Theorem rpnnen2 13959

Description: The other half of rpnnen 13960, where we show an injection from sets of positive integers to real numbers. The obvious choice for this is binary expansion, but it has the unfortunate property that it does not produce an injection on numbers which end with all 0's or all 1's (the more well-known decimal version of this is 0.999... 13690). Instead, we opt for a ternary expansion, which produces (a scaled version of) the Cantor set. Since the Cantor set is riddled with gaps, we can show that any two sequences that are not equal must differ somewhere, and when they do, they are placed a finite distance apart, thus ensuring that the map is injective.

Our map assigns to each subset of the positive integers the number , where ((`A)`)=if(e.A,(3-u),0)) (rpnnen2lem1 13948). This is an infinite sum of real numbers (rpnnen2lem2 13949), and since implies (rpnnen2lem4 13951) and converges to (rpnnen2lem3 13950) by geoisum1 13688, the sum is convergent to some real (rpnnen2lem5 13952 and rpnnen2lem6 13953) by the comparison test for convergence cvgcmp 13630. The comparison test also tells us that implies sum_(`A)sum_(`) (rpnnen2lem7 13954).

Putting it all together, if we have two sets , there must differ somewhere, and so there must be an such that A.(e.x<->e.) but or vice versa. In this case, we split off the first terms (rpnnen2lem8 13955) and cancel them (rpnnen2lem10 13957), since these are the same for both sets. For the remaining terms, we use the subset property to establish that sum_(`)sum_(`(\{})) and sum_(`{})sum_(`x) (where these sums are only over ), and since sum_(`(\{}))=(3-u)2 (rpnnen2lem9 13956) and sum_(`{})=(3-u), we establish that sum_(`)sum_(`x) (rpnnen2lem11 13958) so that they must be different. By contraposition, we find that this map is an injection. (Contributed by Mario Carneiro, 13-May-2013.) (Proof shortened by Mario Carneiro, 30-Apr-2014.)

Hypothesis

Ref	Expression
rpnnen2.1

Assertion

Ref	Expression
rpnnen2

Distinct variable group:   ,

Proof of Theorem rpnnen2

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 6324	. 2
2		elpwi 4021	. . . . 5
3		nnuz 11145	. . . . . . 7
4	3	sumeq1i 13520	. . . . . 6
5		1nn 10572	. . . . . . 7
6		rpnnen2.1	. . . . . . . 8
7	6	rpnnen2lem6 13953	. . . . . . 7
8	5, 7	mpan2 671	. . . . . 6
9	4, 8	syl5eqel 2549	. . . . 5
10	2, 9	syl 16	. . . 4
11		1zzd 10920	. . . . 5
12		eqidd 2458	. . . . 5
13	6	rpnnen2lem2 13949	. . . . . . 7
14	2, 13	syl 16	. . . . . 6
15	14	ffvelrnda 6031	. . . . 5
16	6	rpnnen2lem5 13952	. . . . . 6
17	2, 5, 16	sylancl 662	. . . . 5
18		ssid 3522	. . . . . . . 8
19	6	rpnnen2lem4 13951	. . . . . . . 8
20	18, 19	mp3an2 1312	. . . . . . 7
21	20	simpld 459	. . . . . 6
22	2, 21	sylan 471	. . . . 5
23	3, 11, 12, 15, 17, 22	isumge0 13581	. . . 4
24		halfre 10779	. . . . . 6
25	24	a1i 11	. . . . 5
26		1re 9616	. . . . . 6
27	26	a1i 11	. . . . 5
28	6	rpnnen2lem7 13954	. . . . . . . . 9
29	18, 5, 28	mp3an23 1316	. . . . . . . 8
30	2, 29	syl 16	. . . . . . 7
31		eqid 2457	. . . . . . . 8
32		eqidd 2458	. . . . . . . 8
33		elnnuz 11146	. . . . . . . . . 10
34	6	rpnnen2lem2 13949	. . . . . . . . . . . . 13
35	18, 34	ax-mp 5	. . . . . . . . . . . 12
36	35	ffvelrni 6030	. . . . . . . . . . 11
37	36	recnd 9643	. . . . . . . . . 10
38	33, 37	sylbir 213	. . . . . . . . 9
39	38	adantl 466	. . . . . . . 8
40	6	rpnnen2lem3 13950	. . . . . . . . 9
41	40	a1i 11	. . . . . . . 8
42	31, 11, 32, 39, 41	isumclim 13572	. . . . . . 7
43	30, 42	breqtrd 4476	. . . . . 6
44	4, 43	syl5eqbr 4485	. . . . 5
45		halflt1 10782	. . . . . . 7
46	24, 26, 45	ltleii 9728	. . . . . 6
47	46	a1i 11	. . . . 5
48	10, 25, 27, 44, 47	letrd 9760	. . . 4
49		0re 9617	. . . . 5
50	49, 26	elicc2i 11619	. . . 4
51	10, 23, 48, 50	syl3anbrc 1180	. . 3
52		elpwi 4021	. . . . . . . . . . 11
53		ssdifss 3634	. . . . . . . . . . . 12
54		ssdifss 3634	. . . . . . . . . . . 12
55		unss 3677	. . . . . . . . . . . . 13
56	55	biimpi 194	. . . . . . . . . . . 12
57	53, 54, 56	syl2an 477	. . . . . . . . . . 11
58	2, 52, 57	syl2an 477	. . . . . . . . . 10
59		eqss 3518	. . . . . . . . . . . . 13
60		ssdif0 3885	. . . . . . . . . . . . . 14
61		ssdif0 3885	. . . . . . . . . . . . . 14
62	60, 61	anbi12i 697	. . . . . . . . . . . . 13
63		un00 3862	. . . . . . . . . . . . 13
64	59, 62, 63	3bitri 271	. . . . . . . . . . . 12
65	64	necon3bii 2725	. . . . . . . . . . 11
66	65	biimpi 194	. . . . . . . . . 10
67		nnwo 11176	. . . . . . . . . 10
68	58, 66, 67	syl2an 477	. . . . . . . . 9
69	68	ex 434	. . . . . . . 8
70	58	sselda 3503	. . . . . . . . . 10
71		df-ral 2812	. . . . . . . . . . . 12
72		con34b 292	. . . . . . . . . . . . . 14
73		eldif 3485	. . . . . . . . . . . . . . . . . 18
74		eldif 3485	. . . . . . . . . . . . . . . . . 18
75	73, 74	orbi12i 521	. . . . . . . . . . . . . . . . 17
76		elun 3644	. . . . . . . . . . . . . . . . 17
77		xor 891	. . . . . . . . . . . . . . . . 17
78	75, 76, 77	3bitr4ri 278	. . . . . . . . . . . . . . . 16
79	78	con1bii 331	. . . . . . . . . . . . . . 15
80	79	imbi2i 312	. . . . . . . . . . . . . 14
81	72, 80	bitri 249	. . . . . . . . . . . . 13
82	81	albii 1640	. . . . . . . . . . . 12
83	71, 82	bitri 249	. . . . . . . . . . 11
84		alral 2822	. . . . . . . . . . . 12
85		nnre 10568	. . . . . . . . . . . . . . 15
86		nnre 10568	. . . . . . . . . . . . . . 15
87		ltnle 9685	. . . . . . . . . . . . . . 15
88	85, 86, 87	syl2anr 478	. . . . . . . . . . . . . 14
89	88	imbi1d 317	. . . . . . . . . . . . 13
90	89	ralbidva 2893	. . . . . . . . . . . 12
91	84, 90	syl5ibr 221	. . . . . . . . . . 11
92	83, 91	syl5bi 217	. . . . . . . . . 10
93	70, 92	syl 16	. . . . . . . . 9
94	93	reximdva 2932	. . . . . . . 8
95	69, 94	syld 44	. . . . . . 7
96		rexun 3683	. . . . . . 7
97	95, 96	syl6ib 226	. . . . . 6
98		simpll 753	. . . . . . . . . 10
99		simplr 755	. . . . . . . . . 10
100		simprl 756	. . . . . . . . . 10
101		simprr 757	. . . . . . . . . 10
102		biid 236	. . . . . . . . . 10
103	6, 98, 99, 100, 101, 102	rpnnen2lem11 13958	. . . . . . . . 9
104	103	rexlimdvaa 2950	. . . . . . . 8
105		simplr 755	. . . . . . . . . 10
106		simpll 753	. . . . . . . . . 10
107		simprl 756	. . . . . . . . . 10
108		simprr 757	. . . . . . . . . . 11
109		bicom 200	. . . . . . . . . . . . 13
110	109	imbi2i 312	. . . . . . . . . . . 12
111	110	ralbii 2888	. . . . . . . . . . 11
112	108, 111	sylibr 212	. . . . . . . . . 10
113		eqcom 2466	. . . . . . . . . 10
114	6, 105, 106, 107, 112, 113	rpnnen2lem11 13958	. . . . . . . . 9
115	114	rexlimdvaa 2950	. . . . . . . 8
116	104, 115	jaod 380	. . . . . . 7
117	2, 52, 116	syl2an 477	. . . . . 6
118	97, 117	syld 44	. . . . 5
119	118	necon4ad 2677	. . . 4
120		fveq2 5871	. . . . . 6
121	120	fveq1d 5873	. . . . 5
122	121	sumeq2sdv 13526	. . . 4
123	119, 122	impbid1 203	. . 3
124	51, 123	dom2 7578	. 2
125	1, 124	ax-mp 5	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  \/wo 368  /\wa 369  A.wal 1393  =wceq 1395  e.wcel 1818  =/=wne 2652  A.wral 2807  E.wrex 2808  cvv 3109  \cdif 3472  u.cun 3473  C_wss 3475  c0 3784  ifcif 3941  ~Pcpw 4012   class class class wbr 4452  e.cmpt 4510  domcdm 5004  -->wf 5589  `cfv 5593  (class class class)co 6296  cdom 7534  cc 9511  cr 9512  0cc0 9513  1c1 9514  caddc 9516  clt 9649  cle 9650  cdiv 10231  cn 10561  2c2 10610  3c3 10611  cuz 11110  cicc 11561  seqcseq 12107  cexp 12166  cli 13307  sum_csu 13508

This theorem is referenced by:  rpnnen  13960  opnreen  21336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592  ax-inf2 8079  ax-cnex 9569  ax-resscn 9570  ax-1cn 9571  ax-icn 9572  ax-addcl 9573  ax-addrcl 9574  ax-mulcl 9575  ax-mulrcl 9576  ax-mulcom 9577  ax-addass 9578  ax-mulass 9579  ax-distr 9580  ax-i2m1 9581  ax-1ne0 9582  ax-1rid 9583  ax-rnegex 9584  ax-rrecex 9585  ax-cnre 9586  ax-pre-lttri 9587  ax-pre-lttrn 9588  ax-pre-ltadd 9589  ax-pre-mulgt0 9590  ax-pre-sup 9591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-int 4287  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6257  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6701  df-1st 6800  df-2nd 6801  df-recs 7061  df-rdg 7095  df-1o 7149  df-oadd 7153  df-er 7330  df-pm 7442  df-en 7537  df-dom 7538  df-sdom 7539  df-fin 7540  df-sup 7921  df-oi 7956  df-card 8341  df-pnf 9651  df-mnf 9652  df-xr 9653  df-ltxr 9654  df-le 9655  df-sub 9830  df-neg 9831  df-div 10232  df-nn 10562  df-2 10619  df-3 10620  df-n0 10821  df-z 10890  df-uz 11111  df-rp 11250  df-ico 11564  df-icc 11565  df-fz 11702  df-fzo 11825  df-fl 11929  df-seq 12108  df-exp 12167  df-hash 12406  df-cj 12932  df-re 12933  df-im 12934  df-sqrt 13068  df-abs 13069  df-limsup 13294  df-clim 13311  df-rlim 13312  df-sum 13509