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Mathbox for Jim Kingdon |
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Mirrors > Home > ILE Home > Th. List > Mathboxes > dceqnconst | Unicode version |
Description: Decidability of real number equality implies the existence of a certain non-constant function from real numbers to integers. Variation of Exercise 11.6(i) of [HoTT], p. (varies). See redcwlpo 14773 for more discussion of decidability of real number equality. (Contributed by BJ and Jim Kingdon, 24-Jun-2024.) (Revised by Jim Kingdon, 23-Jul-2024.) |
Ref | Expression |
---|---|
dceqnconst |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reex 7944 |
. . . 4
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2 | 1 | mptex 5742 |
. . 3
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3 | 2 | a1i 9 |
. 2
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4 | 0zd 9264 |
. . . . 5
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5 | 1zzd 9279 |
. . . . 5
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6 | eqeq1 2184 |
. . . . . . 7
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7 | 6 | dcbid 838 |
. . . . . 6
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8 | 7 | rspccva 2840 |
. . . . 5
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9 | 4, 5, 8 | ifcldcd 3570 |
. . . 4
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10 | 9 | fmpttd 5671 |
. . 3
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11 | 0re 7956 |
. . . . . 6
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12 | 0zd 9264 |
. . . . . . . 8
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13 | 1zzd 9279 |
. . . . . . . 8
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14 | eqid 2177 |
. . . . . . . . . . 11
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15 | 14 | orci 731 |
. . . . . . . . . 10
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16 | df-dc 835 |
. . . . . . . . . 10
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17 | 15, 16 | mpbir 146 |
. . . . . . . . 9
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18 | 17 | a1i 9 |
. . . . . . . 8
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19 | 12, 13, 18 | ifcldcd 3570 |
. . . . . . 7
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20 | 19 | mptru 1362 |
. . . . . 6
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21 | eqeq1 2184 |
. . . . . . . 8
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22 | 21 | ifbid 3555 |
. . . . . . 7
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23 | eqid 2177 |
. . . . . . 7
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24 | 22, 23 | fvmptg 5592 |
. . . . . 6
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25 | 11, 20, 24 | mp2an 426 |
. . . . 5
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26 | 14 | iftruei 3540 |
. . . . 5
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27 | 25, 26 | eqtri 2198 |
. . . 4
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28 | 27 | a1i 9 |
. . 3
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29 | 1ne0 8986 |
. . . . . 6
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30 | eqeq1 2184 |
. . . . . . . . . 10
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31 | 30 | ifbid 3555 |
. . . . . . . . 9
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32 | rpre 9659 |
. . . . . . . . . 10
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33 | 32 | adantl 277 |
. . . . . . . . 9
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34 | 0zd 9264 |
. . . . . . . . . 10
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35 | 1zzd 9279 |
. . . . . . . . . 10
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36 | eqeq1 2184 |
. . . . . . . . . . . 12
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37 | 36 | dcbid 838 |
. . . . . . . . . . 11
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38 | simpl 109 |
. . . . . . . . . . 11
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39 | 37, 38, 33 | rspcdva 2846 |
. . . . . . . . . 10
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40 | 34, 35, 39 | ifcldcd 3570 |
. . . . . . . . 9
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41 | 23, 31, 33, 40 | fvmptd3 5609 |
. . . . . . . 8
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42 | rpne0 9668 |
. . . . . . . . . . 11
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43 | 42 | neneqd 2368 |
. . . . . . . . . 10
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44 | 43 | iffalsed 3544 |
. . . . . . . . 9
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45 | 44 | adantl 277 |
. . . . . . . 8
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46 | 41, 45 | eqtrd 2210 |
. . . . . . 7
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47 | 46 | neeq1d 2365 |
. . . . . 6
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48 | 29, 47 | mpbiri 168 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
49 | 48 | ralrimiva 2550 |
. . . 4
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50 | fveq2 5515 |
. . . . . 6
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51 | 50 | neeq1d 2365 |
. . . . 5
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52 | 51 | cbvralv 2703 |
. . . 4
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53 | 49, 52 | sylib 122 |
. . 3
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54 | 10, 28, 53 | 3jca 1177 |
. 2
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55 | feq1 5348 |
. . 3
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56 | fveq1 5514 |
. . . 4
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57 | 56 | eqeq1d 2186 |
. . 3
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58 | fveq1 5514 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
59 | 58 | neeq1d 2365 |
. . . 4
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60 | 59 | ralbidv 2477 |
. . 3
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61 | 55, 57, 60 | 3anbi123d 1312 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
62 | 3, 54, 61 | elabd 2882 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-sub 8129 df-neg 8130 df-inn 8919 df-z 9253 df-rp 9653 |
This theorem is referenced by: dcapnconstALT 14779 |
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