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Theorem rpnnen1 11096
 Description: One half of rpnnen 13628, where we show an injection from the real numbers to sequences of rational numbers. Specifically, we map a real number to the sequence such that is the largest rational number with denominator that is strictly less than . In this manner, we get a monotonically increasing sequence that converges to , and since each sequence converges to a unique real number, this mapping from reals to sequences of rational numbers is injective. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 16-Jun-2013.)
Hypotheses
Ref Expression
rpnnen1.1
rpnnen1.2
Assertion
Ref Expression
rpnnen1
Distinct variable groups:   ,,,   ,
Allowed substitution hints:   (,)

Proof of Theorem rpnnen1
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ovex 6226 . 2
2 rpnnen1.1 . . . 4
3 rpnnen1.2 . . . 4
42, 3rpnnen1lem1 11091 . . 3
5 rneq 5174 . . . . . 6
65supeq1d 7808 . . . . 5
72, 3rpnnen1lem5 11095 . . . . . 6
8 fveq2 5800 . . . . . . . . . 10
98rneqd 5176 . . . . . . . . 9
109supeq1d 7808 . . . . . . . 8
11 id 22 . . . . . . . 8
1210, 11eqeq12d 2476 . . . . . . 7
1312, 7vtoclga 3142 . . . . . 6
147, 13eqeqan12d 2477 . . . . 5
156, 14syl5ib 219 . . . 4
1615, 8impbid1 203 . . 3
174, 16dom2 7463 . 2
181, 17ax-mp 5 1
 Colors of variables: wff setvar class Syntax hints:   wa 369   wceq 1370   wcel 1758  crab 2803  cvv 3078   class class class wbr 4401   cmpt 4459   crn 4950  cfv 5527  (class class class)co 6201   cmap 7325   cdom 7419  csup 7802  cr 9393   clt 9530   cdiv 10105  cn 10434  cz 10758  cq 11065 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-sup 7803  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-n0 10692  df-z 10759  df-q 11066 This theorem is referenced by:  reexALT  11097  rpnnen  13628
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