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Theorem rpnnen2 12817
 Description: The other half of rpnnen 12818, where we show an injection from sets of natural numbers 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... 12650). 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 natural numbers the number , where (rpnnen2lem1 12806). This is an infinite sum of real numbers (rpnnen2lem2 12807), and since implies (rpnnen2lem4 12809) and converges to (rpnnen2lem3 12808) by geoisum1 12648, the sum is convergent to some real (rpnnen2lem5 12810 and rpnnen2lem6 12811) by the comparison test for convergence cvgcmp 12587. The comparison test also tells us that implies (rpnnen2lem7 12812). Putting it all together, if we have two sets , there must differ somewhere, and so there must be an such that but or vice versa. In this case, we split off the first terms (rpnnen2lem8 12813) and cancel them (rpnnen2lem10 12815), since these are the same for both sets. For the remaining terms, we use the subset property to establish that and (where these sums are only over ), and since (rpnnen2lem9 12814) and , we establish that (rpnnen2lem11 12816) 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:   ,
Allowed substitution hints:   (,)

Proof of Theorem rpnnen2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6098 . 2
2 elpwi 3799 . . . . 5
3 nnuz 10513 . . . . . . 7
43sumeq1i 12484 . . . . . 6
5 1nn 10003 . . . . . . 7
6 rpnnen2.1 . . . . . . . 8
76rpnnen2lem6 12811 . . . . . . 7
85, 7mpan2 653 . . . . . 6
94, 8syl5eqel 2519 . . . . 5
102, 9syl 16 . . . 4
11 1z 10303 . . . . . 6
1211a1i 11 . . . . 5
13 eqidd 2436 . . . . 5
146rpnnen2lem2 12807 . . . . . . 7
152, 14syl 16 . . . . . 6
1615ffvelrnda 5862 . . . . 5
176rpnnen2lem5 12810 . . . . . 6
182, 5, 17sylancl 644 . . . . 5
19 ssid 3359 . . . . . . . 8
206rpnnen2lem4 12809 . . . . . . . 8
2119, 20mp3an2 1267 . . . . . . 7
2221simpld 446 . . . . . 6
232, 22sylan 458 . . . . 5
243, 12, 13, 16, 18, 23isumge0 12542 . . . 4
25 1re 9082 . . . . . . 7
2625rehalfcli 10208 . . . . . 6
2726a1i 11 . . . . 5
2825a1i 11 . . . . 5
296rpnnen2lem7 12812 . . . . . . . . 9
3019, 5, 29mp3an23 1271 . . . . . . . 8
312, 30syl 16 . . . . . . 7
32 eqid 2435 . . . . . . . 8
33 eqidd 2436 . . . . . . . 8
34 elnnuz 10514 . . . . . . . . . 10
356rpnnen2lem2 12807 . . . . . . . . . . . . 13
3619, 35ax-mp 8 . . . . . . . . . . . 12
3736ffvelrni 5861 . . . . . . . . . . 11
3837recnd 9106 . . . . . . . . . 10
3934, 38sylbir 205 . . . . . . . . 9
4039adantl 453 . . . . . . . 8
416rpnnen2lem3 12808 . . . . . . . . 9
4241a1i 11 . . . . . . . 8
4332, 12, 33, 40, 42isumclim 12533 . . . . . . 7
4431, 43breqtrd 4228 . . . . . 6
454, 44syl5eqbr 4237 . . . . 5
46 halflt1 10181 . . . . . . 7
4726, 25, 46ltleii 9188 . . . . . 6
4847a1i 11 . . . . 5
4910, 27, 28, 45, 48letrd 9219 . . . 4
50 0re 9083 . . . . 5
5150, 25elicc2i 10968 . . . 4
5210, 24, 49, 51syl3anbrc 1138 . . 3
53 elpwi 3799 . . . . . . . . . . 11
54 ssdifss 3470 . . . . . . . . . . . 12
55 ssdifss 3470 . . . . . . . . . . . 12
56 unss 3513 . . . . . . . . . . . . 13
5756biimpi 187 . . . . . . . . . . . 12
5854, 55, 57syl2an 464 . . . . . . . . . . 11
592, 53, 58syl2an 464 . . . . . . . . . 10
60 eqss 3355 . . . . . . . . . . . . 13
61 ssdif0 3678 . . . . . . . . . . . . . 14
62 ssdif0 3678 . . . . . . . . . . . . . 14
6361, 62anbi12i 679 . . . . . . . . . . . . 13
64 un00 3655 . . . . . . . . . . . . 13
6560, 63, 643bitri 263 . . . . . . . . . . . 12
6665necon3bii 2630 . . . . . . . . . . 11
6766biimpi 187 . . . . . . . . . 10
68 nnwo 10534 . . . . . . . . . 10
6959, 67, 68syl2an 464 . . . . . . . . 9
7069ex 424 . . . . . . . 8
7159sselda 3340 . . . . . . . . . 10
72 df-ral 2702 . . . . . . . . . . . 12
73 con34b 284 . . . . . . . . . . . . . 14
74 eldif 3322 . . . . . . . . . . . . . . . . . 18
75 eldif 3322 . . . . . . . . . . . . . . . . . 18
7674, 75orbi12i 508 . . . . . . . . . . . . . . . . 17
77 elun 3480 . . . . . . . . . . . . . . . . 17
78 xor 862 . . . . . . . . . . . . . . . . 17
7976, 77, 783bitr4ri 270 . . . . . . . . . . . . . . . 16
8079con1bii 322 . . . . . . . . . . . . . . 15
8180imbi2i 304 . . . . . . . . . . . . . 14
8273, 81bitri 241 . . . . . . . . . . . . 13
8382albii 1575 . . . . . . . . . . . 12
8472, 83bitri 241 . . . . . . . . . . 11
85 alral 2756 . . . . . . . . . . . 12
86 nnre 9999 . . . . . . . . . . . . . . 15
87 nnre 9999 . . . . . . . . . . . . . . 15
88 ltnle 9147 . . . . . . . . . . . . . . 15
8986, 87, 88syl2anr 465 . . . . . . . . . . . . . 14
9089imbi1d 309 . . . . . . . . . . . . 13
9190ralbidva 2713 . . . . . . . . . . . 12
9285, 91syl5ibr 213 . . . . . . . . . . 11
9384, 92syl5bi 209 . . . . . . . . . 10
9471, 93syl 16 . . . . . . . . 9
9594reximdva 2810 . . . . . . . 8
9670, 95syld 42 . . . . . . 7
97 rexun 3519 . . . . . . 7
9896, 97syl6ib 218 . . . . . 6
99 simpll 731 . . . . . . . . . 10
100 simplr 732 . . . . . . . . . 10
101 simprl 733 . . . . . . . . . 10
102 simprr 734 . . . . . . . . . 10
103 biid 228 . . . . . . . . . 10
1046, 99, 100, 101, 102, 103rpnnen2lem11 12816 . . . . . . . . 9
105104rexlimdvaa 2823 . . . . . . . 8
106 simplr 732 . . . . . . . . . 10
107 simpll 731 . . . . . . . . . 10
108 simprl 733 . . . . . . . . . 10
109 simprr 734 . . . . . . . . . . 11
110 bicom 192 . . . . . . . . . . . . 13
111110imbi2i 304 . . . . . . . . . . . 12
112111ralbii 2721 . . . . . . . . . . 11
113109, 112sylibr 204 . . . . . . . . . 10
114 eqcom 2437 . . . . . . . . . 10
1156, 106, 107, 108, 113, 114rpnnen2lem11 12816 . . . . . . . . 9
116115rexlimdvaa 2823 . . . . . . . 8
117105, 116jaod 370 . . . . . . 7
1182, 53, 117syl2an 464 . . . . . 6
11998, 118syld 42 . . . . 5
120119necon4ad 2659 . . . 4
121 fveq2 5720 . . . . . 6
122121fveq1d 5722 . . . . 5
123122sumeq2sdv 12490 . . . 4
124120, 123impbid1 195 . . 3
12552, 124dom2 7142 . 2
1261, 125ax-mp 8 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wo 358   wa 359  wal 1549   wceq 1652   wcel 1725   wne 2598  wral 2697  wrex 2698  cvv 2948   cdif 3309   cun 3310   wss 3312  c0 3620  cif 3731  cpw 3791   class class class wbr 4204   cmpt 4258   cdm 4870  wf 5442  cfv 5446  (class class class)co 6073   cdom 7099  cc 8980  cr 8981  cc0 8982  c1 8983   caddc 8985   clt 9112   cle 9113   cdiv 9669  cn 9992  c2 10041  c3 10042  cz 10274  cuz 10480  cicc 10911   cseq 11315  cexp 11374   cli 12270  csu 12471 This theorem is referenced by:  rpnnen  12818  opnreen  18854 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-sum 12472
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