2238	. . . . . . . . . . . . 13
52		oveq2 6083	. . . . . . . . . . . . . . 15
53	52	oveq2d 6091	. . . . . . . . . . . . . 14
54		fveq2 5690	. . . . . . . . . . . . . 14
55	53, 54	oveq12d 6093	. . . . . . . . . . . . 13
56		simpr 110	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
57	20	a1i 9	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
58	56	nnzd 9746	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
59	57, 58	rpexpcld 11113	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
60	59	rprecred 10088	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
61		0re 8316	. . . . . . . . . . . . . . . 16
62		1re 8315	. . . . . . . . . . . . . . . 16
63		prssi 3868	. . . . . . . . . . . . . . . 16 $/\$
64	61, 62, 63	mp2an 430	. . . . . . . . . . . . . . 15
65	2	adantr 276	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
66	65	ad2antrr 492	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
67	66, 56	ffvelcdmd 5835	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
68	64, 67	sselid 3246	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
69	60, 68	remulcld 8346	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
70	51, 55, 56, 69	fvmptd3 5793	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
71	50, 70	syldan 282	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
72	50, 69	syldan 282	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
73	65	adantr 276	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
74	73, 51	trilpolemclim 16990	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
75		nnuz 9937	. . . . . . . . . . . . 13
76	69	recnd 8344	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
77	70, 76	eqeltrd 2315	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
78	75, 47, 77	iserex 12083	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
79	74, 78	mpbid 147	. . . . . . . . . . 11 $/\$ $/\$ $/\$
80	45, 48, 71, 72, 79	isumrecl 12174	. . . . . . . . . 10 $/\$ $/\$ $/\$
81		1zzd 9650	. . . . . . . . . . . . 13 $/\$ $/\$
82	81, 23	fzfigd 10846	. . . . . . . . . . . 12 $/\$ $/\$
83	82	adantr 276	. . . . . . . . . . 11 $/\$ $/\$ $/\$
84	20	a1i 9	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
85		elfzelz 10407	. . . . . . . . . . . . . 14
86	85	adantl 277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
87	84, 86	rpexpcld 11113	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
88	87	rprecred 10088	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
89	83, 88	fsumrecl 12146	. . . . . . . . . 10 $/\$ $/\$ $/\$
90	50, 60	syldan 282	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
91	50, 68	syldan 282	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
92	59	rpreccld 10087	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
93	50, 92	syldan 282	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
94	93	rpge0d 10080	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
95		0le0 9372	. . . . . . . . . . . . . 14
96		simpr 110	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
97	95, 96	breqtrrid 4163	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
98		0le1 8799	. . . . . . . . . . . . . 14
99		simpr 110	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
100	98, 99	breqtrrid 4163	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
101	73	adantr 276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
102	101, 50	ffvelcdmd 5835	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
103		elpri 3728	. . . . . . . . . . . . . 14 $\/$
104	102, 103	syl 14	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $\/$
105	97, 100, 104	mpjaodan 810	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
106	90, 91, 94, 105	mulge0d 8939	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
107	45, 48, 71, 72, 79, 106	isumge0 12175	. . . . . . . . . 10 $/\$ $/\$ $/\$
108	44, 80, 89, 107	leadd2dd 8878	. . . . . . . . 9 $/\$ $/\$ $/\$
109	89	recnd 8344	. . . . . . . . . . 11 $/\$ $/\$ $/\$
110	109	addridd 8465	. . . . . . . . . 10 $/\$ $/\$ $/\$
111	110	eqcomd 2244	. . . . . . . . 9 $/\$ $/\$ $/\$
112	75, 45, 47, 70, 76, 74	isumsplit 12236	. . . . . . . . . . 11 $/\$ $/\$ $/\$
113	3, 112	eqtrid 2283	. . . . . . . . . 10 $/\$ $/\$ $/\$
114	46	nncnd 9297	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
115		1cnd 8332	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
116	114, 115	pncand 8628	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
117	116	oveq2d 6091	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
118		simpr 110	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
119	118	fvresd 5715	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
120		fveqeq2 5699	. . . . . . . . . . . . . . . 16
121		simplr 533	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
122	120, 121, 118	rspcdva 2934	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
123	119, 122	eqtr3d 2273	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
124	123	oveq2d 6091	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
125	87	rpreccld 10087	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
126	125	rpcnd 10078	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
127	126	mulridd 8333	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
128	124, 127	eqtrd 2271	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
129	117, 128	sumeq12rdv 12117	. . . . . . . . . . 11 $/\$ $/\$ $/\$
130	129	oveq1d 6090	. . . . . . . . . 10 $/\$ $/\$ $/\$
131	113, 130	eqtrd 2271	. . . . . . . . 9 $/\$ $/\$ $/\$
132	108, 111, 131	3brtr4d 4157	. . . . . . . 8 $/\$ $/\$ $/\$
133		geo2sum 12259	. . . . . . . . . 10 $/\$
134	133	breq1d 4135	. . . . . . . . 9 $/\$
135	46, 115, 134	syl2anc 415	. . . . . . . 8 $/\$ $/\$ $/\$
136	132, 135	mpbid 147	. . . . . . 7 $/\$ $/\$ $/\$
137	42, 41, 43, 136	subled 8866	. . . . . 6 $/\$ $/\$ $/\$
138	40, 41, 137	lensymd 8438	. . . . 5 $/\$ $/\$ $/\$
139	39, 138	pm2.21dd 629	. . . 4 $/\$ $/\$ $/\$
140	139	ex 115	. . 3 $/\$ $/\$
141		fveq1 5689	. . . . . . 7
142	141	eqeq1d 2247	. . . . . 6
143	142	rexbidv 2551	. . . . 5
144	141	eqeq1d 2247	. . . . . 6
145	144	ralbidv 2550	. . . . 5
146	143, 145	orbi12d 805	. . . 4 $\/$ $\/$
147		finomni 7470	. . . . . 6 Omni
148	82, 147	syl 14	. . . . 5 $/\$ $/\$ Omni
149		isomninn 16985	. . . . . 6 Omni Omni $\/$
150	148, 149	syl 14	. . . . 5 $/\$ $/\$ Omni $\/$
151	148, 150	mpbid 147	. . . 4 $/\$ $/\$ $\/$
152		fz1ssnn 10440	. . . . . . 7
153	152	a1i 9	. . . . . 6 $/\$ $/\$
154	65, 153	fssresd 5561	. . . . 5 $/\$ $/\$
155		0red 8317	. . . . . . 7 $/\$ $/\$
156		1red 8331	. . . . . . 7 $/\$ $/\$
157		prexg 4344	. . . . . . 7 $/\$
158	155, 156, 157	syl2anc 415	. . . . . 6 $/\$ $/\$
159	158, 82	elmapd 6926	. . . . 5 $/\$ $/\$
160	154, 159	mpbird 167	. . . 4 $/\$ $/\$
161	146, 151, 160	rspcdva 2934	. . 3 $/\$ $/\$ $\/$
162	19, 140, 161	mpjaod 730	. 2 $/\$ $/\$
163	10, 162	rexlimddv 2673	1

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720

wceq 1402

wcel 2209

wral 2528

wrex 2529

cvv 2821

wss 3220

cpr 3706 class class class wbr 4125

cmpt 4187

cdm 4769

cres 4771

wf 5368

cfv 5372 (class class class)co 6075

cmap 6912

cfn 7012 Omnicomni 7464

cc 8167

cr 8168

cc0 8169

c1 8170

caddc 8172

cmul 8174

clt 8350

cle 8351

cmin 8487

cdiv 8992

cn 9283

c2 9334

cn0 9542

cz 9623

cuz 9900

crp 10033

cfz 10390

cseq 10862

cexp 10953

cli 12022

csu 12097

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287 ax-arch 8288 ax-caucvg 8289

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-isom 5381 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-irdg 6631 df-frec 6652 df-1o 6677 df-2o 6678 df-oadd 6681 df-er 6797 df-map 6914 df-en 7013 df-dom 7014 df-fin 7015 df-omni 7465 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-inn 9284 df-2 9342 df-3 9343 df-4 9344 df-n0 9543 df-z 9624 df-uz 9901 df-q 9999 df-rp 10034 df-ico 10275 df-fz 10391 df-fzo 10528 df-seqfrec 10863 df-exp 10954 df-ihash 11193 df-cj 11585 df-re 11586 df-im 11587 df-rsqrt 11742 df-abs 11743 df-clim 12023 df-sumdc 12098

This theorem is referenced by: trilpolemres 16996 neapmkvlem 17022

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