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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 13567 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 7845 |
. . . 4
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2 | 1 | mptex 5686 |
. . 3
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3 | 2 | a1i 9 |
. 2
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4 | 0zd 9158 |
. . . . 5
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5 | 1zzd 9173 |
. . . . 5
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6 | eqeq1 2161 |
. . . . . . 7
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7 | 6 | dcbid 824 |
. . . . . 6
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8 | 7 | rspccva 2812 |
. . . . 5
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9 | 4, 5, 8 | ifcldcd 3536 |
. . . 4
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10 | 9 | fmpttd 5615 |
. . 3
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11 | 0re 7857 |
. . . . . 6
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12 | 0zd 9158 |
. . . . . . . 8
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13 | 1zzd 9173 |
. . . . . . . 8
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14 | eqid 2154 |
. . . . . . . . . . 11
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15 | 14 | orci 721 |
. . . . . . . . . 10
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16 | df-dc 821 |
. . . . . . . . . 10
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17 | 15, 16 | mpbir 145 |
. . . . . . . . 9
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18 | 17 | a1i 9 |
. . . . . . . 8
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19 | 12, 13, 18 | ifcldcd 3536 |
. . . . . . 7
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20 | 19 | mptru 1341 |
. . . . . 6
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21 | eqeq1 2161 |
. . . . . . . 8
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22 | 21 | ifbid 3522 |
. . . . . . 7
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23 | eqid 2154 |
. . . . . . 7
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24 | 22, 23 | fvmptg 5537 |
. . . . . 6
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25 | 11, 20, 24 | mp2an 423 |
. . . . 5
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26 | 14 | iftruei 3507 |
. . . . 5
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27 | 25, 26 | eqtri 2175 |
. . . 4
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28 | 27 | a1i 9 |
. . 3
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29 | 1ne0 8880 |
. . . . . 6
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30 | eqeq1 2161 |
. . . . . . . . . 10
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31 | 30 | ifbid 3522 |
. . . . . . . . 9
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32 | rpre 9545 |
. . . . . . . . . 10
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33 | 32 | adantl 275 |
. . . . . . . . 9
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34 | 0zd 9158 |
. . . . . . . . . 10
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35 | 1zzd 9173 |
. . . . . . . . . 10
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36 | eqeq1 2161 |
. . . . . . . . . . . 12
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37 | 36 | dcbid 824 |
. . . . . . . . . . 11
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38 | simpl 108 |
. . . . . . . . . . 11
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39 | 37, 38, 33 | rspcdva 2818 |
. . . . . . . . . 10
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40 | 34, 35, 39 | ifcldcd 3536 |
. . . . . . . . 9
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41 | 23, 31, 33, 40 | fvmptd3 5554 |
. . . . . . . 8
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42 | rpne0 9554 |
. . . . . . . . . . 11
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43 | 42 | neneqd 2345 |
. . . . . . . . . 10
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44 | 43 | iffalsed 3511 |
. . . . . . . . 9
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45 | 44 | adantl 275 |
. . . . . . . 8
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46 | 41, 45 | eqtrd 2187 |
. . . . . . 7
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47 | 46 | neeq1d 2342 |
. . . . . 6
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48 | 29, 47 | mpbiri 167 |
. . . . 5
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49 | 48 | ralrimiva 2527 |
. . . 4
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50 | fveq2 5461 |
. . . . . 6
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51 | 50 | neeq1d 2342 |
. . . . 5
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52 | 51 | cbvralv 2677 |
. . . 4
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53 | 49, 52 | sylib 121 |
. . 3
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54 | 10, 28, 53 | 3jca 1162 |
. 2
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55 | feq1 5295 |
. . 3
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56 | fveq1 5460 |
. . . 4
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57 | 56 | eqeq1d 2163 |
. . 3
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58 | fveq1 5460 |
. . . . 5
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59 | 58 | neeq1d 2342 |
. . . 4
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60 | 59 | ralbidv 2454 |
. . 3
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61 | 55, 57, 60 | 3anbi123d 1291 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
62 | 3, 54, 61 | elabd 2853 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-13 2127 ax-14 2128 ax-ext 2136 ax-coll 4075 ax-sep 4078 ax-pow 4130 ax-pr 4164 ax-un 4388 ax-setind 4490 ax-cnex 7802 ax-resscn 7803 ax-1cn 7804 ax-1re 7805 ax-icn 7806 ax-addcl 7807 ax-addrcl 7808 ax-mulcl 7809 ax-addcom 7811 ax-addass 7813 ax-distr 7815 ax-i2m1 7816 ax-0lt1 7817 ax-0id 7819 ax-rnegex 7820 ax-cnre 7822 ax-pre-ltirr 7823 ax-pre-ltwlin 7824 ax-pre-lttrn 7825 ax-pre-ltadd 7827 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-nel 2420 df-ral 2437 df-rex 2438 df-reu 2439 df-rab 2441 df-v 2711 df-sbc 2934 df-csb 3028 df-dif 3100 df-un 3102 df-in 3104 df-ss 3111 df-if 3502 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-uni 3769 df-int 3804 df-iun 3847 df-br 3962 df-opab 4022 df-mpt 4023 df-id 4248 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-rn 4590 df-res 4591 df-ima 4592 df-iota 5128 df-fun 5165 df-fn 5166 df-f 5167 df-f1 5168 df-fo 5169 df-f1o 5170 df-fv 5171 df-riota 5770 df-ov 5817 df-oprab 5818 df-mpo 5819 df-pnf 7893 df-mnf 7894 df-xr 7895 df-ltxr 7896 df-le 7897 df-sub 8027 df-neg 8028 df-inn 8813 df-z 9147 df-rp 9539 |
This theorem is referenced by: dcapnconstALT 13573 |
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