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Theorem rpnnen1 10605
 Description: One half of rpnnen 12826, 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 6106 . 2
2 rpnnen1.1 . . . 4
3 rpnnen1.2 . . . 4
42, 3rpnnen1lem1 10600 . . 3
5 rneq 5095 . . . . . 6
65supeq1d 7451 . . . . 5
72, 3rpnnen1lem5 10604 . . . . . 6
8 fveq2 5728 . . . . . . . . . 10
98rneqd 5097 . . . . . . . . 9
109supeq1d 7451 . . . . . . . 8
11 id 20 . . . . . . . 8
1210, 11eqeq12d 2450 . . . . . . 7
1312, 7vtoclga 3017 . . . . . 6
147, 13eqeqan12d 2451 . . . . 5
156, 14syl5ib 211 . . . 4
1615, 8impbid1 195 . . 3
174, 16dom2 7150 . 2
181, 17ax-mp 8 1
 Colors of variables: wff set class Syntax hints:   wa 359   wceq 1652   wcel 1725  crab 2709  cvv 2956   class class class wbr 4212   cmpt 4266   crn 4879  cfv 5454  (class class class)co 6081   cmap 7018   cdom 7107  csup 7445  cr 8989   clt 9120   cdiv 9677  cn 10000  cz 10282  cq 10574 This theorem is referenced by:  reexALT  10606  rpnnen  12826 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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068 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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-n0 10222  df-z 10283  df-q 10575
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