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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 15208 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 7965 |
. . . 4
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2 | 1 | mptex 5759 |
. . 3
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3 | 2 | a1i 9 |
. 2
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4 | 0zd 9285 |
. . . . 5
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5 | 1zzd 9300 |
. . . . 5
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6 | eqeq1 2196 |
. . . . . . 7
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7 | 6 | dcbid 839 |
. . . . . 6
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8 | 7 | rspccva 2855 |
. . . . 5
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9 | 4, 5, 8 | ifcldcd 3585 |
. . . 4
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10 | 9 | fmpttd 5688 |
. . 3
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11 | 0re 7977 |
. . . . . 6
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12 | 0zd 9285 |
. . . . . . . 8
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13 | 1zzd 9300 |
. . . . . . . 8
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14 | eqid 2189 |
. . . . . . . . . . 11
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15 | 14 | orci 732 |
. . . . . . . . . 10
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16 | df-dc 836 |
. . . . . . . . . 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 3585 |
. . . . . . 7
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20 | 19 | mptru 1373 |
. . . . . 6
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21 | eqeq1 2196 |
. . . . . . . 8
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22 | 21 | ifbid 3570 |
. . . . . . 7
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23 | eqid 2189 |
. . . . . . 7
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24 | 22, 23 | fvmptg 5609 |
. . . . . 6
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25 | 11, 20, 24 | mp2an 426 |
. . . . 5
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26 | 14 | iftruei 3555 |
. . . . 5
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27 | 25, 26 | eqtri 2210 |
. . . 4
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28 | 27 | a1i 9 |
. . 3
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29 | 1ne0 9007 |
. . . . . 6
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30 | eqeq1 2196 |
. . . . . . . . . 10
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31 | 30 | ifbid 3570 |
. . . . . . . . 9
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32 | rpre 9680 |
. . . . . . . . . 10
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33 | 32 | adantl 277 |
. . . . . . . . 9
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34 | 0zd 9285 |
. . . . . . . . . 10
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35 | 1zzd 9300 |
. . . . . . . . . 10
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36 | eqeq1 2196 |
. . . . . . . . . . . 12
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37 | 36 | dcbid 839 |
. . . . . . . . . . 11
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38 | simpl 109 |
. . . . . . . . . . 11
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39 | 37, 38, 33 | rspcdva 2861 |
. . . . . . . . . 10
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40 | 34, 35, 39 | ifcldcd 3585 |
. . . . . . . . 9
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41 | 23, 31, 33, 40 | fvmptd3 5626 |
. . . . . . . 8
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42 | rpne0 9689 |
. . . . . . . . . . 11
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43 | 42 | neneqd 2381 |
. . . . . . . . . 10
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44 | 43 | iffalsed 3559 |
. . . . . . . . 9
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45 | 44 | adantl 277 |
. . . . . . . 8
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46 | 41, 45 | eqtrd 2222 |
. . . . . . 7
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47 | 46 | neeq1d 2378 |
. . . . . 6
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48 | 29, 47 | mpbiri 168 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
49 | 48 | ralrimiva 2563 |
. . . 4
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50 | fveq2 5531 |
. . . . . 6
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51 | 50 | neeq1d 2378 |
. . . . 5
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52 | 51 | cbvralv 2718 |
. . . 4
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53 | 49, 52 | sylib 122 |
. . 3
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54 | 10, 28, 53 | 3jca 1179 |
. 2
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55 | feq1 5364 |
. . 3
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56 | fveq1 5530 |
. . . 4
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57 | 56 | eqeq1d 2198 |
. . 3
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58 | fveq1 5530 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
59 | 58 | neeq1d 2378 |
. . . 4
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60 | 59 | ralbidv 2490 |
. . 3
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61 | 55, 57, 60 | 3anbi123d 1323 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
62 | 3, 54, 61 | elabd 2897 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7922 ax-resscn 7923 ax-1cn 7924 ax-1re 7925 ax-icn 7926 ax-addcl 7927 ax-addrcl 7928 ax-mulcl 7929 ax-addcom 7931 ax-addass 7933 ax-distr 7935 ax-i2m1 7936 ax-0lt1 7937 ax-0id 7939 ax-rnegex 7940 ax-cnre 7942 ax-pre-ltirr 7943 ax-pre-ltwlin 7944 ax-pre-lttrn 7945 ax-pre-ltadd 7947 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5234 df-fn 5235 df-f 5236 df-f1 5237 df-fo 5238 df-f1o 5239 df-fv 5240 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-pnf 8014 df-mnf 8015 df-xr 8016 df-ltxr 8017 df-le 8018 df-sub 8150 df-neg 8151 df-inn 8940 df-z 9274 df-rp 9674 |
This theorem is referenced by: dcapnconstALT 15215 |
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